Freefall chord through the Earth -- calculus proof

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tagList--thread-1066771 "> <dt> <i class="fa--xf fal fa-tags" aria-hidden="true" title="Tags"></i> <span class="u-srOnly">Tags</span> </dt> <dd> <span class="js-tagList"> <a href="/tags/chord/" data-xf-init="preview-tooltip" data-preview-url="/tags/chord/preview" class="tagItem tagItem--tag_chord" dir="auto"> Chord </a> <a href="/tags/earth/" data-xf-init="preview-tooltip" data-preview-url="/tags/earth/preview" class="tagItem tagItem--tag_earth" dir="auto"> Earth </a> <a href="/tags/freefall/" data-xf-init="preview-tooltip" data-preview-url="/tags/freefall/preview" class="tagItem tagItem--tag_freefall" dir="auto"> Freefall </a> </span> </dd> </dl> </li> </ul> </div> </div> <script src="https://www.physicsforums.com/check-for-tex.js" defer></script> <div class="block block--messages" data-xf-init="" data-type="post" data-href="/inline-mod/" data-search-target="*"> <span class="u-anchorTarget" id="posts"></span> <div class="block-outer"><div class="block-outer-main"> <nav 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class="u-anchorTarget" id="post-7130703"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-06T22:00:51-0600" data-time="1730952051" data-date-string="Nov 6, 2024" data-time-string="10:00 PM" title="Nov 6, 2024 at 10:00 PM">Nov 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130703/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #1 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" > <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576">achap01</span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130703" data-lb-caption-desc="achap01 · Nov 6, 2024 at 10:00 PM"> <div class="message-fields message-fields--before"> <dl class="pairs pairs--columns pairs--fixedSmall pairs--customField" data-field="problemstatement"> <dt>Homework Statement</dt> <dd>I've gotten homework relating to proving that at any angle, freefalling through the earth will take the same amount of time.</dd> </dl> <dl class="pairs pairs--columns pairs--fixedSmall pairs--customField" data-field="equations"> <dt>Relevant Equations</dt> <dd>L = 2rsin(θ) (chord length)</dd> </dl> </div> <article class="message-body js-selectToQuote"> <div > <div class="bbWrapper">I haven't really attempted anything worth showing on the calculus side</div> </div> <div class="js-selectToQuoteEnd"> </div> <div class="googleads_fixed"> <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js?client=ca-pub-6580726045122001" crossorigin="anonymous"></script>  <ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-6580726045122001" data-ad-slot="3049969336" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script> </div> </article> </div> </div> <footer class="message-footer"> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <div class="message-cell message-cell--main" id="post-physorg"> <a href="https://phys.org/physics-news/" target="_blank">Physics news</a> on Phys.org <ul class="phys_items"> <li><a href="https://phys.org/news/2025-02-unexpected-nucleus-prompts-reevaluation-atomic.html" target="_blank">Unexpected shape of lead-208 nucleus prompts reevaluation of atomic nuclei models</a></li> <li><a href="https://phys.org/news/2025-02-scientists-elusive-liquid-transition-deep.html" target="_blank">Scientists map elusive liquid-liquid transition point using deep neural network</a></li> <li><a href="https://phys.org/news/2025-02-links-disparate-pion-reactions-nuclear.html" target="_blank">New calculation links disparate pion reactions in nuclear physics</a></li> </ul> </div> <article class="message message--post js-post js-inlineModContainer " data-author="haruspex" data-content="post-7130705" id="js-post-7130705" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130705/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130705"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-06T22:07:57-0600" data-time="1730952477" data-date-string="Nov 6, 2024" data-time-string="10:07 PM" title="Nov 6, 2024 at 10:07 PM" itemprop="datePublished">Nov 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span 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data-xf-init="tooltip" title="Messages"></i></dt> <dd>42,311</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>10,277</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130705" data-lb-caption-desc="haruspex · Nov 6, 2024 at 10:07 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 754576" data-quote="achap01" data-source="post: 7130703" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> achap01 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> <b>Homework Statement:</b> I've gotten homework relating to proving that at any angle, freefalling through the earth will take the same amount of time.<br /> <b>Relevant Equations:</b> L = 2rsin(θ) (chord length)<br /> <br /> I haven't really attempted anything worth showing on the calculus side </div> </div> </blockquote>I am sure you can think of more relevant equations. How about something concerning gravity and (uniform) spheres?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130707" id="js-post-7130707" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130707/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130707"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-06T22:12:41-0600" data-time="1730952761" data-date-string="Nov 6, 2024" data-time-string="10:12 PM" title="Nov 6, 2024 at 10:12 PM" itemprop="datePublished">Nov 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130707/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #3 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130707" data-lb-caption-desc="achap01 · Nov 6, 2024 at 10:12 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 334404" data-quote="haruspex" data-source="post: 7130705" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> haruspex said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I am sure you can think of more relevant equations. How about something concerning gravity and (uniform) spheres? </div> </div> </blockquote>Ive derived the newtonian motion eqs but i can't seem to understand how to get the derivative of acceleration with respect to anything relevant</div> </div> <div class="js-selectToQuoteEnd"> </div> <div class="googleads_fixed"> <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js?client=ca-pub-6580726045122001" crossorigin="anonymous"></script>  <ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-6580726045122001" data-ad-slot="5961342082" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" 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PM" title="Nov 6, 2024 at 10:12 PM" itemprop="datePublished">Nov 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130708/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #4 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/haruspex.334404/"> <meta itemprop="url" content="https://www.physicsforums.com/members/haruspex.334404/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/haruspex.334404/" class="username " dir="auto" data-user-id="334404" data-xf-init="member-tooltip"><span itemprop="name">haruspex</span></a></span>    <div 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class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> achap01 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Ive derived the newtonian motion eqs but i can't seem to understand how to get the derivative of acceleration with respect to anything relevant </div> </div> </blockquote>Please post your work so far (as required by the forum rules).</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130709" id="js-post-7130709" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130709/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130709"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-06T22:19:02-0600" data-time="1730953142" data-date-string="Nov 6, 2024" data-time-string="10:19 PM" title="Nov 6, 2024 at 10:19 PM" itemprop="datePublished">Nov 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130709/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #5 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130709" data-lb-caption-desc="achap01 · Nov 6, 2024 at 10:19 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 334404" data-quote="haruspex" data-source="post: 7130708" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> haruspex said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Please post your work so far (as required by the forum rules). </div> </div> </blockquote><script class="js-extraPhrases" type="application/json"> { "lightbox_close": "Close", "lightbox_next": "Next", "lightbox_previous": "Previous", "lightbox_error": "The requested content cannot be loaded. 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listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-06T22:32:36-0600" data-time="1730953956" data-date-string="Nov 6, 2024" data-time-string="10:32 PM" title="Nov 6, 2024 at 10:32 PM" itemprop="datePublished">Nov 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130710/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #6 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/haruspex.334404/"> <meta itemprop="url" content="https://www.physicsforums.com/members/haruspex.334404/" /> <div class="message-userDetails"> <span class="message-name"><a 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aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-trophy" data-xf-init="tooltip" title="2024 Award Winner"> <i class="fa--xf fal fa-trophy" aria-hidden="true"></i><span> 2024 Award</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>42,311</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>10,277</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130710" data-lb-caption-desc="haruspex · Nov 6, 2024 at 10:32 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper">Your second equation is a SUVAT equation. They assume constant acceleration.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130711" id="js-post-7130711" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130711/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130711"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-06T22:37:47-0600" data-time="1730954267" data-date-string="Nov 6, 2024" data-time-string="10:37 PM" title="Nov 6, 2024 at 10:37 PM" itemprop="datePublished">Nov 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130711/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #7 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130711" data-lb-caption-desc="achap01 · Nov 6, 2024 at 10:37 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 334404" data-quote="haruspex" data-source="post: 7130710" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> haruspex said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Your second equation is a SUVAT equation. They assume constant acceleration. </div> </div> </blockquote>Is there a proof of this problem somewhere online that I can reference?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="haruspex" data-content="post-7130715" id="js-post-7130715" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130715/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130715"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-06T22:52:12-0600" data-time="1730955132" data-date-string="Nov 6, 2024" data-time-string="10:52 PM" title="Nov 6, 2024 at 10:52 PM" itemprop="datePublished">Nov 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130715/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #8 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/haruspex.334404/"> <meta itemprop="url" content="https://www.physicsforums.com/members/haruspex.334404/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/haruspex.334404/" class="username " dir="auto" data-user-id="334404" data-xf-init="member-tooltip"><span itemprop="name">haruspex</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/haruspex/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-trophy" data-xf-init="tooltip" title="2024 Award Winner"> <i class="fa--xf fal fa-trophy" aria-hidden="true"></i><span> 2024 Award</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>42,311</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>10,277</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130715" data-lb-caption-desc="haruspex · Nov 6, 2024 at 10:52 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 754576" data-quote="achap01" data-source="post: 7130711" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> achap01 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Is there a proof of this problem somewhere online that I can reference? </div> </div> </blockquote>It's not that hard, and quite interesting.<br /> When it is at distance x from the midpoint of the shaft, what are the forces on it?<br /> What acceleration will that produce?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130716" id="js-post-7130716" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130716/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130716"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-06T22:55:06-0600" data-time="1730955306" data-date-string="Nov 6, 2024" data-time-string="10:55 PM" title="Nov 6, 2024 at 10:55 PM" itemprop="datePublished">Nov 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130716/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #9 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130716" data-lb-caption-desc="achap01 · Nov 6, 2024 at 10:55 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 334404" data-quote="haruspex" data-source="post: 7130715" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> haruspex said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> It's not that hard, and quite interesting.<br /> When it is at distance x from the midpoint of the shaft, what are the forces on it?<br /> What acceleration will that produce? </div> </div> </blockquote>I just think it'd be easier for me to learn with a video or something I can reference online rather than talking about it with you over forums</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="haruspex" data-content="post-7130717" id="js-post-7130717" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130717/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130717"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-06T22:58:06-0600" data-time="1730955486" data-date-string="Nov 6, 2024" data-time-string="10:58 PM" title="Nov 6, 2024 at 10:58 PM" itemprop="datePublished">Nov 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130717/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #10 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/haruspex.334404/"> <meta itemprop="url" content="https://www.physicsforums.com/members/haruspex.334404/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/haruspex.334404/" class="username " dir="auto" data-user-id="334404" data-xf-init="member-tooltip"><span itemprop="name">haruspex</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/haruspex/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-trophy" data-xf-init="tooltip" title="2024 Award Winner"> <i class="fa--xf fal fa-trophy" aria-hidden="true"></i><span> 2024 Award</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>42,311</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>10,277</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130717" data-lb-caption-desc="haruspex · Nov 6, 2024 at 10:58 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 754576" data-quote="achap01" data-source="post: 7130716" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> achap01 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I just think it'd be easier for me to learn with a video or something I can reference online rather than talking about it with you over forums </div> </div> </blockquote>But that won't teach you how to think about such problems. It will only show you how to do that particular problem.<br /> What are the forces?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130718" id="js-post-7130718" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130718/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130718"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-06T23:01:03-0600" data-time="1730955663" data-date-string="Nov 6, 2024" data-time-string="11:01 PM" title="Nov 6, 2024 at 11:01 PM" itemprop="datePublished">Nov 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130718/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #11 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130718" data-lb-caption-desc="achap01 · Nov 6, 2024 at 11:01 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 334404" data-quote="haruspex" data-source="post: 7130717" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> haruspex said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> But that won't teach you how to think about such problems. It will only show you how to do that particular problem.<br /> What are the forces? </div> </div> </blockquote>I'm saying it'd be faster and easier for me to figure out the problem, and learn from a process someone else used to think about such problems.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="Orodruin" data-content="post-7130719" id="js-post-7130719" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130719/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130719"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-06T23:06:21-0600" data-time="1730955981" data-date-string="Nov 6, 2024" data-time-string="11:06 PM" title="Nov 6, 2024 at 11:06 PM" itemprop="datePublished">Nov 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130719/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #12 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/orodruin.510075/"> <meta itemprop="url" content="https://www.physicsforums.com/members/orodruin.510075/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/orodruin.510075/" class="username " dir="auto" data-user-id="510075" data-xf-init="member-tooltip"><span itemprop="name">Orodruin</span></a></span>    <div class="post-badges"> <div class="post-badge badge-shield" data-xf-init="tooltip" title="Retired Mentor"> <i class="fa--xf fal fa-shield" aria-hidden="true" style="font-size:12px;color:#990033;"></i><span> Staff Emeritus</span> </div> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> 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data-xf-init="tooltip" title="Messages"></i></dt> <dd>22,638</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>14,498</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130719" data-lb-caption-desc="Orodruin · Nov 6, 2024 at 11:06 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 754576" data-quote="achap01" data-source="post: 7130718" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> achap01 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I'm saying it'd be faster and easier for me to figure out the problem, and learn from a process someone else used to think about such problems. </div> </div> </blockquote>Then you’ll only be learning how someone else solved a problem. You will not be learning to solve problems yourself. It is a really bad way to learn physics. You might think it is faster and easier for you, but you will not actually be learning. Learning requires effort and working through the problem yourself. Not looking up the solution. <br /> <br /> We are happy to help you learn, not to help you get the illusion of learning.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="haruspex" data-content="post-7130720" id="js-post-7130720" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130720/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130720"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-06T23:06:44-0600" data-time="1730956004" data-date-string="Nov 6, 2024" data-time-string="11:06 PM" title="Nov 6, 2024 at 11:06 PM" itemprop="datePublished">Nov 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130720/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #13 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/haruspex.334404/"> <meta itemprop="url" content="https://www.physicsforums.com/members/haruspex.334404/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/haruspex.334404/" class="username " dir="auto" data-user-id="334404" data-xf-init="member-tooltip"><span itemprop="name">haruspex</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/haruspex/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-trophy" data-xf-init="tooltip" title="2024 Award Winner"> <i class="fa--xf fal fa-trophy" aria-hidden="true"></i><span> 2024 Award</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>42,311</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>10,277</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130720" data-lb-caption-desc="haruspex · Nov 6, 2024 at 11:06 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 754576" data-quote="achap01" data-source="post: 7130718" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> achap01 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I'm saying it'd be faster and easier for me to figure out the problem, and learn from a process someone else used to think about such problems. </div> </div> </blockquote>I can tell you the first step they made: they figured out which forces apply. But seeing what they got for that won’t show you how they found them.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130721" id="js-post-7130721" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130721/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130721"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-06T23:12:23-0600" data-time="1730956343" data-date-string="Nov 6, 2024" data-time-string="11:12 PM" title="Nov 6, 2024 at 11:12 PM" itemprop="datePublished">Nov 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130721/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #14 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130721" data-lb-caption-desc="achap01 · Nov 6, 2024 at 11:12 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 334404" data-quote="haruspex" data-source="post: 7130720" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> haruspex said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I can tell you the first step they made: they figured out which forces apply. But seeing what they got for that won’t show you how they found them. </div> </div> </blockquote>the forces would just be gravity(and normal force perpendicular to the chord) gravity would just be linear so (dist from center)/R * weight(at surface). To get an eq relating gravitational force towards the center with force parallel to the chord would require both time and dist which would have to be related seperately right?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="haruspex" data-content="post-7130722" id="js-post-7130722" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130722/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130722"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-06T23:50:15-0600" data-time="1730958615" data-date-string="Nov 6, 2024" data-time-string="11:50 PM" title="Nov 6, 2024 at 11:50 PM" itemprop="datePublished">Nov 6, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130722/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #15 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/haruspex.334404/"> <meta itemprop="url" content="https://www.physicsforums.com/members/haruspex.334404/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/haruspex.334404/" class="username " dir="auto" data-user-id="334404" data-xf-init="member-tooltip"><span itemprop="name">haruspex</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/haruspex/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-trophy" data-xf-init="tooltip" title="2024 Award Winner"> <i class="fa--xf fal fa-trophy" aria-hidden="true"></i><span> 2024 Award</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>42,311</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>10,277</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130722" data-lb-caption-desc="haruspex · Nov 6, 2024 at 11:50 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 754576" data-quote="achap01" data-source="post: 7130721" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> achap01 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> To get an eq relating gravitational force towards the center with force parallel to the chord would require both time and dist which would have to be related seperately right? </div> </div> </blockquote>Don't worry about the time.<br /> If at distance r from the Earth's centre and distance x from the midpoint of the shaft, what is the acceleration along the shaft?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130723" id="js-post-7130723" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130723/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130723"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T00:00:59-0600" data-time="1730959259" data-date-string="Nov 7, 2024" data-time-string="12:00 AM" title="Nov 7, 2024 at 12:00 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130723/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #16 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130723" data-lb-caption-desc="achap01 · Nov 7, 2024 at 12:00 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 334404" data-quote="haruspex" data-source="post: 7130722" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> haruspex said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Don't worry about the time.<br /> If at distance r from the Earth's centre and distance x from the midpoint of the shaft, what is the acceleration along the shaft? </div> </div> </blockquote>force of gravity towards the center would be (r/R(radius of earth))*weight. Angle of Fg would be θx/2rsin(θ/2), so force parallel to chord would be w(r/R)*sin(θx/2rsin(θ/2)). Then plug in to f=ma</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="haruspex" data-content="post-7130724" id="js-post-7130724" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130724/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130724"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T00:05:10-0600" data-time="1730959510" data-date-string="Nov 7, 2024" data-time-string="12:05 AM" title="Nov 7, 2024 at 12:05 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130724/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #17 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/haruspex.334404/"> <meta itemprop="url" content="https://www.physicsforums.com/members/haruspex.334404/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/haruspex.334404/" class="username " dir="auto" data-user-id="334404" data-xf-init="member-tooltip"><span itemprop="name">haruspex</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/haruspex/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-trophy" data-xf-init="tooltip" title="2024 Award Winner"> <i class="fa--xf fal fa-trophy" aria-hidden="true"></i><span> 2024 Award</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>42,311</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>10,277</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130724" data-lb-caption-desc="haruspex · Nov 7, 2024 at 12:05 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 754576" data-quote="achap01" data-source="post: 7130723" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> achap01 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Angle of Fg would be θx/2rsin(θ/2), </div> </div> </blockquote>That's a very strange formula.<br /> Is that ##\theta## still how you defined it in post #1?<br /> If so, ignore that angle and consider the angle between the distances x and r. Call that ##\phi##. What fraction of ##F_g## acts along x?<br /> <br /> (No idea why the LaTeX has stopped working.)</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130726" id="js-post-7130726" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130726/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130726"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T00:11:04-0600" data-time="1730959864" data-date-string="Nov 7, 2024" data-time-string="12:11 AM" title="Nov 7, 2024 at 12:11 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130726/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #18 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130726" data-lb-caption-desc="achap01 · Nov 7, 2024 at 12:11 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 334404" data-quote="haruspex" data-source="post: 7130724" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> haruspex said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> That's a very strange formula.<br /> Is that ##\theta## still how you defined it in post #1? </div> </div> </blockquote>I mistypes in the first post. it should be L = 2rsin(θ/2). I got the angle by finding the fraction of the total chord length x is, and then multiplying that with the central angle</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130727" id="js-post-7130727" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130727/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130727"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T00:13:04-0600" data-time="1730959984" data-date-string="Nov 7, 2024" data-time-string="12:13 AM" title="Nov 7, 2024 at 12:13 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130727/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #19 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130727" data-lb-caption-desc="achap01 · Nov 7, 2024 at 12:13 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 334404" data-quote="haruspex" data-source="post: 7130724" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> haruspex said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> If so, ignore that angle and consider the angle between the distances x and r. Call that ##\phi##. What fraction of ##F_g## acts along x? </div> </div> </blockquote>cos(θ) of Fg</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="haruspex" data-content="post-7130728" id="js-post-7130728" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130728/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130728"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T00:13:49-0600" data-time="1730960029" data-date-string="Nov 7, 2024" data-time-string="12:13 AM" title="Nov 7, 2024 at 12:13 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130728/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #20 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/haruspex.334404/"> <meta itemprop="url" content="https://www.physicsforums.com/members/haruspex.334404/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/haruspex.334404/" class="username " dir="auto" data-user-id="334404" data-xf-init="member-tooltip"><span itemprop="name">haruspex</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/haruspex/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-trophy" data-xf-init="tooltip" title="2024 Award Winner"> <i class="fa--xf fal fa-trophy" aria-hidden="true"></i><span> 2024 Award</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>42,311</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>10,277</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130728" data-lb-caption-desc="haruspex · Nov 7, 2024 at 12:13 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 754576" data-quote="achap01" data-source="post: 7130726" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> achap01 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I mistypes in the first post. it should be L = 2rsin(θ/2). I got the angle by finding the fraction of the total chord length x is, and then multiplying that with the central angle </div> </div> </blockquote>As I wrote, forget about the angle the entire chord subtends at the centre. Concentrate on the angle between r (not R) and x. If the gravitational force acts along r, what fraction of it acts along x?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130729" id="js-post-7130729" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130729/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130729"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T00:14:26-0600" data-time="1730960066" data-date-string="Nov 7, 2024" data-time-string="12:14 AM" title="Nov 7, 2024 at 12:14 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130729/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #21 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130729" data-lb-caption-desc="achap01 · Nov 7, 2024 at 12:14 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 334404" data-quote="haruspex" data-source="post: 7130728" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> haruspex said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> As I wrote, forget about the angle the entire chord subtends at the centre. Concentrate on the angle between r (not R) and x. If the gravitational force acts along r, what fraction of it acts along x? </div> </div> </blockquote>cos(theta) of Fg. Is there any way I could see a reference for this proof bc I feel like this convo is going to take a while with the responses. I can ask you about the reference too, but I feel like I'd learn much better with it.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="haruspex" data-content="post-7130730" id="js-post-7130730" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130730/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130730"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T00:16:25-0600" data-time="1730960185" data-date-string="Nov 7, 2024" data-time-string="12:16 AM" title="Nov 7, 2024 at 12:16 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130730/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #22 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/haruspex.334404/"> <meta itemprop="url" content="https://www.physicsforums.com/members/haruspex.334404/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/haruspex.334404/" class="username " dir="auto" data-user-id="334404" data-xf-init="member-tooltip"><span itemprop="name">haruspex</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/haruspex/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-trophy" data-xf-init="tooltip" title="2024 Award Winner"> <i class="fa--xf fal fa-trophy" aria-hidden="true"></i><span> 2024 Award</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>42,311</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>10,277</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130730" data-lb-caption-desc="haruspex · Nov 7, 2024 at 12:16 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 754576" data-quote="achap01" data-source="post: 7130729" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> achap01 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> cos(theta) of Fg. Is there any way I could see a reference for this proof bc I feel like this convo is going to take a while with the responses. I can ask you about the reference too </div> </div> </blockquote>Where theta is what, now? If the angle between x and r, yes.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130731" id="js-post-7130731" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130731/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130731"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T00:18:05-0600" data-time="1730960285" data-date-string="Nov 7, 2024" data-time-string="12:18 AM" title="Nov 7, 2024 at 12:18 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130731/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #23 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130731" data-lb-caption-desc="achap01 · Nov 7, 2024 at 12:18 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 334404" data-quote="haruspex" data-source="post: 7130730" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> haruspex said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Where theta is what, now? </div> </div> </blockquote>the only way i see to derive that theta(angles between x and r) is with the angle I mentioned previously, but by subtracting it from 90deg. So theta =90-(θx/2rsin(θ/2)). I don't understand how to get it without the central angle</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="haruspex" data-content="post-7130732" id="js-post-7130732" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130732/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130732"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T00:20:15-0600" data-time="1730960415" data-date-string="Nov 7, 2024" data-time-string="12:20 AM" title="Nov 7, 2024 at 12:20 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130732/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #24 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/haruspex.334404/"> <meta itemprop="url" content="https://www.physicsforums.com/members/haruspex.334404/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/haruspex.334404/" class="username " dir="auto" data-user-id="334404" data-xf-init="member-tooltip"><span itemprop="name">haruspex</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/haruspex/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-trophy" data-xf-init="tooltip" title="2024 Award Winner"> <i class="fa--xf fal fa-trophy" aria-hidden="true"></i><span> 2024 Award</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>42,311</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>10,277</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130732" data-lb-caption-desc="haruspex · Nov 7, 2024 at 12:20 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 754576" data-quote="achap01" data-source="post: 7130731" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> achap01 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> the only way i see to derive that theta is with the angle I mentioned previously, but by subtracting it from 90deg. So theta =90-(θx/2rsin(θ/2)). I don't understand how to get it without the central angle </div> </div> </blockquote>You have a right angled triangle where r is the hypotenuse and x is the "adjacent " and (this) theta is the angle between them. What equation relates them?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130734" id="js-post-7130734" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130734/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130734"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T00:21:50-0600" data-time="1730960510" data-date-string="Nov 7, 2024" data-time-string="12:21 AM" title="Nov 7, 2024 at 12:21 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130734/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #25 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130734" data-lb-caption-desc="achap01 · Nov 7, 2024 at 12:21 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper">arccos(x/r)=theta<br /> cos(theta)=x/r</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="haruspex" data-content="post-7130735" id="js-post-7130735" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130735/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130735"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T00:22:50-0600" data-time="1730960570" data-date-string="Nov 7, 2024" data-time-string="12:22 AM" title="Nov 7, 2024 at 12:22 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130735/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #26 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/haruspex.334404/"> <meta itemprop="url" content="https://www.physicsforums.com/members/haruspex.334404/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/haruspex.334404/" class="username " dir="auto" data-user-id="334404" data-xf-init="member-tooltip"><span itemprop="name">haruspex</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/haruspex/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-trophy" data-xf-init="tooltip" title="2024 Award Winner"> <i class="fa--xf fal fa-trophy" aria-hidden="true"></i><span> 2024 Award</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>42,311</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>10,277</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130735" data-lb-caption-desc="haruspex · Nov 7, 2024 at 12:22 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 754576" data-quote="achap01" data-source="post: 7130734" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> achap01 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> arccos(x/r)=theta<br /> cos(theta)=x/r </div> </div> </blockquote>Right, so express the component of the gravitational force along x in terms of x and r.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130736" id="js-post-7130736" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130736/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130736"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T00:26:24-0600" data-time="1730960784" data-date-string="Nov 7, 2024" data-time-string="12:26 AM" title="Nov 7, 2024 at 12:26 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130736/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #27 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130736" data-lb-caption-desc="achap01 · Nov 7, 2024 at 12:26 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper">cos(arccos(x/r))<br /> = x/r<br /> <br /> If this is right im gonna feel so braindead</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130739" id="js-post-7130739" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130739/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130739"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T00:41:42-0600" data-time="1730961702" data-date-string="Nov 7, 2024" data-time-string="12:41 AM" title="Nov 7, 2024 at 12:41 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130739/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #28 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130739" data-lb-caption-desc="achap01 · Nov 7, 2024 at 12:41 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 334404" data-quote="haruspex" data-source="post: 7130735" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> haruspex said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Right, so express the component of the gravitational force along x in terms of x and r. </div> </div> </blockquote>is there a proof online bc i think im close now</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130741" id="js-post-7130741" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130741/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130741"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T00:54:12-0600" data-time="1730962452" data-date-string="Nov 7, 2024" data-time-string="12:54 AM" title="Nov 7, 2024 at 12:54 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130741/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #29 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130741" data-lb-caption-desc="achap01 · Nov 7, 2024 at 12:54 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 334404" data-quote="haruspex" data-source="post: 7130735" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> haruspex said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Right, so express the component of the gravitational force along x in terms of x and r. </div> </div> </blockquote>I go to 9.8/R=a but i dont know what to do now</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" 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AM" title="Nov 7, 2024 at 1:16 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130743/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #30 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/haruspex.334404/"> <meta itemprop="url" content="https://www.physicsforums.com/members/haruspex.334404/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/haruspex.334404/" class="username " dir="auto" data-user-id="334404" data-xf-init="member-tooltip"><span itemprop="name">haruspex</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/haruspex/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-trophy" data-xf-init="tooltip" title="2024 Award Winner"> <i class="fa--xf fal fa-trophy" aria-hidden="true"></i><span> 2024 Award</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>42,311</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>10,277</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130743" data-lb-caption-desc="haruspex · Nov 7, 2024 at 1:16 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 754576" data-quote="achap01" data-source="post: 7130741" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> achap01 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I go to 9.8/R=a but i dont know what to do now </div> </div> </blockquote>That is dimensionally wrong, so cannot be right.<br /> There should be an x in there. Check your working. If you can’t find the error, post it.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="Orodruin" data-content="post-7130744" id="js-post-7130744" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130744/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130744"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T01:20:01-0600" data-time="1730964001" data-date-string="Nov 7, 2024" data-time-string="1:20 AM" title="Nov 7, 2024 at 1:20 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130744/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #31 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/orodruin.510075/"> <meta itemprop="url" content="https://www.physicsforums.com/members/orodruin.510075/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/orodruin.510075/" class="username " dir="auto" data-user-id="510075" data-xf-init="member-tooltip"><span itemprop="name">Orodruin</span></a></span>    <div class="post-badges"> <div class="post-badge badge-shield" data-xf-init="tooltip" title="Retired Mentor"> <i class="fa--xf fal fa-shield" aria-hidden="true" style="font-size:12px;color:#990033;"></i><span> Staff Emeritus</span> </div> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/Orodruin/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-trophy" data-xf-init="tooltip" title="2024 Award Winner"> <i class="fa--xf fal fa-trophy" aria-hidden="true"></i><span> 2024 Award</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>22,638</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>14,498</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130744" data-lb-caption-desc="Orodruin · Nov 7, 2024 at 1:20 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper">I would suggest you need to work more methodically rather than in short snippets. You have to answer the following questions:<br /> <ol> <li data-xf-list-type="ol">What is the magnitude of the force on a mass ##m## at a distance ##r## from the center?</li> <li data-xf-list-type="ol">What are its components in the different directions?</li> <li data-xf-list-type="ol">Can you write down Newton's second law for each component of the motion?</li> </ol>You should note that when you write down 3, you obtain equations of motion for each component that is independent of the other components. This is key. The resulting differential equation for each component should be easily solvable if you are familiar with basic ODEs.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="1" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList is-active"> <ul class="reactionSummary"> <li><span class="reaction reaction--small reaction--1" data-reaction-id="1"><i aria-hidden="true"></i><img src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" class="reaction-sprite js-reaction" alt="Like" title="Like" /></span></li> </ul> <span class="u-srOnly">Likes</span> <bdi>SammyS</bdi> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130790" id="js-post-7130790" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130790/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130790"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T06:17:05-0600" data-time="1730981825" data-date-string="Nov 7, 2024" data-time-string="6:17 AM" title="Nov 7, 2024 at 6:17 AM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130790/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #32 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130790" data-lb-caption-desc="achap01 · Nov 7, 2024 at 6:17 AM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 334404" data-quote="haruspex" data-source="post: 7130743" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> haruspex said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> That is dimensionally wrong, so cannot be right.<br /> There should be an x in there. Check your working. If you can’t find the error, post it. </div> </div> </blockquote>sorry i mean 9.8x/R=a</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="haruspex" data-content="post-7130877" id="js-post-7130877" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130877/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130877"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T13:30:16-0600" data-time="1731007816" data-date-string="Nov 7, 2024" data-time-string="1:30 PM" title="Nov 7, 2024 at 1:30 PM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130877/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #33 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/haruspex.334404/"> <meta itemprop="url" content="https://www.physicsforums.com/members/haruspex.334404/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/haruspex.334404/" class="username " dir="auto" data-user-id="334404" data-xf-init="member-tooltip"><span itemprop="name">haruspex</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/haruspex/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-trophy" data-xf-init="tooltip" title="2024 Award Winner"> <i class="fa--xf fal fa-trophy" aria-hidden="true"></i><span> 2024 Award</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>42,311</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>10,277</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130877" data-lb-caption-desc="haruspex · Nov 7, 2024 at 1:30 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 754576" data-quote="achap01" data-source="post: 7130790" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> achap01 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> sorry i mean 9.8x/R=a </div> </div> </blockquote>Ok, nearly there!<br /> So write the differential equation relating ##x## to ##\ddot x##.<br /> <br /> But it is better to leave it as "g". If you substitute a numeric value you should include units.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="achap01" data-content="post-7130888" id="js-post-7130888" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130888/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130888"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T14:28:32-0600" data-time="1731011312" data-date-string="Nov 7, 2024" data-time-string="2:28 PM" title="Nov 7, 2024 at 2:28 PM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130888/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #34 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/achap01.754576/"> <meta itemprop="url" content="https://www.physicsforums.com/members/achap01.754576/" /> <div class="message-userDetails"> <span class="message-name"><span class="username " dir="auto" data-user-id="754576"><span itemprop="name">achap01</span></span></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>   </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>21</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130888" data-lb-caption-desc="achap01 · Nov 7, 2024 at 2:28 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 334404" data-quote="haruspex" data-source="post: 7130877" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> haruspex said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Ok, nearly there!<br /> So write the differential equation relating ##x## to ##\ddot x##.<br /> <br /> But it is better to leave it as "g". If you substitute a numeric value you should include units. </div> </div> </blockquote>would ẍ just be the second derivative of x with respect to time, so ẍ=a=9.8x/R?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="haruspex" data-content="post-7130890" id="js-post-7130890" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/7130890/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/freefall-chord-through-the-earth-calculus-proof.1066771/" /> <span class="u-anchorTarget" id="post-7130890"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2024-11-07T14:59:39-0600" data-time="1731013179" data-date-string="Nov 7, 2024" data-time-string="2:59 PM" title="Nov 7, 2024 at 2:59 PM" itemprop="datePublished">Nov 7, 2024</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/7130890/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #35 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/haruspex.334404/"> <meta itemprop="url" content="https://www.physicsforums.com/members/haruspex.334404/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/haruspex.334404/" class="username " dir="auto" data-user-id="334404" data-xf-init="member-tooltip"><span itemprop="name">haruspex</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> <div class="post-badge badge-file" data-xf-init="tooltip" title="Click to View My Articles"> <a href="https://www.physicsforums.com/insights/author/haruspex/"><i class="fa--xf fal fa-file-alt" aria-hidden="true"></i><span> Insights Author</span></a> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-trophy" data-xf-init="tooltip" title="2024 Award Winner"> <i class="fa--xf fal fa-trophy" aria-hidden="true"></i><span> 2024 Award</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>42,311</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>10,277</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7130890" data-lb-caption-desc="haruspex · Nov 7, 2024 at 2:59 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="member: 754576" data-quote="achap01" data-source="post: 7130888" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> achap01 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> would ẍ just be the second derivative of x with respect to time, so ẍ=a=9.8x/R? </div> </div> </blockquote>Almost, but watch the signs. If we take x as going from ##x=-L## to ##x=+L## then when ##x<0## the acceleration is positive. Correcting that, do you recognise the form of the ODE?<br /> <br /> Please use "g", not "9.8".</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article> </div> </div> <div class="block-outer block-outer--after"> <div class="block-outer-main"> <nav class="pageNavWrapper pageNavWrapper--logarithmic pageNavWrapper--full " data-block-size="1" data-last-page="2"> <div class="pageNav"> <ul class="pageNav-main"> <li class="pageNav-page pageNav-page--current "><a 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Freefall chord through the Earth -- calculus proof