Hukum gerak Euler - Wikipedia bahasa Indonesia, ensiklopedia bebas

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href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#p-lang" data-mw="interface" data-event-name="menu.languages" title="Bahasa" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet language-selector"> <span class="minerva-icon minerva-icon--language"></span> <span>Bahasa</span> </a></li> <li id="page-actions-watch" class="page-actions-menu__list-item"><a role="button" id="ca-watch" href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Istimewa:Masuk_log&returnto=Hukum+gerak+Euler&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-event-name="menu.watch" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet menu__item--page-actions-watch"> <span class="minerva-icon minerva-icon--star"></span> <span>Pantau</span> </a></li> <li id="page-actions-edit" class="page-actions-menu__list-item"><a role="button" id="ca-edit" href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Hukum_gerak_Euler&action=edit&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-event-name="menu.edit" data-mw="interface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> <span class="minerva-icon minerva-icon--edit"></span> <span>Sunting</span> </a></li> </ul> </nav> <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="id" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <p>Dalam <a href="https://id-m-wikipedia-org.translate.goog/wiki/Mekanika_klasik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mekanika klasik">mekanika klasik</a>, <b>hukum gerak Euler</b> (<a href="https://id-m-wikipedia-org.translate.goog/wiki/Bahasa_Inggris?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bahasa Inggris">Bahasa Inggris</a>: <i>Euler's Laws of Motion</i>) adalah persamaan gerak yang memperluas <a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Newton?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hukum gerak Newton">hukum gerak Newton</a> untuk <a href="https://id-m-wikipedia-org.translate.goog/wiki/Partikel_titik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Partikel titik">partikel titik</a> ke gerak <a href="https://id-m-wikipedia-org.translate.goog/wiki/Benda_tegar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Benda tegar">benda tegar</a>.<sup id="cite_ref-:0_1-0" class="reference"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Hukum ini dirumuskan oleh <a href="https://id-m-wikipedia-org.translate.goog/wiki/Leonhard_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Leonhard Euler">Leonhard Euler</a> sekitar 50 tahun setelah <a href="https://id-m-wikipedia-org.translate.goog/wiki/Isaac_Newton?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Isaac Newton">Isaac Newton</a> merumuskan hukum-hukumnya.</p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="id" dir="ltr"> <h2 id="mw-toc-heading">Daftar isi</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Ikhtisar"><span class="tocnumber">1</span> <span class="toctext">Ikhtisar</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Hukum_pertama_Euler"><span class="tocnumber">1.1</span> <span class="toctext">Hukum pertama Euler</span></a></li> <li class="toclevel-2 tocsection-3"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Hukum_kedua_Euler"><span class="tocnumber">1.2</span> <span class="toctext">Hukum kedua Euler</span></a></li> </ul></li> <li class="toclevel-1 tocsection-4"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Penjelasan_dan_turunan"><span class="tocnumber">2</span> <span class="toctext">Penjelasan dan turunan</span></a></li> <li class="toclevel-1 tocsection-5"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Lihat_juga"><span class="tocnumber">3</span> <span class="toctext">Lihat juga</span></a></li> <li class="toclevel-1 tocsection-6"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Referensi"><span class="tocnumber">4</span> <span class="toctext">Referensi</span></a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Ikhtisar">Ikhtisar</h2><span class="mw-editsection"> <a role="button" href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Hukum_gerak_Euler&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sunting bagian: Ikhtisar" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>sunting</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <div class="mw-heading mw-heading3"> <h3 id="Hukum_pertama_Euler">Hukum pertama Euler</h3><span class="mw-editsection"> <a role="button" href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Hukum_gerak_Euler&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sunting bagian: Hukum pertama Euler" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>sunting</span> </a> </span> </div> <p><b>Hukum pertama Euler</b> menyatakan bahwa laju perubahan <a href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Momentum_linier&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Momentum linier (halaman belum tersedia)">momentum linier</a> <b>p</b> dari sebuah benda tegar sama dengan resultan semua gaya eksternal <b>F</b><sub>ext</sub> yang bekerja pada benda tersebut.<sup id="cite_ref-2" class="reference"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></p> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\text{ext}}={\frac {d\mathbf {p} }{dt}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> ext </mtext> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> p </mi> </mrow> </mrow> <mrow> <mi> d </mi> <mi> t </mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F_{\text{ext}}={\frac {d\mathbf {p} }{dt}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d55873992aba0bc5415ad042c2192cc2bc2215" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.6ex; height:5.509ex;" alt="{\displaystyle F_{\text{ext}}={\frac {d\mathbf {p} }{dt}}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.6ex;height: 5.509ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d55873992aba0bc5415ad042c2192cc2bc2215" data-alt="{\displaystyle F_{\text{ext}}={\frac {d\mathbf {p} }{dt}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>Gaya internal antar partikel yang membentuk sebuah benda tidak berkontribusi dalam mengubah momentum benda karena ada gaya yang sama dan berlawanan sehingga tidak ada efek bersih.<sup id="cite_ref-3" class="reference"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></p> <p>Momentum linier benda tegar adalah hasil kali antara massa benda dan kecepatan <a href="https://id-m-wikipedia-org.translate.goog/wiki/Pusat_massa?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Pusat massa">pusat massa</a>nya, <b>v</b><sub>cm</sub>.<sup id="cite_ref-:0_1-1" class="reference"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_4-0" class="reference"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_5-0" class="reference"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></p> <div class="mw-heading mw-heading3"> <h3 id="Hukum_kedua_Euler">Hukum kedua Euler</h3><span class="mw-editsection"> <a role="button" href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Hukum_gerak_Euler&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sunting bagian: Hukum kedua Euler" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>sunting</span> </a> </span> </div> <p><b>Hukum kedua Euler</b> menyatakan bahwa laju perubahan <a href="https://id-m-wikipedia-org.translate.goog/wiki/Momentum_sudut?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Momentum sudut">momentum sudu</a>t <b>L</b> terhadap suatu titik yang ditetapkan dalam <a href="https://id-m-wikipedia-org.translate.goog/wiki/Kerangka_acuan_inersia?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kerangka acuan inersia">kerangka acuan inersia</a> (sering kali merupakan pusat massa benda), sama dengan jumlah momen gaya eksternal (<a href="https://id-m-wikipedia-org.translate.goog/wiki/Torsi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Torsi">torsi</a>) yang bekerja pada benda <b>M</b> terhadap titik tersebut.<sup id="cite_ref-:0_1-2" class="reference"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_4-1" class="reference"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_5-1" class="reference"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></p> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} ={d\mathbf {L} \over dt}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> M </mi> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> L </mi> </mrow> </mrow> <mrow> <mi> d </mi> <mi> t </mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {M} ={d\mathbf {L} \over dt}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f55baaef350eba63d5d4b72f179e6f94bc7000e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.296ex; height:5.509ex;" alt="{\displaystyle \mathbf {M} ={d\mathbf {L} \over dt}}"> </noscript><span class="lazy-image-placeholder" style="width: 9.296ex;height: 5.509ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f55baaef350eba63d5d4b72f179e6f94bc7000e" data-alt="{\displaystyle \mathbf {M} ={d\mathbf {L} \over dt}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>Perhatikan bahwa rumus di atas hanya berlaku jika <b>M</b> dan <b>L</b> dihitung terhadap kerangka inersia tetap atau kerangka yang sejajar dengan kerangka inersia tetapi tetap pada pusat massa. Untuk benda tegar yang bertranslasi dan berotasi hanya dalam dua dimensi, hal ini dapat dinyatakan sebagai berikut.<sup id="cite_ref-6" class="reference"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></p> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} =\mathbf {r} _{\rm {cm}}\times \mathbf {a} _{\rm {cm}}m+I{\boldsymbol {\alpha }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> M </mi> </mrow> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> c </mi> <mi mathvariant="normal"> m </mi> </mrow> </mrow> </msub> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> a </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> c </mi> <mi mathvariant="normal"> m </mi> </mrow> </mrow> </msub> <mi> m </mi> <mo> + </mo> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {M} =\mathbf {r} _{\rm {cm}}\times \mathbf {a} _{\rm {cm}}m+I{\boldsymbol {\alpha }}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9652e226d389e102323ca6dc91828839cb03541b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.361ex; height:2.509ex;" alt="{\displaystyle \mathbf {M} =\mathbf {r} _{\rm {cm}}\times \mathbf {a} _{\rm {cm}}m+I{\boldsymbol {\alpha }}}"> </noscript><span class="lazy-image-placeholder" style="width: 23.361ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9652e226d389e102323ca6dc91828839cb03541b" data-alt="{\displaystyle \mathbf {M} =\mathbf {r} _{\rm {cm}}\times \mathbf {a} _{\rm {cm}}m+I{\boldsymbol {\alpha }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>dimana:</p> <ul> <li><b>r</b><sub>cm</sub> adalah vektor posisi pusat massa benda terhadap titik dimana momen dijumlahkan,</li> <li><b>a</b><sub>cm</sub> adalah percepatan linier pusat massa benda,</li> <li><b>m</b> adalah massa benda,</li> <li><b>α</b> adalah <a href="https://id-m-wikipedia-org.translate.goog/wiki/Percepatan_sudut?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Percepatan sudut">percepatan sudut</a> benda, dan</li> <li><b>I</b> adalah <a href="https://id-m-wikipedia-org.translate.goog/wiki/Momen_inersia?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Momen inersia">momen inersia</a> benda terhadap pusat massanya.</li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Penjelasan_dan_turunan">Penjelasan dan turunan</h2><span class="mw-editsection"> <a role="button" href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Hukum_gerak_Euler&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sunting bagian: Penjelasan dan turunan" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>sunting</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>Distribusi gaya internal dalam benda yang dapat berubah bentuk tidak selalu sama di seluruh bagian, yaitu tekanan bervariasi dari satu titik ke titik berikutnya. Variasi gaya internal di seluruh benda ini diatur oleh <a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Newton?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hukum gerak Newton">hukum gerak kedua Newton</a> tentang kekekalan <a href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Momentum_linier&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Momentum linier (halaman belum tersedia)">momentum linier</a> dan <a href="https://id-m-wikipedia-org.translate.goog/wiki/Momentum_sudut?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Momentum sudut">momentum sudut</a>, dimana penggunaan yang paling sederhana diterapkan pada partikel bermassa, tetapi diperluas dalam <a href="https://id-m-wikipedia-org.translate.goog/wiki/Mekanika_kontinuum?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mekanika kontinuum">mekanika kontinum</a> menjadi benda bermassa yang terdistribusi secara kontinu. Untuk benda kontinu, hukum-hukum ini disebut <b>hukum gerak Euler</b>.<sup id="cite_ref-7" class="reference"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></p> <p>Gaya total benda yang diterapkan pada benda kontinu dengan massa <i>m</i>, <a href="https://id-m-wikipedia-org.translate.goog/wiki/Massa_jenis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Massa jenis">massa jenis</a> <i>ρ</i>, dan volume <i>V</i>, adalah <a href="https://id-m-wikipedia-org.translate.goog/wiki/Integral_volume?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Integral volume">integral volume</a> yang diintegrasikan pada volume benda:</p> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{B}=\int _{V}\mathbf {b} \,dm=\int _{V}\mathbf {b} \rho \,dV}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> B </mi> </mrow> </msub> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> b </mi> </mrow> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> m </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> b </mi> </mrow> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {F} _{B}=\int _{V}\mathbf {b} \,dm=\int _{V}\mathbf {b} \rho \,dV} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c794458953062f48e78af3a901df549ba096189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.917ex; height:5.676ex;" alt="{\displaystyle \mathbf {F} _{B}=\int _{V}\mathbf {b} \,dm=\int _{V}\mathbf {b} \rho \,dV}"> </noscript><span class="lazy-image-placeholder" style="width: 26.917ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c794458953062f48e78af3a901df549ba096189" data-alt="{\displaystyle \mathbf {F} _{B}=\int _{V}\mathbf {b} \,dm=\int _{V}\mathbf {b} \rho \,dV}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>dimana <b>b</b> adalah gaya yang bekerja pada benda per satuan massa (<a href="https://id-m-wikipedia-org.translate.goog/wiki/Dimensi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dimensi">dimensi</a> percepatan, yang secara keliru disebut "gaya benda"), dan <i>dm = ρ dV</i> adalah elemen massa benda yang sangat kecil.</p> <p>Gaya benda dan gaya kontak yang bekerja pada benda menyebabkan momen (<a href="https://id-m-wikipedia-org.translate.goog/wiki/Torsi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Torsi">torsi</a>) yang sesuai dari gaya-gaya tersebut relatif terhadap titik tertentu. Dengan demikian, torsi total yang diterapkan <b>M</b> tentang titik asal berasal dari:</p> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} =\mathbf {M} _{B}+\mathbf {M} _{C}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> M </mi> </mrow> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> M </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> B </mi> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> M </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> C </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {M} =\mathbf {M} _{B}+\mathbf {M} _{C}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8790179d7bcfb6b691fb7e0aa51e11da5715c241" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.512ex; height:2.509ex;" alt="{\displaystyle \mathbf {M} =\mathbf {M} _{B}+\mathbf {M} _{C}}"> </noscript><span class="lazy-image-placeholder" style="width: 16.512ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8790179d7bcfb6b691fb7e0aa51e11da5715c241" data-alt="{\displaystyle \mathbf {M} =\mathbf {M} _{B}+\mathbf {M} _{C}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>dimana <b>M</b><sub><i>B</i></sub> dan <b>M</b><sub><i>C</i></sub> masing-masing menunjukkan momen yang disebabkan oleh benda dan gaya kontak.</p> <p>Dengan demikian, jumlah semua gaya dan torsi yang diterapkan (sehubungan dengan asal sistem koordinat) yang bekerja pada benda dapat diberikan sebagai jumlah volume dan <a href="https://id-m-wikipedia-org.translate.goog/wiki/Integral_permukaan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Integral permukaan">integral permukaan</a>:</p> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =\int _{V}\mathbf {a} \,dm=\int _{V}\mathbf {a} \rho \,dV=\int _{S}\mathbf {t} \,dS+\int _{V}\mathbf {b} \rho \,dV}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> a </mi> </mrow> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> m </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> a </mi> </mrow> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> V </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> S </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> t </mi> </mrow> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> S </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> b </mi> </mrow> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {F} =\int _{V}\mathbf {a} \,dm=\int _{V}\mathbf {a} \rho \,dV=\int _{S}\mathbf {t} \,dS+\int _{V}\mathbf {b} \rho \,dV} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651922868b07be2854bf7cf5ce6308962c7a64cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.371ex; height:5.676ex;" alt="{\displaystyle \mathbf {F} =\int _{V}\mathbf {a} \,dm=\int _{V}\mathbf {a} \rho \,dV=\int _{S}\mathbf {t} \,dS+\int _{V}\mathbf {b} \rho \,dV}"> </noscript><span class="lazy-image-placeholder" style="width: 47.371ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651922868b07be2854bf7cf5ce6308962c7a64cb" data-alt="{\displaystyle \mathbf {F} =\int _{V}\mathbf {a} \,dm=\int _{V}\mathbf {a} \rho \,dV=\int _{S}\mathbf {t} \,dS+\int _{V}\mathbf {b} \rho \,dV}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} =\mathbf {M} _{B}+\mathbf {M} _{C}=\int _{S}\mathbf {r} \times \mathbf {t} \,dS+\int _{V}\mathbf {r} \times \mathbf {b} \rho \,dV}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> M </mi> </mrow> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> M </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> B </mi> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> M </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> C </mi> </mrow> </msub> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> S </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> t </mi> </mrow> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> S </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> b </mi> </mrow> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {M} =\mathbf {M} _{B}+\mathbf {M} _{C}=\int _{S}\mathbf {r} \times \mathbf {t} \,dS+\int _{V}\mathbf {r} \times \mathbf {b} \rho \,dV} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73f527ae79f5c85569abe556b5e5d5bf1c073e51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.702ex; height:5.676ex;" alt="{\displaystyle \mathbf {M} =\mathbf {M} _{B}+\mathbf {M} _{C}=\int _{S}\mathbf {r} \times \mathbf {t} \,dS+\int _{V}\mathbf {r} \times \mathbf {b} \rho \,dV}"> </noscript><span class="lazy-image-placeholder" style="width: 46.702ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73f527ae79f5c85569abe556b5e5d5bf1c073e51" data-alt="{\displaystyle \mathbf {M} =\mathbf {M} _{B}+\mathbf {M} _{C}=\int _{S}\mathbf {r} \times \mathbf {t} \,dS+\int _{V}\mathbf {r} \times \mathbf {b} \rho \,dV}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>dimana <b>t</b> = <b>t</b>(<b>n</b>) disebut <a href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Traksi_permukaan&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Traksi permukaan (halaman belum tersedia)">traksi permukaan</a>, diintegrasikan di atas permukaan benda, dimana <b>n</b> menunjukkan <a href="https://id-m-wikipedia-org.translate.goog/wiki/Vektor_satuan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vektor satuan">vektor satuan</a> normal dan diarahkan ke luar permukaan <i>S</i>.</p> <p>Misalkan sistem koordinat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2},x_{3})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{1},x_{2},x_{3})} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c299961b19740135135d0ef0289b16b6095048d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.029ex; height:2.843ex;" alt="{\displaystyle (x_{1},x_{2},x_{3})}"> </noscript><span class="lazy-image-placeholder" style="width: 11.029ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c299961b19740135135d0ef0289b16b6095048d3" data-alt="{\displaystyle (x_{1},x_{2},x_{3})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> adalah <a href="https://id-m-wikipedia-org.translate.goog/wiki/Kerangka_acuan_inersia?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kerangka acuan inersia">kerangka acuan inersia</a>, <b>r</b> adalah vektor posisi partikel titik dalam benda kontinu sehubungan dengan asal sistem koordinat, dan <b>v</b> = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {dr \over dx}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mi> r </mi> </mrow> <mrow> <mi> d </mi> <mi> x </mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {dr \over dx}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a864ef68fa48b9e44be23c70b94b07af950fd64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.382ex; height:5.509ex;" alt="{\displaystyle {dr \over dx}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.382ex;height: 5.509ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a864ef68fa48b9e44be23c70b94b07af950fd64" data-alt="{\displaystyle {dr \over dx}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> adalah vektor kecepatan dari titik tersebut.</p> <p><b>Aksioma atau hukum pertama Euler</b> (hukum keseimbangan momentum linier atau keseimbangan gaya) menyatakan bahwa dalam kerangka inersia, laju waktu perubahan momentum linier <b>p</b> dari bagian sembarang benda kontinu sama dengan total gaya yang diterapkan <b>F</b> yang bekerja pada bagian itu, dan dinyatakan sebagai berikut.</p> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d\mathbf {p} }{dt}}&=\mathbf {F} \\{\frac {d}{dt}}\int _{V}\rho \mathbf {v} \,dV&=\int _{S}\mathbf {t} \,dS+\int _{V}\mathbf {b} \rho \,dV\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> p </mi> </mrow> </mrow> <mrow> <mi> d </mi> <mi> t </mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> V </mi> </mtd> <mtd> <mi></mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> S </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> t </mi> </mrow> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> S </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> b </mi> </mrow> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> V </mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\frac {d\mathbf {p} }{dt}}&=\mathbf {F} \\{\frac {d}{dt}}\int _{V}\rho \mathbf {v} \,dV&=\int _{S}\mathbf {t} \,dS+\int _{V}\mathbf {b} \rho \,dV\end{aligned}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff86cd6b0e5cb6dbe6c3ff29b21d9d6c8daceb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.902ex; margin-bottom: -0.269ex; width:35.514ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {d\mathbf {p} }{dt}}&=\mathbf {F} \\{\frac {d}{dt}}\int _{V}\rho \mathbf {v} \,dV&=\int _{S}\mathbf {t} \,dS+\int _{V}\mathbf {b} \rho \,dV\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 35.514ex;height: 11.509ex;vertical-align: -4.902ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff86cd6b0e5cb6dbe6c3ff29b21d9d6c8daceb6" data-alt="{\displaystyle {\begin{aligned}{\frac {d\mathbf {p} }{dt}}&=\mathbf {F} \\{\frac {d}{dt}}\int _{V}\rho \mathbf {v} \,dV&=\int _{S}\mathbf {t} \,dS+\int _{V}\mathbf {b} \rho \,dV\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p><b>Aksioma atau hukum kedua Euler</b> (hukum keseimbangan momentum sudut atau keseimbangan torsi) menyatakan bahwa dalam kerangka inersia, laju waktu perubahan momentum sudut <b>L</b> dari bagian sembarang benda kontinu sama dengan torsi total yang diterapkan <b>M</b> yang bekerja pada bagian itu, dan dinyatakan sebagai berikut.</p> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d\mathbf {L} }{dt}}&=\mathbf {M} \\{\frac {d}{dt}}\int _{V}\mathbf {r} \times \rho \mathbf {v} \,dV&=\int _{S}\mathbf {r} \times \mathbf {t} \,dS+\int _{V}\mathbf {r} \times \mathbf {b} \rho \,dV\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> d </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> L </mi> </mrow> </mrow> <mrow> <mi> d </mi> <mi> t </mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> M </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo> × </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> V </mi> </mtd> <mtd> <mi></mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> S </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> t </mi> </mrow> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> S </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> b </mi> </mrow> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> V </mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\frac {d\mathbf {L} }{dt}}&=\mathbf {M} \\{\frac {d}{dt}}\int _{V}\mathbf {r} \times \rho \mathbf {v} \,dV&=\int _{S}\mathbf {r} \times \mathbf {t} \,dS+\int _{V}\mathbf {r} \times \mathbf {b} \rho \,dV\end{aligned}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0050c8148823718acd12626d02dadf20509e96c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:47.342ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {d\mathbf {L} }{dt}}&=\mathbf {M} \\{\frac {d}{dt}}\int _{V}\mathbf {r} \times \rho \mathbf {v} \,dV&=\int _{S}\mathbf {r} \times \mathbf {t} \,dS+\int _{V}\mathbf {r} \times \mathbf {b} \rho \,dV\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 47.342ex;height: 11.509ex;vertical-align: -5.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0050c8148823718acd12626d02dadf20509e96c" data-alt="{\displaystyle {\begin{aligned}{\frac {d\mathbf {L} }{dt}}&=\mathbf {M} \\{\frac {d}{dt}}\int _{V}\mathbf {r} \times \rho \mathbf {v} \,dV&=\int _{S}\mathbf {r} \times \mathbf {t} \,dS+\int _{V}\mathbf {r} \times \mathbf {b} \rho \,dV\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>dimana <b>v</b> adalah kecepatan, <i><b>V</b></i> adalah volume, dan turunan dari <b>p</b> dan <b>L</b> adalah <a href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Turunan_material&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Turunan material (halaman belum tersedia)">turunan material</a>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Lihat_juga">Lihat juga</h2><span class="mw-editsection"> <a role="button" href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Hukum_gerak_Euler&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sunting bagian: Lihat juga" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>sunting</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <p><a href="https://id-m-wikipedia-org.translate.goog/wiki/Daftar_hal-hal_yang_dinamai_dari_Leonhard_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Daftar hal-hal yang dinamai dari Leonhard Euler">Daftar hal-hal yang dinamai dari Leonhard Euler</a></p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Referensi">Referensi</h2><span class="mw-editsection"> <a role="button" href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Hukum_gerak_Euler&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sunting bagian: Referensi" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>sunting</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a> <a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:0_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite class="citation book">McGill, David J.; King, Wilton W.; McGill, David J.; McGill, David J. (1995). <i>Engineering mechanics: statics and an introduction to dynamics</i>. PWS series in engineering (edisi ke-3. ed). Boston: PWS Publ: ITP. <a href="https://id-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="https://id-m-wikipedia-org.translate.goog/wiki/Istimewa:Sumber_buku/978-0-534-93399-9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Istimewa:Sumber buku/978-0-534-93399-9">978-0-534-93399-9</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+mechanics%3A+statics+and+an+introduction+to+dynamics&rft.place=Boston%3A+PWS+Publ&rft.series=PWS+series+in+engineering&rft.edition=3.+ed&rft.pub=ITP&rft.date=1995&rft.isbn=978-0-534-93399-9&rft.aulast=McGill&rft.aufirst=David+J.&rft.au=King%2C+Wilton+W.&rft.au=McGill%2C+David+J.&rft.au=McGill%2C+David+J.&rfr_id=info%3Asid%2Fid.wikipedia.org%3AHukum+gerak+Euler" class="Z3988"><span style="display:none;"> </span></span><span class="citation-comment" style="display:none; color:#33aa33; margin-left:0.3em">Pemeliharaan CS1: Teks tambahan (<a href="https://id-m-wikipedia-org.translate.goog/wiki/Kategori:Pemeliharaan_CS1:_Teks_tambahan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kategori:Pemeliharaan CS1: Teks tambahan">link</a>) </span></span></li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://emweb.unl.edu/NEGAHBAN/EM373/note19/note19.htm">"Equations of motion for a rigid body"</a>. <i>emweb.unl.edu</i><span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2024-02-24</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=emweb.unl.edu&rft.atitle=Equations+of+motion+for+a+rigid+body&rft_id=http%3A%2F%2Femweb.unl.edu%2FNEGAHBAN%2FEM373%2Fnote19%2Fnote19.htm&rfr_id=info%3Asid%2Fid.wikipedia.org%3AHukum+gerak+Euler" class="Z3988"><span style="display:none;"> </span></span></span></li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation book">Gray, Gary L.; Costanzo, Francesco; Plesha, Michael E. (2010). <i>Engineering mechanics. 2: Dynamics / Gary L. Gray; Francesco Costanzo; Michael E. Plesha</i> (edisi ke-SI ed). Boston, Mass.: McGraw-Hill. <a href="https://id-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="https://id-m-wikipedia-org.translate.goog/wiki/Istimewa:Sumber_buku/978-0-07-282871-9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Istimewa:Sumber buku/978-0-07-282871-9">978-0-07-282871-9</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+mechanics.+2%3A+Dynamics+%2F+Gary+L.+Gray%3B+Francesco+Costanzo%3B+Michael+E.+Plesha&rft.place=Boston%2C+Mass.&rft.edition=SI+ed&rft.pub=McGraw-Hill&rft.date=2010&rft.isbn=978-0-07-282871-9&rft.aulast=Gray&rft.aufirst=Gary+L.&rft.au=Costanzo%2C+Francesco&rft.au=Plesha%2C+Michael+E.&rfr_id=info%3Asid%2Fid.wikipedia.org%3AHukum+gerak+Euler" class="Z3988"><span style="display:none;"> </span></span><span class="citation-comment" style="display:none; color:#33aa33; margin-left:0.3em">Pemeliharaan CS1: Teks tambahan (<a href="https://id-m-wikipedia-org.translate.goog/wiki/Kategori:Pemeliharaan_CS1:_Teks_tambahan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kategori:Pemeliharaan CS1: Teks tambahan">link</a>) </span></span></li> <li id="cite_note-:1-4"><span class="mw-cite-backlink">^ <a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:1_4-0"><sup><i><b>a</b></i></sup></a> <a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:1_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation book"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.bookrags.com/research/eulers-laws-of-motion-wom/"><i>Euler</i></a> (dalam bahasa Inggris).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Euler&rft_id=https%3A%2F%2Fwww.bookrags.com%2Fresearch%2Feulers-laws-of-motion-wom%2F&rfr_id=info%3Asid%2Fid.wikipedia.org%3AHukum+gerak+Euler" class="Z3988"><span style="display:none;"> </span></span></span></li> <li id="cite_note-:2-5"><span class="mw-cite-backlink">^ <a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:2_5-0"><sup><i><b>a</b></i></sup></a> <a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:2_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation book">Rao, Anil V.; Rao, Anil Vithala (2006). <i>Dynamics of particles and rigid bodies: a systematic approach</i> (edisi ke-1. publ). New York: Cambridge University Press. <a href="https://id-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="https://id-m-wikipedia-org.translate.goog/wiki/Istimewa:Sumber_buku/978-0-521-85811-3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Istimewa:Sumber buku/978-0-521-85811-3">978-0-521-85811-3</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamics+of+particles+and+rigid+bodies%3A+a+systematic+approach&rft.place=New+York&rft.edition=1.+publ&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=978-0-521-85811-3&rft.aulast=Rao&rft.aufirst=Anil+V.&rft.au=Rao%2C+Anil+Vithala&rfr_id=info%3Asid%2Fid.wikipedia.org%3AHukum+gerak+Euler" class="Z3988"><span style="display:none;"> </span></span></span></li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-6">^</a></b></span> <span class="reference-text"><cite class="citation book">Ruina, Andy: Rudra Pratap (2002). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://ruina.tam.cornell.edu/Book/RuinaPratapNoProblems.pdf"><i>Introduction to Statics and Dynamics</i></a> <span style="font-size:85%;">(PDF)</span>. Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Statics+and+Dynamics&rft.pub=Oxford+University+Press&rft.date=2002&rft.aulast=Ruina&rft.aufirst=Andy%3A+Rudra+Pratap&rft_id=http%3A%2F%2Fruina.tam.cornell.edu%2FBook%2FRuinaPratapNoProblems.pdf&rfr_id=info%3Asid%2Fid.wikipedia.org%3AHukum+gerak+Euler" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">Parameter <code style="color:inherit; border:inherit; padding:inherit;">|url-status=</code> yang tidak diketahui akan diabaikan (<a href="https://id-m-wikipedia-org.translate.goog/wiki/Bantuan:Galat_CS1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#parameter_ignored" title="Bantuan:Galat CS1">bantuan</a>)</span></span></li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="https://id-m-wikipedia-org.translate.goog/wiki/Hukum_gerak_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-7">^</a></b></span> <span class="reference-text"><cite class="citation book">Lubliner, Jacob (2008). <i>Plasticity theory</i>. Dover books on engineering (edisi ke-Dover ed., rev. and corr. republ). Mineola: Dover Publ. <a href="https://id-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="https://id-m-wikipedia-org.translate.goog/wiki/Istimewa:Sumber_buku/978-0-486-46290-5?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Istimewa:Sumber buku/978-0-486-46290-5">978-0-486-46290-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Plasticity+theory&rft.place=Mineola&rft.series=Dover+books+on+engineering&rft.edition=Dover+ed.%2C+rev.+and+corr.+republ&rft.pub=Dover+Publ&rft.date=2008&rft.isbn=978-0-486-46290-5&rft.aulast=Lubliner&rft.aufirst=Jacob&rfr_id=info%3Asid%2Fid.wikipedia.org%3AHukum+gerak+Euler" class="Z3988"><span style="display:none;"> </span></span></span></li> </ol>   </section> </div> <noscript> <img 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