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srcset="https://media.springernature.com/w144/springer-static/cover-hires/book/978-3-030-15195-9?as=webp 2x" alt=""> </picture> <span class="app-article-masthead__journal-title ">Principles of Mechanics</span> </a> <span class="app-article-masthead__conference-info"></span> </div> </div> </div> </section> </div> <div class="c-article-main u-container u-mt-24 u-mb-32 l-with-sidebar" id="main-content" data-component="article-container"> <main class="js-main-column u-serif c-chapter-body" data-track-component="chapter"> <div class="c-context-bar u-hide" data-test="context-bar" data-context-bar aria-hidden="true"> <div class="c-context-bar__container u-container" data-track-context="sticky banner"> <div class="c-context-bar__title"> Oscillatory Motion </div> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both"> <a href="/content/pdf/10.1007/978-3-030-15195-9.pdf" rel="noopener" class="u-button u-button--full-width u-button--primary u-justify-content-space-between 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focusable="false" width="16" height="16" class="u-icon"> <use xlink:href="#icon-eds-i-download-medium"/> </svg> </a> </div> </div> </div> </div> <article lang="en"> <div class="c-article-header"> <header> <div class="app-overview-section"> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Salma-Alrasheed" data-author-popup="auth-Salma-Alrasheed" data-corresp-id="c1">Salma Alrasheed<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><sup class="u-js-hide"><a href="#Aff20">20</a></sup> </li></ul> <div class="app-overview-section__separator app-overview-section__book-series"> <div class="app-book-series-listing"> <div> <svg class="app-book-series-listing__icon" width="24" height="24" aria-hidden="true" focusable="false"><use href="#icon-eds-i-book-series-medium"></use></svg> </div> <div> <p data-test="series-link"> <span class="app-book-series-listing__description">Part of the book series:</span> <a href="https://www.springer.com/series/15883" data-track="click" data-track-action="open book series" data-track-label="link">Advances in Science, Technology & Innovation</a> ((ASTI)) </p> </div> </div> </div> <div class="app-overview-section__separator" data-test="article-metrics"> <div id="altmetric-container"> <ul class="app-article-metrics-bar u-list-reset" data-test="article-metrics"> <li class="app-article-metrics-bar__item" data-test="access-count"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-accesses-medium"></use> </svg>83k <span 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js-c-reading-companion-sections-item"><span class="c-article-section__title-number"> </span>Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>A motion repeating itself is referred to as periodic or oscillatory motion. An object in such motion oscillates about an equilibrium position due to a restoring force or torque. Such force or torque tends to restore (return) the system toward its equilibrium position no matter in which direction the system is displaced. This motion is important to study many phenomena including electromagnetic waves, alternating current circuits, and molecules. For a vibration to occur, two quantities are necessary to be present—stiffness and inertia.</p></div></div></section> <div data-test="cobranding-download"> <div class="note test-pdf-link" id="cobranding-and-download-availability-text"> <div class="c-article-access-provider" data-component="provided-by-box"> <p class="c-article-access-provider__text c-article-access-provider__text--chapter"> You have full access to this open access chapter,  <a href="/content/pdf/10.1007/978-3-030-15195-9_10.pdf?pdf=inline%20link" class="c-pdf-download__link" id="js-body-chapter-download" style="display: inline; padding:0px!important;" target="_blank" rel="noopener" data-track="content_download" data-track-context="article body" data-track-type="chapter PDF download" data-track-action="Pdf download" data-track-label="inline link" download>Download chapter PDF</a> <svg width="24" height="24" focusable="false" role="img" aria-hidden="true" class="c-download-pdf-icon-large"> <use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use> </svg> </p> </div> </div> </div> <div class="main-content"> <section data-title="Oscillatory Motion"><div class="c-article-section" id="Sec1-section"><h2 id="Sec1" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number">10.1 </span>Oscillatory Motion</h2><div class="c-article-section__content" id="Sec1-content"><p>A motion repeating itself is referred to as periodic or oscillatory motion. An object in such motion oscillates about an equilibrium position due to a restoring force or torque. Such force or torque tends to restore (return) the system toward its equilibrium position no matter in which direction the system is displaced. This motion is important to study many phenomena including electromagnetic waves, alternating current circuits, and molecules. For a vibration to occur, two quantities are necessary to be present—stiffness and inertia.</p></div></div></section><section data-title="Free Vibrations"><div class="c-article-section" id="Sec2-section"><h2 id="Sec2" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number">10.2 </span>Free Vibrations</h2><div class="c-article-section__content" id="Sec2-content"><p>When a system vibrates, a restoring force must be present. In addition to that force, there is always a retarding or damping force such as friction. If the effect of the damping force is small and can be neglected, then the motion is classified as free and undamped motion. Otherwise, the motion is classified as free damped motion. In both cases, the motion is known as free vibration since no forces other than the restoring and damping forces exist during vibration. If a driving force that does positive work on the system exists, the motion is classified as forced vibration.</p><p>This force may be applied externally to the system or sometimes is produced within the system. In this chapter, the case in which a restoring force is directly proportional to the displacement is considered. The resulting motion is then known as a harmonic vibration and the system is said to be linear. If the restoring force depends on the displacement in some other way, the resulting motion is known as anharmonic vibration and the system is said to be nonlinear.</p></div></div></section><section data-title="Free Undamped Vibrations"><div class="c-article-section" id="Sec3-section"><h2 id="Sec3" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number">10.3 </span>Free Undamped Vibrations</h2><div class="c-article-section__content" id="Sec3-content"><p>This kind of motion is known as the simple harmonic motion. Next, we will examine examples of such motion in physics.</p><h3 class="c-article__sub-heading" id="Sec4"><span class="c-article-section__title-number">10.3.1 </span>Mass Attached to a Spring</h3><p>Consider a block of mass <i>m</i> attached to a light spring of spring constant <i>k</i> that is fixed at the other end (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig1">10.1</a>). Suppose that the system lies on a frictionless horizontal surface. For small displacements, the restoring force acting on the block by the spring is given by Hook’s law</p><div id="Equ23" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ F_{s}=-kx $$</span></div></div><p>As we’ve mentioned in Sect. <a href="https://doi.org/10.1007/978-3-030-15195-9_4">4.1</a>, if the block is displaced slightly to the right (for example to <span class="mathjax-tex">$x=A$</span>), the restoring spring force will accelerate the block to the left transferring its potential energy into kinetic energy As the block reaches its equilibrium position <span class="mathjax-tex">$x=0$</span>, all of its potential energy will be transformed into kinetic energy and it will overshoot to the other side. Again, as it moves left, the spring force decelerates the block to the right, transferring its kinetic energy into potential energy until all of its energy is potential at <span class="mathjax-tex">$x= -A$</span> where it comes to rest. At that point, it accelerates back to <span class="mathjax-tex">$x=0$</span> and regains all of its kinetic energy where it overshoots again to <span class="mathjax-tex">$x=A$</span>. Therefore, stiffness restores the mass where inertia is responsible for the mass to overshoot. From Newton’s second law we, have</p><div id="Equ24" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ ma=-kx $$</span></div></div><p>or</p><div id="Equ25" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ m\frac{d^{2}x}{dt^{2}}+kx=0 $$</span></div></div><p>or</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \displaystyle \frac{d^{2}x}{dt^{2}}+\omega _{n}^{2}x=0 \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.1) </div></div><p>where <span class="mathjax-tex">$\omega _{n}=\sqrt{k/m}$</span> is called the natural angular frequency of the system. The general solution of this equation is of the form</p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} x(t)=A_{1}\cos \omega _{n}t+A_{2}\sin \omega _{n}t \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.2) </div></div><p>where <span class="mathjax-tex">$A_{1}$</span> and <span class="mathjax-tex">$A_{2}$</span> are arbitrary constants that can be found from the initial conditions. Therefore, there are many possible motions with the same angular frequency <span class="mathjax-tex">$\omega _{n}$</span>. By multiplying and dividing Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ2">10.2</a> by <span class="mathjax-tex">$\sqrt{A_{1}^{2}+A_{2}^{2}}$</span>, you can show that the solution may be written as</p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} x(t)=A\cos (\omega _{n}t-\phi ) \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.3) </div></div><p>where <span class="mathjax-tex">$A=\sqrt{A_{1}^{2}+A_{2}^{2}}$</span> is called the amplitude of motion and <span class="mathjax-tex">$\phi =\tan ^{-1}A_{2}/A_{1}$</span> is called the phase constant. In general, <span class="mathjax-tex">$\phi $</span> is chosen such that <span class="mathjax-tex">$0\le \phi \le \pi . A$</span> and <span class="mathjax-tex">$\phi $</span> can be determined from the initial conditions, i.e., from the values of the displacement and velocity when the motion starts. The mass therefore oscillates between <i>A</i> and <span class="mathjax-tex">$-A$</span>. The quantity <span class="mathjax-tex">$(\omega _{n}t-\phi )$</span> is called the phase angle. If this angle is increased by <span class="mathjax-tex">$ 2\pi $</span>, all physical quantities such as the displacement, velocity, and acceleration repeat themselves. The plot of <i>x</i> versus <i>t</i> is shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig2">10.2</a>. If <i>A</i> is fixed and <span class="mathjax-tex">$\phi $</span> is changed the motion will be the same except that the same physical quantities will appear either earlier or later than the preceding motion.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-1" data-title="Fig. 10.1"><figure><figcaption><b id="Fig1" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.1</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/1" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig1_HTML.png?as=webp"><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig1_HTML.png" alt="figure 1" loading="lazy" width="685" height="200"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p>A block of mass <i>m</i> attached to a light spring of spring constant <i>k</i> that is fixed at the other end</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/1" data-track-dest="link:Figure1 Full size image" aria-label="Full size image figure 1" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-2" data-title="Fig. 10.2"><figure><figcaption><b id="Fig2" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.2</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/2" rel="nofollow"><picture><img aria-describedby="Fig2" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig2_HTML.png" alt="figure 2" loading="lazy" width="685" height="293"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-2-desc"><p>Plot of <i>x</i> versus <i>t</i> for a simple harmonic oscillator</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/2" data-track-dest="link:Figure2 Full size image" aria-label="Full size image figure 2" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec5"><span class="c-article-section__title-number">10.3.1.1 </span>The Period and Frequency of Motion</h4><p>The period of motion is the time required for one complete cycle or oscillation. Since the phase angle is changed by <span class="mathjax-tex">$ 2\pi $</span> after one complete cycle, we have for the mass–spring system,</p><div id="Equ26" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \omega _{n}t+2\pi =\omega _{n}(t+T) $$</span></div></div><p>or</p><div id="Equ27" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ T=\frac{2\pi }{\omega _{n}}=2\pi \sqrt{\frac{m}{k}} $$</span></div></div><p>The frequency is defined as the number of complete cycles per unit time</p><div id="Equ28" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ f_{n}=\frac{1}{T}=\frac{\omega _{n}}{2\pi } $$</span></div></div><p>This frequency is called the natural frequency of the motion. The unit of the frequency is cycles/s or hertz (Hz).</p><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec6"><span class="c-article-section__title-number">10.3.1.2 </span>The Phase Difference</h4><p>The phase constant <span class="mathjax-tex">$\phi $</span> is important when comparing two or more oscillations of the same frequency Suppose a certain vibration has <span class="mathjax-tex">$\phi =0$</span>, this means that at <span class="mathjax-tex">$t=0$</span> the displacement is maximum <span class="mathjax-tex">$x=A$</span>. If a second vibration has also <span class="mathjax-tex">$\phi =0$</span>, then the two vibrations are said to be in phase (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig3">10.3</a> part a). Otherwise, the two vibrations are out of phase. If the phase constant of the second vibration is <span class="mathjax-tex">$\phi >0$</span>, then the second vibration is leading the first vibration in phase by <span class="mathjax-tex">$\phi $</span>. If <span class="mathjax-tex">$\phi <0$</span>, then the second vibration is lagging the first by <span class="mathjax-tex">$\phi $</span>. If <span class="mathjax-tex">$\phi =\pm \pi $</span>, the two vibrations are said to be in antiphase with each other (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig3">10.3</a> part b).</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-3" data-title="Fig. 10.3"><figure><figcaption><b id="Fig3" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.3</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/3" rel="nofollow"><picture><img aria-describedby="Fig3" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig3_HTML.png" alt="figure 3" loading="lazy" width="685" height="748"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-3-desc"><p><b>a</b> Two simple harmonic motions of the same frequency and same phase constant <span class="mathjax-tex">$\pi =0$</span> but differing in amplitude. <b>b</b> Two simple harmonic motions of the same frequency and amplitude but differing in phase by <span class="mathjax-tex">$\phi =\pm \pi $</span></p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/3" data-track-dest="link:Figure3 Full size image" aria-label="Full size image figure 3" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec7"><span class="c-article-section__title-number">10.3.1.3 </span>The Velocity and Acceleration</h4><p>The velocity of the mass is</p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} v(t)=\displaystyle \frac{dx}{dt}=-\omega _{n}A\sin (\omega _{n}t-\phi ) \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.4) </div></div><p>This can also be written as</p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} v(t)=\displaystyle \omega _{n}A\cos \bigg (\omega _{n}t-\phi +\frac{\pi }{2}\bigg ) \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.5) </div></div><p>The acceleration of the mass is</p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} a(t)=\displaystyle \frac{dv}{dt}=-\omega _{n}^{2}A\cos (\omega _{n}t-\phi ) \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.6) </div></div><p>or</p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} a(t)=\displaystyle \frac{dv}{dt}=\omega _{n}^{2}A\cos (\omega _{n}t-\phi +\pi ) \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.7) </div></div><p>Hence, the velocity and acceleration also vary harmonically with time with amplitudes <span class="mathjax-tex">$\omega _{n}A$</span> and <span class="mathjax-tex">$\omega _{n}^{2}A$</span>, respectively, but they all have the same angular frequency From Eqs. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ5">10.5</a> and <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ7">10.7</a> you can see that the velocity leads the displacement by <span class="mathjax-tex">$\pi /2$</span> or 90. The acceleration on the other hand leads the velocity by <span class="mathjax-tex">$\pi /2$</span> and the displacement by <span class="mathjax-tex">$\pi $</span> or 180. Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig4">10.4</a> shows the displacement, velocity, and acceleration versus time.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-4" data-title="Fig. 10.4"><figure><figcaption><b id="Fig4" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.4</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/4" rel="nofollow"><picture><img aria-describedby="Fig4" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig4_HTML.png" alt="figure 4" loading="lazy" width="685" height="1117"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-4-desc"><p>The displacement, velocity and acceleration versus time</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/4" data-track-dest="link:Figure4 Full size image" aria-label="Full size image figure 4" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec8"><span class="c-article-section__title-number">10.3.1.4 </span>Boundary Conditions</h4><p>Boundary conditions are used to find <i>A</i> and <span class="mathjax-tex">$\phi $</span> for a specific vibration. Suppose that the vibration is measured when the stopwatch is set to zero, i.e., at <span class="mathjax-tex">$t=0$</span> and that at that instant the mass is released from rest at a distance of <span class="mathjax-tex">$x=A_{1}$</span> from its equilibrium position. Substituting these conditions into Eqs. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ3">10.3</a> and <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ4">10.4</a>, we have</p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} x=A\cos \phi =A_{1} \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.8) </div></div> <div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} v=v_{0}=-\omega _{n}A\sin \phi \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.9) </div></div><p>Dividing Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ9">10.9</a> by Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ8">10.8</a> gives</p><div id="Equ29" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \tan \phi =\frac{-v_{0}}{\omega _{n}A_{1}} $$</span></div></div><p>Squaring and adding Eqs. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ9">10.9</a> and <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ8">10.8</a> gives</p><div id="Equ30" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ A_{1}^{2}+\bigg (\frac{v_{0}}{\omega _{n}}\bigg )^{2}=A^{2}\cos ^{2}\phi +A^{2}\sin ^{2}\phi $$</span></div></div><p>or</p><div id="Equ31" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ A=\sqrt{A_{1}^{2}+\bigg (\frac{v_{0}}{\omega _{n}}\bigg )^{2}} $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar1">Example 10.1</h3> <p>An object oscillates in simple harmonic motion according to the expression <span class="mathjax-tex">$x=(3\mathrm {m})\cos (\pi t+\pi /3)$</span>. Find (a) the amplitude, phase constant, period, and frequency of motion; (b) the displacement, velocity, and acceleration of the object at <span class="mathjax-tex">$t=0.5\mathrm {s}(\mathrm {c})$</span> the time when the object first reach <span class="mathjax-tex">$x=-1.5 \; \mathrm {m}.$</span></p> <h3 class="c-article__sub-heading" id="FPar2">Solution 10.1</h3> <p>(a)</p><div id="Equ32" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ A=3\,\mathrm {m} $$</span></div></div> <div id="Equ33" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \phi =\frac{\pi }{3} $$</span></div></div> <div id="Equ34" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ T=\frac{2\pi }{\omega _{n}}=\frac{(2\pi )}{\pi }=2 \; \mathrm {s} $$</span></div></div><p>and</p><div id="Equ35" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ f_{n}=\displaystyle \frac{1}{T}=\frac{1}{(2\mathrm {s})}=0.5 \; \text {Hz} $$</span></div></div><p>(b) At <span class="mathjax-tex">$t=0.5\,\mathrm {s}$</span></p><div id="Equ36" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=(3 \; \mathrm {m})\cos \bigg (\pi (0.5 \; \mathrm {s})+\frac{\pi }{3}\bigg )=-2.6 \; \mathrm {m} $$</span></div></div> <div id="Equ37" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ v=-(3\pi \; \mathrm {m}/\mathrm {s})\sin \bigg (\pi t+\frac{\pi }{3}\bigg ) $$</span></div></div><p>At <span class="mathjax-tex">$t=0.5\,\mathrm {s}$</span></p><div id="Equ38" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ v=(-3\pi \; \mathrm {m}/\mathrm {s})\sin \bigg (\pi (0.5 \; \mathrm {s})+\frac{\pi }{3}\bigg )=-4.7 \; \mathrm {m}/\mathrm {s} $$</span></div></div> <div id="Equ39" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ a=(-3\pi ^{2} \; \mathrm {m}/\mathrm {s}^{2})\cos \bigg (\pi t+\frac{\pi }{3}\bigg ) $$</span></div></div><p>at <span class="mathjax-tex">$t=0.5 \; \mathrm {s}$</span></p><div id="Equ40" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ a=(-3\pi ^{2} \; \mathrm {m}/\mathrm {s}^{2})\cos \bigg (\pi (0.5 \; \mathrm {s})+\frac{\pi }{3}\bigg )=25.6 \; \mathrm {m}/\mathrm {s}^{2} $$</span></div></div><p>(c) at <span class="mathjax-tex">$x=-1.5 \; \mathrm {m}$</span></p><div id="Equ41" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ (-1.5 \; \mathrm {m})=(3 \; \mathrm {m})\cos \bigg (\pi t+\frac{\pi }{3}\bigg ) $$</span></div></div><p>or</p><div id="Equ42" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{2\pi }{3}=\pi t+\frac{\pi }{3} $$</span></div></div><p>that gives <span class="mathjax-tex">$t=0.3 \; \mathrm {s}.$</span></p> <h3 class="c-article__sub-heading" id="FPar3">Example 10.2</h3> <p>A 9 kg object is moving along the <span class="mathjax-tex">$\mathrm {x}$</span>-axis under the influence of a force given by <span class="mathjax-tex">$F=(-3x)$</span> N. Find (a) the equation of motion; (b) the displacement of the mass at any time if at <span class="mathjax-tex">$t=0, x=5 \; \mathrm {m}$</span> and <span class="mathjax-tex">$v=0.$</span></p> <h3 class="c-article__sub-heading" id="FPar4">Solution 10.2</h3> <p>(a)</p><div id="Equ43" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ F=-3x=ma=m\frac{d^{2}x}{dt^{2}} $$</span></div></div><p>hence,</p><div id="Equ44" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{d^{2}x}{dt^{2}}+3x=0 $$</span></div></div><p>(b) The general solution of this equation is</p><div id="Equ45" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=A\cos \sqrt{3}t+B\sin \sqrt{3}t $$</span></div></div><p>Since at <span class="mathjax-tex">$t=0, \; x=5 \; \mathrm {m}$</span>, then <span class="mathjax-tex">$A=5 \; \mathrm {m}$</span> and</p><div id="Equ46" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=(5\mathrm {m})\cos \sqrt{3}t+B\sin \sqrt{3}t $$</span></div></div><p>also we have at <span class="mathjax-tex">$t=0, dx/dt=0$</span>, or</p><div id="Equ47" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ -5\sqrt{3}\sin \sqrt{3}t+\sqrt{3}B\cos \sqrt{3}t=0 $$</span></div></div><p>and therefore <span class="mathjax-tex">$B=0$</span>. Thus,</p><div id="Equ48" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=(5\mathrm {m})\cos \sqrt{3}t $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar5">Example 10.3</h3> <p>A 0.3 kg block is attached to a spring of force constant 20 <span class="mathjax-tex">$\mathrm {N}/\mathrm {m}$</span> on a frictionless horizontal surface. If the initial displacement and velocity of the system is 0.02 <span class="mathjax-tex">$\mathrm {m}$</span> and 0.2 <span class="mathjax-tex">$\mathrm {m}/\mathrm {s}$</span>, respectively, find the period, amplitude, and phase constant of motion.</p> <h3 class="c-article__sub-heading" id="FPar6">Solution 10.3</h3> <div id="Equ49" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \omega _{n}=\sqrt{\frac{k}{m}}=\sqrt{\frac{(20 \; \mathrm {N}/\mathrm {m})}{(0.3 \; \mathrm {k}\mathrm {g})}}=8.2 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s} $$</span></div></div> <div id="Equ50" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ A=\sqrt{A_{1}^{2}+\bigg (\frac{v_{0}}{\omega _{n}}\bigg )^{2}}=\sqrt{(0.02 \; \mathrm {m})^{2}+\bigg (\frac{(0.2 \; \mathrm {m}/\mathrm {s})}{(82 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s})}\bigg )^{2}}=0.03 \; \mathrm {m} $$</span></div></div> <div id="Equ51" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \tan \phi =\frac{-v_{0}}{\omega _{n}A_{1}}=\frac{-(0.2 \; \mathrm {m}/.\mathrm {s})}{(8.2 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s})(0.03 \; \mathrm {m})}=-0.8 $$</span></div></div> <div id="Equ52" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \phi =-38.7^{\circ } $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar7">Example 10.4</h3> <p>A particle of mass <i>m</i> is dropped in a straight tunnel that is drilled through the earth and which passes through the center of earth as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig5">10.5</a>. Show that the motion of the particle is simple harmonic motion and find its period.</p> <h3 class="c-article__sub-heading" id="FPar8">Solution 10.4</h3> <p>Assuming that the earth is a perfect sphere of uniform density and since the particle is inside the earth, then from Sect. <a href="https://doi.org/10.1007/978-3-030-15195-9_9">9.2</a>, the gravitational force exerted on the particle by the earth is</p><div id="Equ53" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ F=-\left( \frac{GmM_{E}}{R_{E}^{3}}\right) r=-kr $$</span></div></div><p>Because this force is directly proportional to the displacement and is opposite to it, then the particle will move in simple harmonic motion about the center of the earth. The equation of motion is</p><div id="Equ54" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{dr^{2}}{dt^{2}}+\bigg (\frac{GM_{E}}{R_{E}^{3}}\bigg )r=0 $$</span></div></div><p>hence,</p><div id="Equ55" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \omega _{n}=\sqrt{\frac{GM_{E}}{R_{E}^{3}}}=\sqrt{\frac{(6.67\times 10^{-11} \; \mathrm {N}\mathrm {m}^{2}/\mathrm {k}\mathrm {g}^{2})(5.98\times 10^{24} \; \mathrm {k}\mathrm {g})}{(6.37\times \mathrm {l0}^{6} \; \mathrm {m})^{3}}}=1.24\times 10^{-3} \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s} $$</span></div></div> <div id="Equ56" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ T=\frac{2\pi }{\omega _{n}}=\frac{2(3.14)}{(1.24\times 10^{-3} \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s})}=5055.4 \; \mathrm {s}=84.25 \; \min $$</span></div></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-5" data-title="Fig. 10.5"><figure><figcaption><b id="Fig5" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.5</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/5" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig5_HTML.png?as=webp"><img aria-describedby="Fig5" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig5_HTML.png" alt="figure 5" loading="lazy" width="685" height="685"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-5-desc"><p>A particle of mass <i>m</i> is dropped in a straight tunnel that is drilled through the earth and which passes through the center of earth</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/5" data-track-dest="link:Figure5 Full size image" aria-label="Full size image figure 5" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="FPar9">Example 10.5</h3> <p>A 0.4 kg block is connected to two springs of force constants <span class="mathjax-tex">$k_{1}=20 \; \mathrm {N}/\mathrm {m}$</span> and <span class="mathjax-tex">$k_{2}=50 \; \mathrm {N}/\mathrm {m}$</span> as in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig6">10.6</a>. Find (a) the total force acting on the block; (b) the period of motion.</p> <h3 class="c-article__sub-heading" id="FPar10">Solution 10.5</h3> <p>The force that each spring exerts on the block acts in the opposite direction of the displacement, therefore we have</p><div id="Equ57" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \sum F=-k_{1}x-k_{2}x=-(k_{1}+k_{2})x=-(70 \; \mathrm {N}/\mathrm {m})x $$</span></div></div><p>Thus the two springs can be considered as one spring of a force constant of <span class="mathjax-tex">$(k_{1}+k_{2})$</span>. The period of motion is therefore</p><div id="Equ58" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ T=2\pi \sqrt{\frac{m}{k_{1}+k_{2}}}=2(3.14)\sqrt{\frac{(0.4 \; \mathrm {k}\mathrm {g})}{(70 \; \mathrm {N}/\mathrm {m})}}=0.5 \; \mathrm {s} $$</span></div></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-6" data-title="Fig. 10.6"><figure><figcaption><b id="Fig6" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.6</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/6" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig6_HTML.png?as=webp"><img aria-describedby="Fig6" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig6_HTML.png" alt="figure 6" loading="lazy" width="685" height="128"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-6-desc"><p>A block connected to two springs</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/6" data-track-dest="link:Figure6 Full size image" aria-label="Full size image figure 6" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-7" data-title="Fig. 10.7"><figure><figcaption><b id="Fig7" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.7</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/7" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig7_HTML.png?as=webp"><img aria-describedby="Fig7" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig7_HTML.png" alt="figure 7" loading="lazy" width="685" height="260"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-7-desc"><p>A second block on top of a block connected to a spring</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/7" data-track-dest="link:Figure7 Full size image" aria-label="Full size image figure 7" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="FPar11">Example 10.6</h3> <p>A 6 kg block is connected to a light spring of force constant of 300 <span class="mathjax-tex">$\mathrm {N}/\mathrm {m}$</span> on a frictionless horizontal surface. On top of it a second block of mass of 2 kg is placed. If the coefficient of static friction between the two blocks is 0.4 (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig7">10.7</a>), find the maximum amplitude the system can have when it is in simple harmonic motion such that there is no slipping between the blocks.</p> <h3 class="c-article__sub-heading" id="FPar12">Solution 10.6</h3> <p>The maximum acceleration of the lower block is <span class="mathjax-tex">$a_{\max }=\omega _{n}^{2}A$</span>. In order for the upper block not to slip, the force of static friction between the two blocks must produce the same acceleration as the lower block. The maximum statistical frictional force that can be exerted on the upper block is <span class="mathjax-tex">$\mu _{s}mg$</span> and hence, the maximum acceleration that the force of static friction can produce is <span class="mathjax-tex">$\mu _{s}g$</span>. Therefore, <span class="mathjax-tex">$\mu _{s}g=a_{\max }=\omega _{n}^{2}A$</span>. Since</p><div id="Equ59" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \omega _{n}=\sqrt{\frac{k}{(m+M)}} $$</span></div></div><p>we have</p><div id="Equ60" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ A=\frac{\mu _{s}g}{\omega _{n}^{2}}=\frac{\mu _{s}g(m+M)}{k}=\frac{(0.4)(9.8 \; \mathrm {m}/\mathrm {s}^{2})(8 \; \mathrm {k}\mathrm {g})}{(300 \; \mathrm {N}/\mathrm {m})}=0.1 \; \mathrm {m} $$</span></div></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-8" data-title="Fig. 10.8"><figure><figcaption><b id="Fig8" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.8</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/8" rel="nofollow"><picture><img aria-describedby="Fig8" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig8_HTML.png" alt="figure 8" loading="lazy" width="685" height="528"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-8-desc"><p>A particle in uniform circular motion</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/8" data-track-dest="link:Figure8 Full size image" aria-label="Full size image figure 8" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="Sec9"><span class="c-article-section__title-number">10.3.2 </span>Simple Harmonic Motion and Uniform Circular Motion</h3><p>Consider a circle of radius <i>A</i> centered at the <span class="mathjax-tex">$\mathrm {x}$</span> and <span class="mathjax-tex">$\mathrm {y}$</span> axes as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig8">10.8</a>. Let A be the position vector of a particle <span class="mathjax-tex">$\mathrm {P}$</span> rotating with a constant angular speed <span class="mathjax-tex">$\omega _{n}$</span> in the anticlockwise direction. The particle is thus in uniform circular motion. Suppose <span class="mathjax-tex">$\mathrm {P}$</span> starts the rotation at <span class="mathjax-tex">$t=0$</span> at an angle of <span class="mathjax-tex">$\phi $</span> measured from the positive <span class="mathjax-tex">$\mathrm {x}$</span>-axis. At any time, the angular position of the particle is given by <span class="mathjax-tex">$(\omega _{n}t+\phi )$</span>, therefore the vector position of the particle at any time is</p><div id="Equ61" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \mathbf {A}=x\mathbf {i}+y\mathbf {j}=A\cos (\omega _{n}t+\phi )\mathbf {i}+A\sin (\omega _{n}t+\phi )\mathbf {j} $$</span></div></div><p>Hence,</p><div id="Equ62" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=A\cos (\omega _{n}t+\phi ) $$</span></div></div><p>and</p><div id="Equ63" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ y=A\sin (\omega _{n}t+\phi ) $$</span></div></div><p>That is, as <span class="mathjax-tex">$\mathrm {P}$</span> moves in uniform circular motion, its projection <span class="mathjax-tex">$\mathrm {P}'$</span> on the x-axis moves in simple harmonic motion where the radius of the circle is equal to the amplitude of motion. The projection of <span class="mathjax-tex">$\mathrm {P}$</span> along the <span class="mathjax-tex">$\mathrm {y}$</span>-axis also undergoes simple harmonic motion. Thus, uniform circular motion may be considered as a combination of the simple harmonic motions of the projections of <span class="mathjax-tex">$\mathrm {P}$</span> on each axis. These two simple harmonic motions have equal amplitudes and angular frequencies but are in quadrature with each other (they differ in phase by <span class="mathjax-tex">$\pi /2$</span>). The linear tangential velocity of the particle in this uniform circular motion is given by</p><div id="Equ64" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ v=A\omega _{n} $$</span></div></div><p>The <span class="mathjax-tex">$\mathrm {x}$</span> component of the velocity is from Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig9">10.9</a> given by</p><div id="Equ65" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ v_{x}=-\omega _{n}A\sin (\omega _{n}t+\phi ) $$</span></div></div><p>The acceleration of the particle in uniform circular motion is just the radial (centripetal) acceleration that is given by</p><div id="Equ66" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ a=\frac{v^{2}}{A}=A\omega _{n}^{2} $$</span></div></div><p>The <span class="mathjax-tex">$\mathrm {x}$</span> components of the acceleration (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig10">10.10</a>) is</p><div id="Equ67" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ a_{x}=-\omega _{n}^{2}A\cos (\omega _{n}t+\phi ) $$</span></div></div><p>Hence as you can see, the displacement, velocity, and acceleration of the projection of <span class="mathjax-tex">$\mathrm {P}$</span> onto the <span class="mathjax-tex">$\mathrm {x}$</span> (or <span class="mathjax-tex">$\mathrm {y}$</span> axis) are the same as that of a simple harmonic motion. From this, we conclude that the simple harmonic motion can be represented as the projection of uniform circular motion along a diameter of the circle.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-9" data-title="Fig. 10.9"><figure><figcaption><b id="Fig9" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.9</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/9" rel="nofollow"><picture><img aria-describedby="Fig9" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig9_HTML.png" alt="figure 9" loading="lazy" width="685" height="528"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-9-desc"><p>The velocity components of the particle</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/9" data-track-dest="link:Figure9 Full size image" aria-label="Full size image figure 9" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-10" data-title="Fig. 10.10"><figure><figcaption><b id="Fig10" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.10</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/10" rel="nofollow"><picture><img aria-describedby="Fig10" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig10_HTML.png" alt="figure 10" loading="lazy" width="685" height="528"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-10-desc"><p>The acceleration components of the particle</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/10" data-track-dest="link:Figure10 Full size image" aria-label="Full size image figure 10" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="Sec10"><span class="c-article-section__title-number">10.3.3 </span>Energy of a Simple Harmonic Oscillator</h3><p>Since in a simple harmonic oscillator, there aren’t any dissipative forces, the total mechanical energy of the system is conserved and is equal to the sum of its kinetic and potential energies, that is</p><div id="Equ68" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ E=K+U $$</span></div></div> <div id="Equ69" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ K=\frac{1}{2}mv^{2}=\frac{1}{2}m\omega _{n}^{2}A^{2}\sin ^{2}(\omega _{n}t+\phi ) $$</span></div></div> <div id="Equ70" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ U=\frac{1}{2}kx^{2}=\frac{1}{2}kA^{2}\cos ^{2}(\omega _{n}t+\phi ) $$</span></div></div><p>Thus,</p><div id="Equ71" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ E=\frac{1}{2}kA^{2}[\sin ^{2}(\omega _{n}t+\phi )+\cos ^{2}(\omega _{n}t+\phi )] $$</span></div></div><p>or</p><div id="Equ72" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ E=\displaystyle \frac{1}{2}kA^{2}= \text {constant} $$</span></div></div><p>The equation of motion of a simple harmonic oscillator can be obtained from the total mechanical energy of the system as follows:</p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} E=\displaystyle \frac{1}{2}m\dot{x}^{2}+\frac{1}{2}kx^{2}=\frac{1}{2}kA^{2} \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.10) </div></div> <div id="Equ73" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{dE}{dt}=m\dot{x}\ddot{x}+kx\dot{x}=0 $$</span></div></div><p>or</p><div id="Equ74" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ m\ddot{x}+kx=0 $$</span></div></div><p>Hence</p><div id="Equ75" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \ddot{x}+\omega _{n}^{2}x=0 $$</span></div></div><p>where <span class="mathjax-tex">$\omega _{n}=\sqrt{k/m}$</span>. As the mass moves, its kinetic energy is transformed into potential energy and vice versa. Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig11">10.11</a> shows the kinetic energy and potential energy of the system as a function of time and as a function of the displacement respectively Note that the variation of <i>U</i> and <i>K</i> with time is at twice the angular frequency of the variation of <i>x</i>, <i>v</i>, and <i>a</i> with time. This is because the potential energy is converted to kinetic energy twice in each cycle. The velocity of the simple harmonic oscillator can be obtained from the total energy of the system. From Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ10">10.10</a>, we have</p><div id="Equ76" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ v=\pm \sqrt{\frac{k}{m}(A^{2}-x^{2})} $$</span></div></div><p>Hence, the maximum speed is at <span class="mathjax-tex">$x=0$</span> and is zero at <span class="mathjax-tex">$x=\pm A$</span> which are called the turning points as discussed in Chap. chap444.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-11" data-title="Fig. 10.11"><figure><figcaption><b id="Fig11" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.11</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/11" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig11_HTML.png?as=webp"><img aria-describedby="Fig11" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig11_HTML.png" alt="figure 11" loading="lazy" width="685" height="610"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-11-desc"><p>As the mass moves, its kinetic energy is transformed into potential energy and vice versa</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/11" data-track-dest="link:Figure11 Full size image" aria-label="Full size image figure 11" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="FPar13">Example 10.7</h3> <p>A 0.3 kg mass is attached to a light spring. If the total energy of the system is 0.025 <span class="mathjax-tex">$\mathrm {J}$</span> and the amplitude of motion is 5 cm, find the period and frequency of motion.</p> <h3 class="c-article__sub-heading" id="FPar14">Solution 10.7</h3> <div id="Equ77" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ E= (0.025\,\displaystyle \mathrm {J})=\frac{1}{2}kA^{2}=\frac{1}{2}k(0.05 \; \mathrm {m})^{2} $$</span></div></div><p>hence</p><div id="Equ78" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ k=20 \; \mathrm {N}/\mathrm {m} $$</span></div></div><p>The period of motion is therefore</p><div id="Equ79" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ T=2\pi \sqrt{\frac{m}{k}}=2(3.14)\sqrt{\frac{(0.3 \; \mathrm {k}\mathrm {g})}{(20 \; \mathrm {N}/\mathrm {m})}}=0.8 \; \mathrm {s} $$</span></div></div><p>and the frequency is</p><div id="Equ80" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$f_{n}=\displaystyle \frac{1}{T}=\frac{1}{(0.8 \; \mathrm {s})}=1.25 \; \mathrm {Hz}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar15">Example 10.8</h3> <p>A 0.2 kg block is attached to a light spring of force constant of 11 <span class="mathjax-tex">$\mathrm {N}/\mathrm {m}$</span> on a horizontal frictionless surface. If the block is displaced a distance of 8 cm from its equilibrium position, find (a) the amplitude, the angular frequency, the period and the frequency of motion when the block is released; (b) the maximum force exerted on the block; (c) the total mechanical energy of the system; (d) the maximum speed and maximum acceleration of the block; (e) the velocity of the block when its displacement is 2 cm; (f) the acceleration of the block when its displacement is 3 cm.</p> <h3 class="c-article__sub-heading" id="FPar16">Solution 10.8</h3> <p>(a)</p><div id="Equ81" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ A=8\,\mathrm {c}\mathrm {m} $$</span></div></div> <div id="Equ82" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \omega _{n}=\sqrt{\frac{k}{m}}=\sqrt{\frac{(11 \; \mathrm {N}/\mathrm {m})}{(0.2 \; \mathrm {k}\mathrm {g})}}=7.4 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s} $$</span></div></div> <div id="Equ83" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ T=\frac{2\pi }{\omega _{n}}=\frac{2(3.14)}{(7.4 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s})}=0.85 \; \mathrm {s} $$</span></div></div> <div id="Equ84" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ f_{n}=\displaystyle \frac{1}{T}=\frac{1}{(0.85 \; \mathrm {s})}=1.2 \; \mathrm {Hz} $$</span></div></div><p>(b)</p><div id="Equ85" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ |F|=kA=(11\,\mathrm {N}/\mathrm {m})(0.08\,\mathrm {m})=0.9\,\mathrm {N} $$</span></div></div><p>(c)</p><div id="Equ86" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ E=\frac{1}{2}kA^{2}=\frac{1}{2}(11 \; \mathrm {N}/\mathrm {m})(0.08 \; \mathrm {m})^{2}=0.035 \; \mathrm {J} $$</span></div></div><p>(d)</p><div id="Equ87" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ v_{\max }=\omega _{n}A=( 7.4 \; \mathrm {rad/s}) (0.08 \; \mathrm {m})=0.6 \; \mathrm {m}/\mathrm {s} $$</span></div></div> <div id="Equ88" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ a_{\max }=\omega _{n}^{2}A=(7.4 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s})^{2}(0.08 \; \mathrm {m})=4.4 \; \mathrm {m}/\mathrm {s}^{2} $$</span></div></div><p>(e)</p><div id="Equ89" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ v=\pm \sqrt{\frac{k}{m}(A^{2}-x^{2})}=\sqrt{\frac{(11 \; \mathrm {N}/\mathrm {m})}{(0.2 \; \mathrm {k}\mathrm {g})}((0.08 \; \mathrm {m})^2-(0.02 \; \mathrm {m})^2)}=1.8 \; \mathrm {m}/\mathrm {s} $$</span></div></div><p>(f)</p><div id="Equ90" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ a=-\omega _{n}^{2}x=-(7.4 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s})^{2}(0.03 \; \mathrm {m})=-1.6 \; \mathrm {m}/\mathrm {s}^{2} $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar17">Example 10.9</h3> <p>An object connected to a spring is in simple harmonic motion on a frictionless surface. If the object’s displacement when <span class="mathjax-tex">$(2v_{\max }/3)$</span> is <span class="mathjax-tex">$\pm 0.015 \; \mathrm {m}$</span>, find the amplitude of motion.</p> <h3 class="c-article__sub-heading" id="FPar18">Solution 10.9</h3> <div id="Equ91" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{1}{2}kA^{2}=\frac{1}{2}mv^{2}+\frac{1}{2}kx^{2}=\frac{1}{2}m\frac{4\omega _{n}^{2}A^{2}}{9}+\frac{1}{2}kx^{2} $$</span></div></div><p>therefore</p><div id="Equ92" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ A^{2}=\displaystyle \frac{9}{5}x^{2}=\frac{9}{5} (0.015 \; \mathrm {m})^{2} $$</span></div></div> <div id="Equ93" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ A=0.02 \; \mathrm {m} $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar19">Example 10.10</h3> <p>A solid cylinder is connected to a light spring as in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig12">10.12</a>. If the cylinder rolls without slipping along the surface, show that the motion of the cylinder is simple harmonic motion and find its frequency.</p> <h3 class="c-article__sub-heading" id="FPar20">Solution 10.10</h3> <p>At any instant the total mechanical energy is</p><div id="Equ94" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ E=\frac{1}{2}kx^{2}+\frac{1}{2}I_{cm}\omega ^{2}+\frac{1}{2}Mv_{cm}^{2}=\frac{1}{2}kx^{2}+\frac{1}{2}I_{cm}\frac{v_{cm}^{2}}{R^{2}}+\frac{1}{2}Mv_{cm}^{2} $$</span></div></div> <div id="Equ95" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ =\frac{1}{2}kx^{2}+\frac{1}{2}\left( \frac{1}{2}MR^{2}\right) \frac{v_{cm}^{2}}{R^{2}}+\frac{1}{2}Mv_{cm}^{2} $$</span></div></div><p>Since the total mechanical energy is conserved</p><div id="Equ96" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{dE}{dt}=kv_{cm}x+\frac{1}{2}Mv_{cm}a_{cm}+Mv_{cm}a_{cm}=0 $$</span></div></div> <div id="Equ97" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ kv_{cm}x=\frac{-3}{2}Mv_{cm}a_{cm} $$</span></div></div><p>or</p><div id="Equ98" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ a_{cm}=\frac{-2}{3}\frac{k}{M}x $$</span></div></div> <div id="Equ99" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{d^{2}x}{dt^{2}}+\frac{2}{3}\frac{k}{M}x=0 $$</span></div></div><p>this equation is of a simple harmonic motion with</p><div id="Equ100" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \omega _{n}=\sqrt{\frac{2}{3}\frac{k}{M}} $$</span></div></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-12" data-title="Fig. 10.12"><figure><figcaption><b id="Fig12" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.12</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/12" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig12_HTML.png?as=webp"><img aria-describedby="Fig12" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig12_HTML.png" alt="figure 12" loading="lazy" width="685" height="260"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-12-desc"><p>A solid cylinder connected to a light spring</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/12" data-track-dest="link:Figure12 Full size image" aria-label="Full size image figure 12" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="Sec11"><span class="c-article-section__title-number">10.3.4 </span>The Simple Pendulum</h3><p>The simple pendulum is an example of an angular vibration in which the restoring effect is due to a restoring torque. A simple pendulum consists of a mass (called the bob) suspended by a light string of length <i>L</i> that is fixed at the other end (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig13">10.13</a>). If the mass is pulled to the right or left from its equilibrium position and released, then the pendulum will swing in a vertical plane about an axis passing through O. The resulting motion is then a periodic or oscillatory motion. The restoring torque is due to gravity and is given by</p><div id="Equ101" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \tau =-(mg\sin \theta )L $$</span></div></div><p>The minus sign indicates that the torque is a restoring torque, since it always tends to decrease <span class="mathjax-tex">$\theta $</span>. The moment of inertia of the bob about an axis passing through <span class="mathjax-tex">$\mathrm {O}$</span> is</p><div id="Equ102" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ I=mL^{2} $$</span></div></div><p>From Newton’s second law in angular form, we have</p><div id="Equ103" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \tau =I\alpha =I\ddot{\theta } $$</span></div></div><p>Hence,</p><div id="Equ104" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ -mg\sin \theta L=mL^{2}\ddot{\theta } $$</span></div></div><p>or</p><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \displaystyle \ddot{\theta }+\bigg (\frac{g}{L}\bigg )\sin \theta =0 \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.11) </div></div><p>This equation does not represent a harmonic motion. That is because the torque is not directly proportional to the angular displacement. Thus, the system is nonlinear. However for small angular displacements, we have <span class="mathjax-tex">$\sin \theta \approx \theta ($</span>since <span class="mathjax-tex">$\sin \theta =\theta -\theta ^{3}/3!+\theta ^{5}/5!\ldots )$</span> and Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ11">10.11</a> becomes</p><div id="Equ105" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \ddot{\theta }+\bigg (\frac{g}{L}\bigg )\theta =0 $$</span></div></div><p>or</p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \ddot{\theta }+\omega _{n}^{2}\theta =0 \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.12) </div></div><p>where <span class="mathjax-tex">$\omega _{n}=\sqrt{g/L}$</span>. Hence for small angular displacements, the motion is a simple harmonic motion. The solution of Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ12">10.12</a> is of the form</p><div id="Equ106" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \theta =\theta _{m}\cos (\omega _{n}t-\phi ) $$</span></div></div><p>where <span class="mathjax-tex">$\theta _{m}$</span> is the maximum angular displacement and <span class="mathjax-tex">$\phi $</span> is the phase constant. The plot of this equation is shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig14">10.14</a>. The period of the simple pendulum is therefore given by</p><div id="Equ107" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ T=\frac{2\pi }{\omega _{n}}=2\pi \sqrt{\frac{L}{g}} $$</span></div></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-13" data-title="Fig. 10.13"><figure><figcaption><b id="Fig13" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.13</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/13" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig13_HTML.png?as=webp"><img aria-describedby="Fig13" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig13_HTML.png" alt="figure 13" loading="lazy" width="685" height="656"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-13-desc"><p>The simple pendulum</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/13" data-track-dest="link:Figure13 Full size image" aria-label="Full size image figure 13" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-14" data-title="Fig. 10.14"><figure><figcaption><b id="Fig14" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.14</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/14" rel="nofollow"><picture><img aria-describedby="Fig14" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig14_HTML.png" alt="figure 14" loading="lazy" width="685" height="373"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-14-desc"><p>The displacement versus time of a simple pendulum</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/14" data-track-dest="link:Figure14 Full size image" aria-label="Full size image figure 14" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec12"><span class="c-article-section__title-number">10.3.4.1 </span>Energy</h4><p>The kinetic energy of the simple pendulum is</p><div id="Equ108" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ K=\frac{1}{2}mv^{2}=\frac{1}{2}mL^{2}\omega _{n}^{2}=\frac{1}{2}mL\dot{\theta }^{2} $$</span></div></div><p>Taking the reference point of potential energy of the system to be zero when the bob is at the bottom, we have</p><div id="Equ109" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ U=MgL(1-\cos \theta ) $$</span></div></div><p>The total energy is therefore given by</p><div id="Equ110" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ E=K+U=\frac{1}{2}ML^{2}\dot{\theta }^{2}+MgL(1-\cos \theta ) $$</span></div></div><p>For small <span class="mathjax-tex">$\theta $</span>, we have <span class="mathjax-tex">$\displaystyle \cos \theta \approx 1-\frac{\theta ^{2}}{2} $</span>since <span class="mathjax-tex">$\cos \theta =1-\theta ^{2}/2!+\theta ^{4}/4!\ldots )$</span> thus</p><div id="Equ111" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ E=\frac{1}{2}ML^{2}\dot{\theta }^{2}+\frac{1}{2}MgL\theta ^{2} $$</span></div></div><p>Since</p><div id="Equ112" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \dot{\theta }=-\theta _{m}\omega _{n}\sin (\omega _{n}t-\phi ) $$</span></div></div><p>we have</p><div id="Equ113" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ E=\frac{1}{2}ML^{2}\theta _{m}^{2}\omega _{n}^{2}\sin ^{2}(\omega _{n}t-\phi )+\frac{1}{2}MgL\theta _{m}^{2}\cos ^{2}(\omega _{n}t-\phi ) $$</span></div></div><p>or</p><div id="Equ114" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ E=\frac{1}{2}MgL\theta _{m}^{2} $$</span></div></div><p>Therefore, the total energy of the system is constant. Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig15">10.15</a> shows the variation of the kinetic and potential energies with the displacement.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-15" data-title="Fig. 10.15"><figure><figcaption><b id="Fig15" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.15</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/15" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig15_HTML.png?as=webp"><img aria-describedby="Fig15" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig15_HTML.png" alt="figure 15" loading="lazy" width="685" height="302"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-15-desc"><p>The total energy of a simple pendulum</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/15" data-track-dest="link:Figure15 Full size image" aria-label="Full size image figure 15" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>The equation of motion may also be obtained from energy as follows:</p><div id="Equ115" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{dE}{dt}=ML^{2}\dot{\theta }\ddot{\theta }+MgL\theta \dot{\theta }=0 $$</span></div></div><p>or</p><div id="Equ116" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \ddot{\theta }+\bigg (\frac{g}{L}\bigg )\theta =0 $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar21">Example 10.11</h3> <p>A simple pendulum is 0.5 <span class="mathjax-tex">$\mathrm {m}$</span> long. Find its period at the surface of Mars and compare it to its period at the earth’s surface.</p> <h3 class="c-article__sub-heading" id="FPar22">Solution 10.11</h3> <p>At Mars’s surface, the gravitational acceleration is</p><div id="Equ117" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ g_{M}=\frac{GM_{M}}{R_{M}^{2}}=\frac{(6.67\times 10^{-11} \; \mathrm {N}\, mathrm{m}^{2}/\mathrm {k}\mathrm {g}^{2})(6.42\times 10^{23} \; \mathrm {k}\mathrm {g})}{(3.37\times \mathrm {l}0^{6} \; \mathrm {m})^{2}}=3.8 \; \mathrm {m}/\mathrm {s}^{2} $$</span></div></div><p>The period at Mars is therefore</p><div id="Equ118" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ T_{M}=2\pi \sqrt{\frac{L}{g_{M}}}=2(3.14)\sqrt{\frac{(0.5 \; \mathrm {m})}{(3.8 \; \mathrm {m}/\mathrm {s}^{2})}}=2.3 \; \mathrm {s} $$</span></div></div><p>At the earth’s surface,</p><div id="Equ119" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ T_{E}=2\pi \sqrt{\frac{L}{g_{E}}}=2(3.14)\sqrt{\frac{(0.5 \; \mathrm {m})}{(9.8 \; \mathrm {m}/\mathrm {s}^{2})}}=1.4 \; \mathrm {s} $$</span></div></div><p>Thus, <span class="mathjax-tex">$T_{M}=1.6T_{E}.$</span></p> <h3 class="c-article__sub-heading" id="FPar23">Example 10.12</h3> <p>A simple pendulum of length of 2 <span class="mathjax-tex">$\mathrm {m}$</span> is displaced through an angle of <span class="mathjax-tex">$12^{\circ }$</span> and released. Find (a) the angular frequency of motion; (b) the maximum angular speed and maximum angular acceleration.</p> <h3 class="c-article__sub-heading" id="FPar24">Solution 10.12</h3> <p>(a) The amplitude of motion is</p><div id="Equ120" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \displaystyle \theta _{\max }=(12^{\circ })\bigg (\frac{2\pi \; \mathrm {r}\mathrm {a}\mathrm {d}}{360^{\circ } \; \deg }\bigg )=0.21 \; \text {rad} $$</span></div></div><p>The angular frequency is</p><div id="Equ121" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \omega _{n}=\sqrt{\frac{g}{L}}=\sqrt{\frac{(9.8 \; \mathrm {m}/\mathrm {s}^{2})}{(2 \; \mathrm {m})}}=2.2 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s} $$</span></div></div><p>(b) The maximum angular speed is</p><div id="Equ122" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \dot{\theta }_{\max }=\omega _{n}A=( 2.2 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s})(0.21 \; \mathrm {r}\mathrm {a}\mathrm {d}) =0.5 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s} $$</span></div></div><p>The maximum angular acceleration is</p><div id="Equ123" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \ddot{\theta }_{\max }=\omega _{n}^{2}A=(2.2 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s})^{2}(0.21 \; \mathrm {r}\mathrm {a}\mathrm {d} )=1 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}^{2} $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar25">Example 10.13</h3> <p>A simple pendulum 1.4 <span class="mathjax-tex">$\mathrm {m}$</span> in length is displaced through an angle of <span class="mathjax-tex">$10^{\circ }$</span> and released. Find the velocity of the bob when it reaches the bottom.</p> <h3 class="c-article__sub-heading" id="FPar26">Solution 10.13</h3> <div id="Equ124" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \displaystyle \theta =(10^{\circ })\bigg (\frac{2\pi \; \mathrm {r}\mathrm {a}\mathrm {d}}{360^{\circ } \; \deg }\bigg )=0.17 \; \text {rad} $$</span></div></div><p>Taking the potential energy to be zero at the bottom, we have</p><div id="Equ125" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ mgL(1-\cos \theta )=\frac{1}{2}mv^{2} $$</span></div></div><p>Since <span class="mathjax-tex">$\theta $</span> is small, <span class="mathjax-tex">$\cos \theta \approx 1-\theta ^{2}/2$</span> and therefore</p><div id="Equ126" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ mgL\frac{\theta ^{2}}{2}=\frac{1}{2}mv^{2} $$</span></div></div><p>and</p><div id="Equ127" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ v=\sqrt{gL}\theta =\sqrt{(9.8 \; \mathrm {m}/\mathrm {s}^{2})(14 \; \mathrm {m})} (0.17 \; \mathrm {rad}) = \; 0.63\mathrm {m}/\mathrm {s} $$</span></div></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-16" data-title="Fig. 10.16"><figure><figcaption><b id="Fig16" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.16</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/16" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig16_HTML.png?as=webp"><img aria-describedby="Fig16" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig16_HTML.png" alt="figure 16" loading="lazy" width="685" height="412"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-16-desc"><p>The physical pendulum</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/16" data-track-dest="link:Figure16 Full size image" aria-label="Full size image figure 16" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="Sec13"><span class="c-article-section__title-number">10.3.5 </span>The Physical Pendulum</h3><p>The physical pendulum is a rigid body that oscillates about an axis passing through a point in the body other than its center of mass (the center of mass is assumed to be located at the center of gravity). Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig16">10.16</a> shows a rigid body pivoted at point <span class="mathjax-tex">$\mathrm {O}$</span> that is at a distance <i>d</i> from the center of mass. The equilibrium position of the body is when its center of mass is directly below the pivot O. If the body is displaced either to the right or left from the equilibrium position, a restoring torque due to gravity will act on it. As a result, the body will oscillate in a vertical plane where the axis of rotation is perpendicular to the page. The restoring torque is given by</p><div id="Equ128" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \tau =-Mgd\ \sin \theta $$</span></div></div><p>where <i>M</i> is the mass of the body and <i>d</i> is the moment arm of the tangential component of the weight <span class="mathjax-tex">$(Mg\ \sin \theta )$</span>. From Newton’s second law, we have</p><div id="Equ129" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \tau =I\alpha $$</span></div></div> <div id="Equ130" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ -Mgd\sin \theta =I\ddot{\theta } $$</span></div></div><p>For small angular displacements <span class="mathjax-tex">$\sin \theta \approx \theta $</span> and hence</p><div id="Equ131" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \ddot{\theta }+\bigg (\frac{Mgd}{I}\bigg )\theta =0 $$</span></div></div><p>or</p><div id="Equ132" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \ddot{\theta }+\omega _{n}^{2}\theta =0 $$</span></div></div><p>This equation is of a simple harmonic motion with an angular frequency of</p><div id="Equ133" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \omega _{n}=\sqrt{\frac{Mgd}{I}} $$</span></div></div><p>and a period of motion of</p><div id="Equ134" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ T=\frac{2\pi }{\omega _{n}}=2\pi \sqrt{\frac{I}{Mgd}} $$</span></div></div><p>Thus,</p><div id="Equ135" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ I=\frac{T^{2}Mgd}{4\pi ^{2}} $$</span></div></div><p>Therefore, the moment of inertia of a body can be found by measuring its period when it is in simple harmonic motion as a physical pendulum. Note that, the simple pendulum is a special case of the physical pendulum since for a simple pendulum of mass <i>m</i>, the moment of inertia is</p><div id="Equ136" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ I=md^{2} $$</span></div></div><p>and thus, the angular frequency is</p><div id="Equ137" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \omega _{n}=\sqrt{\frac{mgd}{md^{2}}}=\sqrt{\frac{g}{d}} $$</span></div></div><p>This angular frequency is of a simple pendulum where <i>d</i> represents the length of the string.</p> <h3 class="c-article__sub-heading" id="FPar27">Example 10.14</h3> <p>A uniform rod of length of 0.6 <span class="mathjax-tex">$\mathrm {m}$</span> that is suspended at one end oscillates with a small amplitude as in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig17">10.17</a>. Find the frequency of motion.</p> <h3 class="c-article__sub-heading" id="FPar28">Solution 10.14</h3> <div id="Equ138" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$f_{n}=\displaystyle \frac{1}{2\pi }\sqrt{\frac{Mgd}{I}}=\frac{1}{2\pi }\sqrt{\frac{Mg({L}/{2})}{(1/3){ML^{2}}}}=\frac{1}{2\pi }\sqrt{\frac{3g}{2L}}=\frac{1}{2(3.14)}\sqrt{\frac{3(9.8\mathrm {m}/\mathrm {s}^{2})}{2(0.6\mathrm {m})}}=0.8\,\text {Hz} $$</span></div></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-17" data-title="Fig. 10.17"><figure><figcaption><b id="Fig17" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.17</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/17" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig17_HTML.png?as=webp"><img aria-describedby="Fig17" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig17_HTML.png" alt="figure 17" loading="lazy" width="685" height="552"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-17-desc"><p>A uniform rod suspended at one end oscillated with a small amplitude</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/17" data-track-dest="link:Figure17 Full size image" aria-label="Full size image figure 17" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-18" data-title="Fig. 10.18"><figure><figcaption><b id="Fig18" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.18</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/18" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig18_HTML.png?as=webp"><img aria-describedby="Fig18" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig18_HTML.png" alt="figure 18" loading="lazy" width="592" height="513"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-18-desc"><p>A uniform square plate pivoted at one of its corners and oscillates in a vertical plane</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/18" data-track-dest="link:Figure18 Full size image" aria-label="Full size image figure 18" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="FPar29">Example 10.15</h3> <p>A uniform square plate of length <i>a</i> is pivoted at one of its corners and oscillates in a vertical plane as in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig18">10.18</a>. Find the period of motion if the amplitude is small.</p> <h3 class="c-article__sub-heading" id="FPar30">Solution 10.15</h3> <p>The moment of inertia of a uniform rectangular plate about its center of mass is</p><div id="Equ139" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ I_{cm}=\frac{1}{12}M(a^{2}+b^{2}) $$</span></div></div><p>Thus for a uniform square plate, we have</p><div id="Equ140" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ I_{cm}=\frac{1}{6}Ma^{2} $$</span></div></div><p>From the parallel axis theorem, the moment of inertia of the plate about an axis that is parallel to the center of mass axis and passing through one corner <span class="mathjax-tex">$(D=\sqrt{2}a)$</span> is</p><div id="Equ141" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ I=I_{cm}+MD^{2}=\frac{1}{6}\ Ma^{2}+2Ma^{2}=\frac{13}{6}Ma^{2} $$</span></div></div><p>and hence</p><div id="Equ142" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ T=2\pi \sqrt{\frac{I}{Mgd}}=2\pi \sqrt{\frac{(13/6){Ma^{2}}}{Mg\sqrt{2}a}}=2\pi \sqrt{1.5\frac{a}{g}} $$</span></div></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-19" data-title="Fig. 10.19"><figure><figcaption><b id="Fig19" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.19</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/19" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig19_HTML.png?as=webp"><img aria-describedby="Fig19" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig19_HTML.png" alt="figure 19" loading="lazy" width="471" height="542"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-19-desc"><p>The torsional pendulum</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/19" data-track-dest="link:Figure19 Full size image" aria-label="Full size image figure 19" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="Sec14"><span class="c-article-section__title-number">10.3.6 </span>The Torsional Pendulum</h3><p>The torsional pendulum consists of a rigid body suspended by a wire from its center of mass where the other end of the wire is fixed as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig19">10.19</a>. The body is in equilibrium if the wire is untwisted. If the body is rotated through an angle <span class="mathjax-tex">$\theta $</span> it will oscillate about its equilibrium position (the line OP) due to a restoring torque exerted by the twisted wire on the body. This torque is found to be directly proportional to the angular displacement of the body. That is</p><div id="Equ143" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \tau =-k\theta $$</span></div></div><p>where <i>k</i> is called the torsional constant. Its value depends on the property of the wire. Note that this equation is the rotational analogue of Hook’s law in linear form <span class="mathjax-tex">$(F=-kx)$</span>. From Newton’s second law, we have</p><div id="Equ144" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \tau =I\alpha $$</span></div></div><p>or</p><div id="Equ145" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ -k\theta =I\ddot{\theta } $$</span></div></div><p>That gives</p><div id="Equ146" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \ddot{\theta }+\bigg (\frac{k}{I}\bigg )\theta =0 $$</span></div></div><p>or</p><div id="Equ147" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \ddot{\theta }+\omega _{n}^{2}\theta =0 $$</span></div></div><p>where <span class="mathjax-tex">$\omega _{n}=\sqrt{k/I}$</span> and the period is <span class="mathjax-tex">$T=2\pi \sqrt{I/k}.$</span></p> <h3 class="c-article__sub-heading" id="FPar31">Example 10.16</h3> <p>A uniform solid sphere of mass of 4.7 kg and radius of 5 cm is suspended at its midpoint by a light string (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig20">10.20</a>) where it oscillates as a torsional pendulum. If the period of motion is 3.5 <span class="mathjax-tex">$\mathrm {s}$</span>, find the torsion constant.</p> <h3 class="c-article__sub-heading" id="FPar32">Solution 10.16</h3> <div id="Equ148" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ T=2\pi \sqrt{\frac{I}{k}} $$</span></div></div><p>for a uniform solid sphere</p><div id="Equ149" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ I_{cm}=\displaystyle \frac{2}{5}MR^{2}=\frac{2}{5}(4.7 \; \mathrm {k}\mathrm {g})(0.05 \; \mathrm {m})^{2}=4.7\times 10^{-3} \; \mathrm {k}\mathrm {g}\,\mathrm {m}^{2} $$</span></div></div><p>hence,</p><div id="Equ150" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ k=\frac{4\pi ^{2}I_{cm}}{T}=\frac{4(3.14)^{2}(4.7\times 10^{-3} \; \mathrm {k}\mathrm {g}\,\mathrm {m}^{2})}{(3.5 \; \mathrm {s})}=0.05 \; \mathrm {k}\mathrm {g}\,\mathrm {m}^{2}/\mathrm {s}^{2} $$</span></div></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-20" data-title="Fig. 10.20"><figure><figcaption><b id="Fig20" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.20</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/20" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig20_HTML.png?as=webp"><img aria-describedby="Fig20" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig20_HTML.png" alt="figure 20" loading="lazy" width="475" height="679"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-20-desc"><p>A uniform solid sphere suspended at its midpoint by a light string</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/20" data-track-dest="link:Figure20 Full size image" aria-label="Full size image figure 20" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> </div></div></section><section data-title="Damped Free Vibrations"><div class="c-article-section" id="Sec15-section"><h2 id="Sec15" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number">10.4 </span>Damped Free Vibrations</h2><div class="c-article-section__content" id="Sec15-content"><p>In this section, we will discuss the case in which the effect of damping that is due to a nonconservative force cannot be neglected. An example of such a force in mechanical systems is the force of friction. In this case, the mechanical energy of the system will be lost, the amplitude of motion will decrease to zero, and the oscillation dies out eventually. Here, we will discuss damping due to friction in the simplest case, where the frictional force is proportional to the first power of the velocity of the oscillating body. An example of such a frictional force is the force that an object experience when moving in a fluid with a low speed and is given by</p><div id="Equ151" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ F_{D}=-bv $$</span></div></div><p>where <i>b</i> is a positive constant called the damping coefficient. Its SI units is <span class="mathjax-tex">$\mathrm {N}(\mathrm {m}\,\mathrm {s}^{-1})=\mathrm {k}\mathrm {g}\,\mathrm {s}^{-1}$</span>. The negative sign shows that the direction of the force is always opposite to the velocity. Now consider the spring–mass system as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig21">10.21</a>, the cylinder shown in the figure contains a viscous fluid and a piston moving in it. Such device is known as the viscous damper. The net force on the oscillating body is</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-21" data-title="Fig. 10.21"><figure><figcaption><b id="Fig21" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.21</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/21" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig21_HTML.png?as=webp"><img aria-describedby="Fig21" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig21_HTML.png" alt="figure 21" loading="lazy" width="685" height="260"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-21-desc"><p>A mass-spring system with damping</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/21" data-track-dest="link:Figure21 Full size image" aria-label="Full size image figure 21" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div id="Equ152" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \sum F=F_{s}+F_{D}=-kx-bv $$</span></div></div><p>hence</p><div id="Equ153" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ m\ddot{x}+b\dot{x}+kx=0 $$</span></div></div><p>or</p><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \ddot{x}+\gamma \dot{x}+\omega _{n}^{2}x=0 \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.13) </div></div><p>where <span class="mathjax-tex">$\gamma =b/m$</span> and <span class="mathjax-tex">$\omega _{n}=\sqrt{k/m}$</span>. The units of <span class="mathjax-tex">$\gamma $</span> is <span class="mathjax-tex">$\mathrm {s}^{-1}$</span>. This equation is a second order linear differential equation of constant coefficients. We may assume a solution of the form</p><div id="Equ154" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=Ce^{\lambda t} $$</span></div></div><p>Substituting this solution into Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ13">10.13</a> gives the characteristic (auxiliary) equation given by</p><div id="Equ155" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \lambda ^{2}+\gamma \lambda +\omega _{n}^{2}=0 $$</span></div></div><p>The roots of this equation are given by</p><div id="Equ156" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \lambda _{1}=-\frac{\gamma }{2}+\sqrt{\bigg (\frac{\gamma ^{2}}{4}-\omega _{n}^{2}\bigg )} $$</span></div></div><p>and</p><div id="Equ157" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \lambda _{2}=-\frac{\gamma }{2}-\sqrt{\bigg (\frac{\gamma ^{2}}{4}-\omega _{n}^{2}\bigg )} $$</span></div></div><p>From superposition, the general solution is given by</p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} x=C_{1}e^{\lambda _{1}t}+C_{2}e^{\lambda _{2}t} \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.14) </div></div><p>Three possible solutions arise depending on whether the sign of the bracket <span class="mathjax-tex">$(\gamma ^{2}/4-\omega _{n}^{2})$</span> is positive, negative or zero, i.e., depending on the size of the damping force. The roots <span class="mathjax-tex">$\lambda _{1}$</span> and <span class="mathjax-tex">$\lambda _{2}$</span> are either distinct real roots, equal real roots or a conjugate complex roots. Therefore, there are three possible motions of the system.</p><h3 class="c-article__sub-heading" id="Sec16"><span class="c-article-section__title-number">10.4.1 </span>Light Damping (Under-Damped) <span class="mathjax-tex">$(\gamma <2\omega _{n})$</span></h3><p>If <span class="mathjax-tex">$\gamma <2\omega _{n}$</span> the resulting roots are complex roots given by</p><div id="Equ158" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \lambda _{1}=-\frac{\gamma }{2}+i\omega _{D} $$</span></div></div><p>and</p><div id="Equ159" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \lambda _{2}=-\frac{\gamma }{2}-i\omega _{D} $$</span></div></div><p>where</p><div id="Equ160" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \omega _{D}=\bigg (\omega _{n}^{2}-\frac{\gamma ^{2}}{4}\bigg )^{1_{/2}} $$</span></div></div><p>Hence, Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ14">10.14</a> may be written as</p><div id="Equ161" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=\bigg [C_{1}e^{i\omega _{D}t}+C_{2}e^{-i\omega _{D}t}\bigg ]e^{\frac{-\gamma }{2}t} $$</span></div></div><p>Since <span class="mathjax-tex">$e^{\pm ix}=\cos x\pm i\sin x$</span> we have</p><div id="Equ162" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=[C_{1}(\cos \omega _{D}t+i\sin \omega _{D}t)+C_{2}(\cos \omega _{D}t-i\sin \omega _{D}t)]e^{\frac{-\gamma }{2}t} $$</span></div></div> <div id="Equ163" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ =[(C_{1}+C_{2})\cos \omega _{D}t+i(C_{1}-C_{2})\sin \omega _{D}t]e^{\frac{-\gamma }{2}t} $$</span></div></div> <div id="Equ15" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} =[A_{1}\cos \omega _{D}t+A_{2}\sin \omega _{D}t]e^{\frac{-\gamma }{2}t} \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.15) </div></div><p>where <span class="mathjax-tex">$A_{1}=C_{1}+C_{2}$</span> and <span class="mathjax-tex">$A_{2}=i(C_{1}-C_{2})$</span>. As mentioned earlier Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ15">10.15</a> can be written as</p><div id="Equ16" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} x=A\cos (\omega _{D}t-\phi )e^{\frac{-\gamma }{2}t} \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.16) </div></div><p>where <i>A</i> is the initial amplitude of motion. <span class="mathjax-tex">$Ae^{\frac{-\gamma }{2}t}$</span> is called the amplitude of motion and <span class="mathjax-tex">$\phi $</span> is the phase constant and <span class="mathjax-tex">$\omega _{D}$</span> is the angular frequency of the damped motion. This equation shows that the system oscillates in a decreasing harmonic motion where the amplitude of motion decreases exponentially with time until eventually the oscillation dies out (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig22">10.22</a>). The dashed lines in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig22">10.22</a> are called the envelope of the oscillation curve. The period of motion in light damping is therefore given by</p><div id="Equ164" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \tau _{D}=\frac{2\pi }{\omega _{D}}=\frac{2\pi }{\sqrt{\omega _{n}^{2}-\frac{\gamma ^{2}}{4}}} $$</span></div></div><p>If <span class="mathjax-tex">$b=0$</span> and thus <span class="mathjax-tex">$\gamma =0$</span> the period of motion is reduced to that of a simple harmonic oscillator. If <span class="mathjax-tex">$\gamma \ll \omega _{D}$</span>, the situation is referred to as very light damping and <span class="mathjax-tex">$\omega _{D}\approx \omega _{n}$</span>. Furthermore if there are two amplitudes <span class="mathjax-tex">$A_{a}$</span> and <span class="mathjax-tex">$A_{b}$</span> separated by the period of motion, then their ratio is given by</p><div id="Equ165" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{A_{a}}{A_{b}}=\frac{Ae^{-\frac{\gamma }{2}t_{1}}}{Ae^{-\frac{\gamma }{2}(t_{1}+\tau _{D})}}=e^{\frac{\gamma }{2}\tau _{D}} $$</span></div></div><p>A quantity known as the logarithmic decrement is defined as</p><div id="Equ166" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \delta =\ln \bigg (\frac{A_{a}}{A_{b}}\bigg )=\frac{\gamma }{2}\tau _{D} $$</span></div></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-22" data-title="Fig. 10.22"><figure><figcaption><b id="Fig22" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.22</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/22" rel="nofollow"><picture><img aria-describedby="Fig22" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig22_HTML.png" alt="figure 22" loading="lazy" width="685" height="470"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-22-desc"><p>In A lightly damped oscillator, the system oscillates in a decreasing harmonic motion where the amplitude of motion decreases exponentially with time until eventually the oscillation dies out</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/22" data-track-dest="link:Figure22 Full size image" aria-label="Full size image figure 22" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="FPar33">Example 10.17</h3> <p>An 8 kg block is attached to a light spring and a light viscous damper. If at <span class="mathjax-tex">$t=0, x=0.12 \; \mathrm {m}$</span> and <span class="mathjax-tex">$v=0$</span>, find (a) the displacement at any time; (b) the logarithmic decrement. <span class="mathjax-tex">$(k=30 \; \mathrm {N}/\mathrm {m},\ b=20 \; \mathrm {N}\,\mathrm {s}/\mathrm {m})$</span>.</p> <h3 class="c-article__sub-heading" id="FPar34">Solution 10.17</h3> <p>(a)</p><div id="Equ167" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \omega _{n}=\sqrt{\frac{k}{m}}=\sqrt{\frac{(30 \; \mathrm {N}/\mathrm {m})}{(8 \; \mathrm {k}\mathrm {g})}}=1.9 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s} $$</span></div></div> <div id="Equ168" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \gamma =\frac{b}{m}=\frac{(20 \; \mathrm {N}\,\mathrm {s}/\mathrm {m})}{(8 \; \mathrm {k}\mathrm {g})}=2.5 \; \mathrm {s}^{-1} $$</span></div></div><p>and</p><div id="Equ169" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \displaystyle \omega _{D}=\bigg (\omega _{n}^{2}-\frac{\gamma ^{2}}{4}\bigg )^{1_{/2}}=((1.9 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s})^{2}-(2.5 \; \mathrm {N} \mathrm {s}/\mathrm {m}\,\mathrm {k}\mathrm {g})^{2}4)^{1_{/2}}=1.43 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s} $$</span></div></div><p>since <span class="mathjax-tex">$\gamma <2\omega _{\mathrm {n}}$</span>, the damping is light. The displacement as a function of time is given by</p><div id="Equ170" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=A\cos (\omega _{D}t-\phi )e^{\frac{-\gamma }{2}t} $$</span></div></div><p>or</p><div id="Equ171" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=A\cos (1.43t-\phi )e^{-1.25t} $$</span></div></div><p>since at <span class="mathjax-tex">$t=0, x=0.12 \; \mathrm {m}$</span>, then</p><div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} (0.12 \; \mathrm {m})=A\cos \phi \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.17) </div></div><p>the velocity of the block at any time is</p><div id="Equ172" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \dot{x}=-1.43A\sin (1.43t-\phi )e^{-1.25t}-1.25A\cos (1.43t-\phi )e^{-1.25t} $$</span></div></div><p>at <span class="mathjax-tex">$t=0, v=0$</span> and thus</p><div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} 0=-1.43A\sin \phi -1.25A\cos \phi \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.18) </div></div><p>Solving Eqs. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ17">10.17</a> and <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ18">10.18</a> for <i>A</i> and <span class="mathjax-tex">$\phi $</span> gives <span class="mathjax-tex">$\phi =-0.7$</span> rad and <span class="mathjax-tex">$A=0.17 \; \mathrm {m}.$</span> Therefore,</p><div id="Equ173" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=0.17\cos (1.43t-0.7)e^{-1.25t} $$</span></div></div><p>(b)</p><div id="Equ174" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \tau _{D}=\frac{2\pi }{\omega _{D}}=\frac{2\pi }{(1.43 \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s})}=4.4 \; \mathrm {s} $$</span></div></div> <div id="Equ175" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \delta =\frac{\gamma }{2}\tau _{D}=(1.25 \; \mathrm {s}^{-1})(4.4 \; \mathrm {s})=5.5 $$</span></div></div> <h3 class="c-article__sub-heading" id="Sec17"><span class="c-article-section__title-number">10.4.2 </span>Critically Damped Motion <span class="mathjax-tex">$(\gamma =2\omega _{n})$</span></h3><p>If <span class="mathjax-tex">$\gamma =2\omega _{n}$</span>, then the roots are equal real roots</p><div id="Equ176" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \lambda _{1}=\lambda _{2}=-\frac{\gamma }{2}=-\omega _{n} $$</span></div></div><p>In that case, the motion decays without oscillation (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig23">10.23</a>) and the general solution of Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ13">10.13</a> is</p><div id="Equ177" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=(C_{1}+C_{2}\omega _{n}t)e^{-\omega _{n}t} $$</span></div></div><p><span class="mathjax-tex">$C_{1}$</span> and <span class="mathjax-tex">$C_{2}$</span> are found from boundary conditions. If at <span class="mathjax-tex">$t=0, x=A$</span>, and <span class="mathjax-tex">$v=0,$</span> then</p><div id="Equ178" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x(0)=C_{1}=A $$</span></div></div><p>and</p><div id="Equ179" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ v(0)=\omega _{n}C_{2}-\omega _{n}C_{1}=0 $$</span></div></div><p>or</p><div id="Equ180" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ C_{1}=C_{2}=A $$</span></div></div><p>That gives</p><div id="Equ181" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=A(1+\omega _{n}t)e^{-\omega _{n}t} $$</span></div></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-23" data-title="Fig. 10.23"><figure><figcaption><b id="Fig23" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.23</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/23" rel="nofollow"><picture><img aria-describedby="Fig23" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig23_HTML.png" alt="figure 23" loading="lazy" width="685" height="356"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-23-desc"><p>In a critically damped motion, the motion decays without oscillation</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/23" data-track-dest="link:Figure23 Full size image" aria-label="Full size image figure 23" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="Sec18"><span class="c-article-section__title-number">10.4.3 </span>Over Damped Motion (Heavy Damping) <span class="mathjax-tex">$(\gamma >2\omega _{n})$</span></h3><p>If <span class="mathjax-tex">$\gamma >2\omega _{n}$</span>, the roots are distinct real roots given by</p><div id="Equ182" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \lambda _{1}=-\frac{\gamma }{2}+\sqrt{\bigg (\frac{\gamma ^{2}}{4}-\omega _{n}^{2}\bigg )} $$</span></div></div><p>and</p><div id="Equ183" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \lambda _{2}=-\frac{\gamma }{2}-\sqrt{\bigg (\frac{\gamma ^{2}}{4}-\omega _{n}^{2}\bigg )} $$</span></div></div><p>The general solution is given by</p><div id="Equ184" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=C_{1}e^{\lambda _{1}t}+C_{2}e^{\lambda _{2}t} $$</span></div></div><p>or</p><div id="Equ185" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=(C_{1}e^{\alpha t}+C_{2}e^{-\alpha t})e^{-\frac{\gamma }{2}t} $$</span></div></div><p>where</p><div id="Equ186" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \alpha =\sqrt{\bigg (\frac{\gamma ^{2}}{4}-\omega _{n}^{2}\bigg )} $$</span></div></div><p><span class="mathjax-tex">$C_{1}$</span> and <span class="mathjax-tex">$C_{2}$</span> are found from boundary conditions. As critical damping, the resulting motion here is nonperiodic but the system returns to its equilibrium position at large values of <i>t</i> unlike critical damping (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig24">10.24</a>).</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-24" data-title="Fig. 10.24"><figure><figcaption><b id="Fig24" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.24</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/24" rel="nofollow"><picture><img aria-describedby="Fig24" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig24_HTML.png" alt="figure 24" loading="lazy" width="685" height="356"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-24-desc"><p>As critical damping, the resulting motion here is non-periodic but the system returns to its equilibrium position at large values of <i>t</i> unlike critical damping</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/24" data-track-dest="link:Figure24 Full size image" aria-label="Full size image figure 24" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="FPar35">Example 10.18</h3> <p>In Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar33">10.17</a>, find the range of values of the damping coefficient for the system to be: (a) over damped; (b) critically damped.</p> <h3 class="c-article__sub-heading" id="FPar36">Solution 10.18</h3> <p>(a) over damped if <span class="mathjax-tex">$\gamma >2\omega _{n}$</span>, i.e., if <span class="mathjax-tex">$\gamma >3.8\mathrm {s}^{-1}(\mathrm {b})$</span> critically damped if <span class="mathjax-tex">$\gamma =3.8\mathrm {s}^{-1}.$</span></p> <h3 class="c-article__sub-heading" id="Sec19"><span class="c-article-section__title-number">10.4.4 </span>Energy Decay</h3><p>In damped free vibrations, the total mechanical energy is not constant since the damping force opposes the motion and dissipates the energy of the system. Now, consider the mass–spring system, the total mechanical energy of the system is</p><div id="Equ187" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ E=K+U=\frac{1}{2}m\dot{x}^{2}+\frac{1}{2}kx^{2} $$</span></div></div><p>The rate of change of energy is</p><div id="Equ188" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{dE}{dt}=(m\ddot{x}+kx)\dot{x} $$</span></div></div><p>For damped vibrations in which the damping force is directly proportional to the velocity, we have</p><div id="Equ189" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ m\ddot{x}+kx=-b\dot{x} $$</span></div></div><p>Hence,</p><div id="Equ190" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \frac{dE}{dt}=-b\dot{x}^{2}\le 0 $$</span></div></div><p>Thus, the energy decreases with time in any damped motion and the rate in which it decreases is not uniform.</p></div></div></section><section data-title="Forced Vibrations"><div class="c-article-section" id="Sec20-section"><h2 id="Sec20" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number">10.5 </span>Forced Vibrations</h2><div class="c-article-section__content" id="Sec20-content"><p>In the previous sections, only free vibrations have been considered (i.e., vibrations in which only a restoring and damping force act within the system during motion). This section considers the case in which an external driving force is applied to the vibrator. This force is given as a function of time and we have</p><div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} m\ddot{x}+b\dot{x}+kx=F(t) \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.19) </div></div><p>Here, we will consider the case in which the force is a simple periodic force given by</p><div id="Equ20" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} F(t)=F_{0}\cos \omega t \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.20) </div></div><p>where <span class="mathjax-tex">$F_{0}$</span> is the amplitude and <span class="mathjax-tex">$\omega $</span> is the driving frequency. This force does positive work on the system to balance the energy loss due to damping. Substituting Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ20">10.20</a> into Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ19">10.19</a> gives</p><div id="Equ21" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} m\ddot{x}+b\dot{x}+kx=F_{0}\cos \omega t \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.21) </div></div><p>or</p><div id="Equ191" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \ddot{x}+\gamma \dot{x}+\omega _{n}^{2}x=\frac{F_{0}\cos \omega t}{m} $$</span></div></div><p>Let us assume that the solution of Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ19">10.19</a> is given by</p><div id="Equ192" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=C_{1}\cos \omega t+C_{2}\sin \omega t $$</span></div></div><p>then, we have</p><div id="Equ193" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \dot{x}=-\omega C_{1}\sin \omega t+\omega C_{2}\cos \omega t $$</span></div></div><p>and</p><div id="Equ194" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \ddot{x}=-\omega ^{2}C_{1}\cos \omega t-\omega ^{2}C_{2}\sin \omega t $$</span></div></div><p>Substituting into Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ19">10.19</a> gives</p><div id="Equ195" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&(-\displaystyle \omega ^{2}C_{1}\cos \omega t-\omega ^{2}C_{2}\sin \omega t)+\gamma (-\omega C_{1}\sin \omega t+\omega C_{2}\cos \omega t) \nonumber \\&+\omega _{n}^{2}(C_{1}\cos \omega t+C_{2}\sin \omega t)=\frac{F_{0}\cos \omega t}{m} \end{aligned}$$</span></div></div><p>That gives</p><div id="Equ196" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ -\omega ^{2}C_{1}+\gamma \omega C_{2}+\omega _{n}^{2}C_{1}=\frac{F_{0}}{m} $$</span></div></div><p>and</p><div id="Equ197" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ -\omega ^{2}C_{2}-\gamma \omega C_{1}+\omega _{n}^{2}C_{2}=0 $$</span></div></div><p>Solving for <span class="mathjax-tex">$C_{1}$</span> and <span class="mathjax-tex">$C_{2}$</span> gives</p><div id="Equ198" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ C_{1}=\frac{({F_{0}}/{m})(\omega _{n}^{2}-\omega ^{2})}{(\omega ^{2}-\omega _{n}^{2})^{2}+\gamma ^{2}\omega ^{2}} $$</span></div></div><p>and</p><div id="Equ199" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ C_{2}=\frac{({F_{0}}/{m})\gamma \omega }{(\omega ^{2}-\omega _{n}^{2})^{2}+\gamma ^{2}\omega ^{2}} $$</span></div></div><p>Hence,</p><div id="Equ200" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=\frac{({F_{0}}/{m})[(\omega _{n}^{2}-\omega ^{2})\cos \omega t+\gamma \omega \sin \omega t]}{(\omega ^{2}-\omega _{n}^{2})^{2}+\gamma ^{2}\omega ^{2}} $$</span></div></div><p>The term in brackets is of the form <span class="mathjax-tex">$A_{1}\cos \omega t+A_{2}\sin \omega t$</span> and thus it can be written as <span class="mathjax-tex">$A'\cos (\omega t-\phi )$</span> where</p><div id="Equ201" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ A'=\sqrt{A_{1}^{2}+A_{2}^{2}} $$</span></div></div><p>i.e.,</p><div id="Equ202" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ A'=((\omega _{n}^{2}-\omega ^{2})^{2}+\gamma ^{2}\omega ^{2})^{\frac{1}{2}} $$</span></div></div><p>and</p><div id="Equ203" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \phi =\tan ^{-1}\frac{A_{2}}{A_{1}}=\tan ^{-1}\frac{\gamma \omega }{(\omega ^{2}-\omega _{n}^{2})} $$</span></div></div><p>where <span class="mathjax-tex">$ 0\le \phi \le \pi $</span>. Hence,</p><div id="Equ22" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} x=\displaystyle \frac{(^{F_{0}}/_{m})}{\sqrt{(\omega ^{2}-\omega _{n}^{2})^{2}+\gamma ^{2}\omega ^{2}}}\cos (\omega t-\phi ) \end{aligned}$$</span></div><div class="c-article-equation__number"> (10.22) </div></div><p>If the driving force is applied for a long time compared with the time that the damped vibration dies out, then the system will eventually vibrate at the same frequency of the deriving force. Therefore, the general solution of Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ13">10.13</a> is called the transient solution since it approaches zero in a relativity short time whereas Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="#Equ21">10.21</a> is called the steady-state solution where the system oscillates with the same frequency as the deriving force. Therefore, the amplitude of a steady-state vibration is</p><div id="Equ204" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ A=\frac{({F_{0}}/_{m})}{\sqrt{(\omega ^{2}-\omega _{n}^{2})^{2}+\gamma ^{2}\omega ^{2}}} $$</span></div></div><p>When the deriving frequency <span class="mathjax-tex">$\omega $</span> approaches the natural frequency of the system <span class="mathjax-tex">$\omega _{D}$</span>, the amplitude of the resulting forced oscillation will increase. This is known as resonance. If the damping is very light, the amplitude reaches its peak when the deriving frequency is nearly equal to the natural frequency <span class="mathjax-tex">$\omega _{n}$</span>. As the damping becomes heavier, the maximum amplitude shifts to lower frequencies (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig25">10.25</a>). In the case where there is no damping at all <span class="mathjax-tex">$(b=0)$</span>, the amplitude of resonance is infinite at <span class="mathjax-tex">$\omega =\omega _{n}.$</span></p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-25" data-title="Fig. 10.25"><figure><figcaption><b id="Fig25" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.25</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/25" rel="nofollow"><picture><img aria-describedby="Fig25" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig25_HTML.png" alt="figure 25" loading="lazy" width="685" height="576"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-25-desc"><p>When the deriving frequency <span class="mathjax-tex">$\omega $</span> approaches the natural frequency of the system <span class="mathjax-tex">$\omega _{D}$</span>, the amplitude of the resulting forced oscillation will increase. This is known as resonance. If the damping is very light the amplitude reaches its peak when the deriving frequency is nearly equal to the natural frequency <span class="mathjax-tex">$\omega _{n}$</span>. As the damping becomes heavier, the maximum amplitude shifts to lower frequencies</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/25" data-track-dest="link:Figure25 Full size image" aria-label="Full size image figure 25" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="FPar37">Example 10.19</h3> <p>In Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar33">10.17</a>, if a driving force of the form <span class="mathjax-tex">$F(t)=5\cos 4t$</span> is applied to the system, find the steady-state displacement as a function of time.</p> <h3 class="c-article__sub-heading" id="FPar38">Solution 10.19</h3> <div id="Equ205" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ A=\frac{({F_{0}}/_{m})}{\sqrt{(\omega ^{2}-\omega _{n}^{2})^{2}+\gamma ^{2}\omega ^{2}}}=\frac{({5}/8)}{\sqrt{((4)^{2}-(1.9)^{2})^{2}+(2.5)^{2}(4)^{2}}}=0.04 \; \mathrm {m} $$</span></div></div> <div id="Equ206" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \phi =\tan ^{-1}\frac{\gamma \omega }{(\omega ^{2}-\omega _{n}^{2})}=\tan ^{-1}\ \frac{(2.5)(4)}{((4)^{2}-(1.9)^{2})}=0.8^{\circ } $$</span></div></div><p>Hence,</p><div id="Equ207" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ x=0.04\cos (4t-0.8) $$</span></div></div><p>Therefore, the forced vibration has the same frequency as the deriving force but lag in phase by <span class="mathjax-tex">$0.8^{\circ }$</span></p> <h3 class="c-article__sub-heading" id="FPar39">Example 10.20</h3> <p>In Example (10.17), find the steady-state displacement as a function of time if there is no damping.</p> <h3 class="c-article__sub-heading" id="FPar40">Solution 10.20</h3> <p>The amplitude of the forced oscillation when the angular frequency <span class="mathjax-tex">$\omega $</span> of the deriving force is varied.</p><div id="Equ208" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ A=\frac{({F_{0}}/_{m})}{\sqrt{(\omega ^{2}-\omega _{n}^{2})^{2}+\gamma ^{2}\omega ^{2}}}=\frac{({5}/{8})}{\sqrt{((4)^{2}-(1.9)^{2})^{2}}}=0.05 \; \mathrm {m} $$</span></div></div><p><span class="mathjax-tex">$x=0.05\cos 4t,\; \phi =0.$</span></p> <p> <b>Problems</b> </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">1.</span> <p>A 2 kg block is fastened to a spring of force constant 98 <span class="mathjax-tex">$\mathrm {N}/\mathrm {m}$</span> on a horizontal frictionless surface. If the block is released a distance of 6 cm from its equilibrium position, find (a) the angular frequency, the frequency and the period of the resulting motion, (b) the time it takes the block to first reach <span class="mathjax-tex">$x=-5$</span> cm and its velocity at that time, (c) the maximum speed and maximum acceleration of the oscillating block, (d) the total mechanical energy of the oscillator.</p> </li> <li> <span class="u-custom-list-number">2.</span> <p>A 10 kg block is attached to a light spring of force constant 200 <span class="mathjax-tex">$\mathrm {N}/\mathrm {m}$</span> on a smooth horizontal surface. Find the amplitude of motion if at <span class="mathjax-tex">$x=0.06 \; \mathrm {m}$</span> the velocity of the block is <span class="mathjax-tex">$v=0.5 \; \mathrm {m}/\mathrm {s}.$</span></p> </li> <li> <span class="u-custom-list-number">3.</span> <p>A particle rotate counterclockwise in a circle of radius 0.2 <span class="mathjax-tex">$\mathrm {m}$</span> with a constant angular speed of 2 <span class="mathjax-tex">$\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}$</span>. If at <span class="mathjax-tex">$t=0$</span> the <span class="mathjax-tex">$\mathrm {x}$</span>-coordinate of the particle is 0.14 <span class="mathjax-tex">$\mathrm {m}$</span>, find the displacement, velocity and acceleration of the particle at any time.</p> </li> <li> <span class="u-custom-list-number">4.</span> <p>If a simple pendulum has a period of 2 <span class="mathjax-tex">$\mathrm {s}$</span>, find its period when its length is increased by <span class="mathjax-tex">$20\%$</span>.</p> </li> <li> <span class="u-custom-list-number">5.</span> <p>A simple pendulum of length lm and mass of 0.4 kg oscillates in a region where <span class="mathjax-tex">$g=9.8 \; \mathrm {m}/\mathrm {s}^{2}$</span>. If the amplitude of oscillation is <span class="mathjax-tex">$10^{\circ }$</span>, find (a) the angular displacement, angular velocity and angular acceleration of the pendulum as a function of time.</p> </li> <li> <span class="u-custom-list-number">6.</span> <p>A uniform solid cylinder of radius <i>R</i> and mass <i>M</i> rolls without slipping on a track of radius 4<i>R</i> as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig26">10.26</a>. Find the period of oscillation when the cylinder is displaced slightly from its equilibrium position.</p> </li> <li> <span class="u-custom-list-number">7.</span> <p>A planer body of mass 3 kg oscillates as a physical pendulum. If the period of oscillation is 3 <span class="mathjax-tex">$\mathrm {s}$</span> and if the pivot point is at 0.2 <span class="mathjax-tex">$\mathrm {m}$</span> from the center of mass, find the moment of inertia of the body.</p> </li> <li> <span class="u-custom-list-number">8.</span> <p>A uniform hollow cylinder of radius <i>R</i> and mass <i>M</i> is suspended at its midpoint from a wire and form a torsional pendulum. If the period of motion is <i>T</i>, find the torsion constant.</p> </li> <li> <span class="u-custom-list-number">9.</span> <p>For the system shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig27">10.27</a>, determine the displacement of the block at any time if at <span class="mathjax-tex">$t=0, x=0$</span> and <span class="mathjax-tex">$v=0.\,(k=200 \; \mathrm {N}/\mathrm {m},\ b=200 \; \mathrm {N}\,\mathrm {s}/\mathrm {m})$</span>.</p> </li> <li> <span class="u-custom-list-number">10.</span> <p>For the system shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig28">10.28</a>, find the steady-state displacement as a function of time.</p> </li> </ol> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-26" data-title="Fig. 10.26"><figure><figcaption><b id="Fig26" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.26</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/26" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig26_HTML.png?as=webp"><img aria-describedby="Fig26" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig26_HTML.png" alt="figure 26" loading="lazy" width="358" height="621"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-26-desc"><p>A uniform solid cylinder of radius <i>R</i> and mass <i>M</i> rolls without slipping on a track of radius 4<i>R</i></p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/26" data-track-dest="link:Figure26 Full size image" aria-label="Full size image figure 26" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-27" data-title="Fig. 10.27"><figure><figcaption><b id="Fig27" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.27</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/27" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig27_HTML.png?as=webp"><img aria-describedby="Fig27" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig27_HTML.png" alt="figure 27" loading="lazy" width="685" height="244"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-27-desc"><p>A damped oscillator</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/27" data-track-dest="link:Figure27 Full size image" aria-label="Full size image figure 27" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-28" data-title="Fig. 10.28"><figure><figcaption><b id="Fig28" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10.28</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/28" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig28_HTML.png?as=webp"><img aria-describedby="Fig28" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-030-15195-9_10/MediaObjects/459974_1_En_10_Fig28_HTML.png" alt="figure 28" loading="lazy" width="685" height="206"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-28-desc"><p>A forced oscillator</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-030-15195-9_10/figures/28" data-track-dest="link:Figure28 Full size image" aria-label="Full size image figure 28" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> </div></div></section> </div> <div id="MagazineFulltextChapterBodySuffix"></div><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 id="author-information" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number"> </span>Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff20"><p class="c-article-author-affiliation__address">Thuwal, Saudi Arabia</p><p class="c-article-author-affiliation__authors-list">Salma 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