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31 32<!-- End Jekyll SEO tag --> 33 34 </head> 35 <body> 36 <header> 37 <img id="logo" src="/assets/images/tob-white.svg" /> 38 <nav> 39 <a href="/" >Home</a><a href="/music" >Music</a><a href="/an/about/page" >About</a> 40</nav> 41 42</header> 43 44 45 46 47 48<div class="h-image"> 49<h1>There oughta be a Lego marble run</h1> 50<img src="/assets/resized/images/2026-08-14/1280/youtube.jpg" srcset=" /assets/resized/images/2026-08-14/640/youtube.jpg 640w, /assets/resized/images/2026-08-14/768/youtube.jpg 768w, /assets/resized/images/2026-08-14/1024/youtube.jpg 1024w, /assets/resized/images/2026-08-14/1280/youtube.jpg 1280w"> 51</div> 52 53 54<p class="date">14 August 2026</p> 55 56<p>My son loves marble runs, but as he became more interested in Lego than the bigger Duplo bricks, I needed a marble run for Lego. So I printed one. This is a rather simple project, but I stumbled onto an interesting physics detail that I wanted to discuss.</p> 57 58<figure class="youtube"> 59<a href="https://youtu.be/SVclwTR1JH4" target="_blank"> 60<img src="/assets/resized/images/2026-08-14/1280/youtube.jpg" style="max-width: min(1920px, 100%)" alt="Thumbnail of the youtube video: Several colorful marble run tracks on a green Lego baseplate. Some pieces form a bridge across another track and a shiny ball bearing is rolling across it." srcset=" /assets/resized/images/2026-08-14/640/youtube.jpg 640w, /assets/resized/images/2026-08-14/768/youtube.jpg 768w, /assets/resized/images/2026-08-14/1024/youtube.jpg 1024w, /assets/resized/images/2026-08-14/1280/youtube.jpg 1280w, /assets/images/2026-08-14/youtube.jpg 1920w" sizes="(max-width: 60rem) 100vw, 60rem" /> 61</a> 62<figcaption>Click the image to see the video on youtube.com.</figcaption> 63</figure> 64 65<p>Most of you probably just want to watch the video, but if you want to dive into the physics with a proper mathematical description, the blog article is just right for you.</p> 66 67<!--more--> 68 69<p>First, letâs get the housekeeping out of the way. If you want to print the marble run, you can get it from any of the following sites:</p> 70 71<ul> 72 <li><a href="https://www.printables.com/model/1810431-slopeless-marble-run-for-lego-bricks" target="_blank">printables.com</a><sup id="fnref:printables"><a href="#fn:printables" class="footnote" rel="footnote" role="doc-noteref">1</a></sup></li> 73 <li><a href="https://www.thingiverse.com/thing:7395779" target="_blank">thingiverse.com</a><sup id="fnref:thingiverse"><a href="#fn:thingiverse" class="footnote" rel="footnote" role="doc-noteref">2</a></sup></li> 74 <li><a href="https://makerworld.com/de/models/3171892-slopeless-marble-run-for-construction-bricks" target="_blank">makerworld.com</a><sup id="fnref:makerworld"><a href="#fn:makerworld" class="footnote" rel="footnote" role="doc-noteref">3</a></sup></li> 75</ul> 76 77<p>Also, since I generate the marble tracks using <a href="https://openscad.org/" target="_blank">OpenSCAD</a>, you can just download the <a href="/assets/openscad/marble-run.scad">scad file</a>. All files are shared under the Creative Commons Attribution licence (CC-BY 4.0).</p> 78 79<p>Use 14mm ball bearings as marbles and print the parts you like. You should not need to print supports, and layer height or material should not matter as long as you donât go too crazy. Thatâs it.</p> 80 81<p>And now for the physicsâ¦</p> 82 83<figure> 84 85<img src="/assets/resized/images/2026-08-14/1280/track.jpg" style="max-width: min(3840px, 75%)" alt="Photo of a marble run assembled from several 3D printed track pieces. They are build on a green Lego base plate with several flat Lego bricks as support. The 3D printed tracks have many different bright colors and form a complicated path with bridges. In the photo there are currently several marbles on the tracks, which are actually shiny metal ball bearings." srcset=" /assets/resized/images/2026-08-14/640/track.jpg 640w, /assets/resized/images/2026-08-14/768/track.jpg 768w, /assets/resized/images/2026-08-14/1024/track.jpg 1024w, /assets/resized/images/2026-08-14/1280/track.jpg 1280w, /assets/images/2026-08-14/track.jpg 3840w" sizes="(max-width: 60rem) calc(75/100 * 100vw), calc(75/100 * 60rem)" /> 86 87<figcaption>A marble run made up of several 3D printed tracks on an original Lego baseplate. The marbles are actually metal ball bearings.</figcaption> 88</figure> 89 90<h2 id="energies-potential-kinetic-rotational-and-translational">Energies: potential, kinetic, rotational and translational</h2> 91 92<p>So, why am I even talking about the physics of a simple marble run? Well, it is just an oddity I stumbled upon while designing the tracks.</p> 93 94<p>Normally, you would design some kind of ramp, place the marble at the top and let it go. At the beginning, at a height <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span>, the marble with mass <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> has the potential energy <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>E</mi><mtext>pot</mtext></msub><mo>=</mo><mi>m</mi><mi>
94g</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">E_\text{pot} = m g h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">pot</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">m</span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mord mathnormal">h</span></span></span></span> (with gravitational acceleration <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span></span></span></span>) and when you let go, potential energy is converted into kinetic energy <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>E</mi><mtext>kin</mtext></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>â</mtext><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">E_\text{kin} = \frac{1}{2}\,m v^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">kin</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>. If the marble rolls down the entire slope across the height <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span>, the entire energy becomes kinetic energy with velocity <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span></span></span>:</p> 95 96<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>E</mi><mtext>pot</mtext></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>E</mi><mtext>kin</mtext></msub></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo>â</mo><mi>m</mi><mi>
96g</mi><mi>h</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>â</mtext><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo>â</mo><mi>v</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msqrt><mrow><mn>2</mn><mi>g</mi><mi>h</mi></mrow></msqrt></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align} 97E_\text{pot} &= E_\text{kin} \\ 98\Rightarrow mgh &= \frac{1}{2}\,m v^2\\ 99\Rightarrow v &= \sqrt{2gh} 100\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.1513em;vertical-align:-2.3257em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.8257em;"><span style="top:-5.3071em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">pot</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.3257em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mrel">â</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">m</span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mord mathnormal">h</span></span></span><span style="top:-1.3558em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mrel">â</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:2.3257em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.8257em;"><span style="top:-5.3071em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">kin</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.3257em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-1.3558em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9839em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0359em;">g</span><span class="mord mathnormal">h</span></span></span><span style="top:-2.9439em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119 101c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120 102c340,-704.7,510.7,-1060.3,512,-1067 103l0 -0 104c4.7,-7.3,11,-11,19,-11 105H40000v40H1012.3 106s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232 107c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 108s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 109c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z 110M1001 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.2561em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:2.3257em;"><span></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.8257em;"><span style="top:-5.3071em;"><span class="pstrut" style="height:3.3214em;"></span><span class="eqn-num"></span></span><span style="top:-3.3257em;"><span class="pstrut" style="height:3.3214em;"></span><span class="eqn-num"></span></span><span style="top:-1.3558em;"><span class="pstrut" style="height:3.3214em;"></span><span class="eqn-num"></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:2.3257em;"><span></span></span></span></span></span></span></span></span> 111 112<p>Thatâs what you might have learned in school, right? Well, depending on how advanced your physics course was, you might also have learned that this is not correct for a solid object. It is only correct for a point mass, a theoretical construct where all mass of the object sits in a single point. If the object has an actual size (like any real object), kinetic energy does not only come from the speed of the objectâs center of mass, but also from its rotation. The famous <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>â</mtext><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\frac{1}{2}\,m v^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> is actually only the translational kinetic energy <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>E</mi><mtext>t</mtext></msub></mrow><annotation encoding="application/x-tex">E_\text{t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">t</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, but there is also rotational kinetic energy <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>E</mi><mtext>r</mtext></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>â</mtext><mi>I</mi><msup><mi>Ï</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">E_\text{r} = \frac{1}{2}\,I \omega^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">r</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">Ï</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>:</p> 113 114<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>E</mi><mtext>kin</mtext></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>E</mi><mtext>t</mtext></msub><mo>+</mo><msub><mi>E</mi><mtext>r</mtext></msub></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>â</mtext><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>â</mtext><mi>I</mi><msup><mi>Ï</mi><mn>2</mn></msup></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align} 115E_\text{kin} &= E_\text{t} + E_\text{r}\\ 116 &= \frac{1}{2}\,m v^2 + \frac{1}{2}\,I \omega^2 117\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.5074em;vertical-align:-1.5037em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0037em;"><span style="top:-4.4852em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">kin</spa
117n></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.5037em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:1.5037em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0037em;"><span style="top:-4.4852em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">t</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">r</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.5037em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">Ï</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:1.5037em;"><span></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0037em;"><span style="top:-4.4852em;"><span class="pstrut" style="height:3.3214em;"></span><span class="eqn-num"></span></span><span style="top:-2.5037em;"><span class="pstrut" style="height:3.3214em;"></span><span class="eqn-num"></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:1.5037em;"><span></span></span></span></span></span></span></span></span> 118 119<p>These formulas look very similar in form, but rotational energy depen
119ds on the rotational counterparts: Velocity becomes angular velocity <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Ï</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">Ï</span></span></span></span> (in radians per second) and mass becomes the moment of inertia <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span></span></span></span>. Resisting the urge to go on a long tangent about the moment of inertia, we can just note that for a solid sphere (like a marble) with radius <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span></span></span></span>, the moment of inertia is</p> 120 121<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>I</mi><mo>=</mo><mfrac><mn>2</mn><mn>5</mn></mfrac><mtext>â</mtext><mi>m</mi><msup><mi>R</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">I = \frac{2}{5}\,m R^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span> 122 123<p>Taking into account that the speed of the marble is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>=</mo><mi>R</mi><mi>Ï</mi></mrow><annotation encoding="application/x-tex">v = R \omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="mord mathnormal" style="margin-right:0.0359em;">Ï</span></span></span></span>, we can express the rotational energy in terms of radius and velocity, just like the translational energy:</p> 124 125<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>E</mi><mtext>r</mtext></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>â</mtext><mi>I</mi><msup><mi>Ï</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mtext>â</mtext><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">E_\text{r} = \frac{1}{2}\,I \omega^2 = \frac{1}{5}\,m v^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">r</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">Ï</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span> 126 127<p>Which leads to a fixed ratio of translational and rotational energy:</p> 128 129<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><msub><mi>E</mi><mtext>t</mtext></msub><msub><mi>E</mi><mtext>r</mtext></msub></mfrac><mo>=</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{E_\text{t}}{E_\text{r}} = \frac{5}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1963em;vertical-align:-0.836em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">r</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">t</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> 130 131<p>In other words: Even for a marble on a regular slope, a part of the energy goes into the rotation, leaving less energy âfor the speedâ.</p> 132 133<h2 id="two-radii">Two radii</h2> 134 135<p>But I did not want to print a slope. I planned for slopes that would work across a third of a Lego brickâs height (the flat Lego bricks), so I would need a rather gentle slope, which is tricky to print. So, instead I decided to design rails (or rather just a gap) that widen along the track. As the rail distance <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span> increases, the marble sinks deeper between the rails, its center of mass sinks deeper and once again, it accelerates as its potential energy is converted into kinetic energy. Just like a marble on a sloped track, right?</p> 136 137<figure> 138 139<img src="/assets/resized/images/2026-08-14/1024/openscad.png" style="max-width: min(1092px, 50%)" alt="Screenshot of a track piece in OpenSCAD. The image shows coordinate axes and markers for scale, but the main feature of the image is the track piece itself. It has the shape of a 4 by 2 Lego brick, but the top is flat with a widening gap in which the marble can run. Below the gap in the top layer is another layer with a smaller gap, which is the track on which the marble runs once the upper track has widened enough for the marble to reach the lower one." srcset=" /assets/resized/images/2026-08-14/640/openscad.png 640w, /assets/resized/images/2026-08-14/768/openscad.png 768w, /assets/resized/images/2026-08-14/1024/openscad.png 1024w, /assets/images/2026-08-14/openscad.png 1092w" sizes="(max-width: 60rem) calc(50/100 * 100vw), calc(50/100 * 60rem)" /> 140 141<figcaption>The top of this 4 by 2 brick is flat, but the marble's center of mass will sink into the widening gap at the top. You can also see a layer below with a smaller gap. When the top gap has widened enough, the marble will sit on the lower track and continue there.</figcaption> 142</figure> 143 144<p>Not quite. There is a small difference. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>E</mi><mtext>r</mtext></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>â</mtext><mi>I</mi><msup><mi>Ï</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">E_\text{r} = \frac{1}{2}\,I \omega^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">r</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.1
1445em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">Ï</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> is universal. Thatâs still correct. And so is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo>=</mo><mfrac><mn>2</mn><mn>5</mn></mfrac><mtext>â</mtext><mi>m</mi><msup><mi>R</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">I = \frac{2}{5}\,m R^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> because we are still looking at a solid sphere. But <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>=</mo><mi>R</mi><mi>Ï</mi></mrow><annotation encoding="application/x-tex">v = R \omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="mord mathnormal" style="margin-right:0.0359em;">Ï</span></span></span></span> does not apply!</p> 145 146<p>As the marble sinks between the tracks, it rolls about a smaller effective radius <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span></span></span></span>. The marble touches the two rails at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>±</mo>
146<mi>w</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">\pm w/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">±</span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mord">/2</span></span></span></span>. Both contact points lie on a sphere with the marbleâs radius <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span></span></span></span>, so the distance from the marbleâs spin axis (through its center) to the contacts is not <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span></span></span></span> but</p> 147 148<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>r</mi><mo>=</mo><msqrt><mrow><msup><mi>R</mi><mn>2</mn></msup><mo>â</mo><msup><mrow><mo fence="true">(</mo><mfrac><mi>w</mi><mn>2</mn></mfrac><mo fence="true">)</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">r = \sqrt{R^2 - \left(\frac{w}{2}\right)^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.44em;vertical-align:-0.7721em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6679em;"><span class="svg-align" style="top:-4.4em;"><span class="pstrut" style="height:4.4em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">â</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.354em;"><span style="top:-3.6029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.6279em;"><span class="pstrut" style="height:4.4em;"></span><span class="hide-tail" style="min-width:1.02em;height:2.48em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="2.48em" viewBox="0 0 400000 2592" preserveAspectRatio="xMinYMin slice"><path d="M424,2478 149c-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514 150c0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20
151s-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121 152s209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081 153l0 -0c4,-6.7,10,-10,18,-10 H400000 154v40H1014.6 155s-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185 156c-2,6,-10,9,-24,9 157c-8,0,-12,-0.7,-12,-2z M1001 80 158h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.7721em;"><span></span></span></span></span></span></span></span></span></span> 159 160<figure> 161 162<img src="/assets/resized/images/2026-08-14/1280/sideviews.png" style="max-width: min(3840px, 100%)" alt="Schematic of the marbles position on the rails. The left half of the image shows a view along the rails where it is clearly visible how the marble sits between the rails. The right half shows a view from the side where it is easier to see how the marble is rolling at an effectively smaller radius." srcset=" /assets/resized/images/2026-08-14/640/sideviews.png 640w, /assets/resized/images/2026-08-14/768/sideviews.png 768w, /assets/resized/images/2026-08-14/1024/sideviews.png 1024w, /assets/resized/images/2026-08-14/1280/sideviews.png 1280w, /assets/images/2026-08-14/sideviews.png 3840w" sizes="(max-width: 60rem) 100vw, 60rem" /> 163 164<figcaption>Left: As the marble sits between the tracks, it rolls with a smaller effective radius. Right: Viewing from the side, one can see that the smaller radius corresponds to a slower speed at the same angular velocity. The distance per rotation is smaller as indicated by the blue dots on the track.</figcaption> 165</figure> 166 167<p>So, our marble still has the same velocity <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span></span></span>, we still have the same translational energy. And at the same angular velocity <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Ï</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">Ï</span></span></span></span>, we still have the same rotational energy. But the link between <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Ï</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">Ï</span></span></span></span> has changed from <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>=</mo><mi>R</mi><mi>Ï</mi></mrow><annotation encoding="application/x-tex">v = R\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="mord mathnormal" style="margin-right:0.0359em;">Ï</span></span></span></span> to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>=</mo><mi>r</mi><mi>Ï</mi></mrow><annotation encoding="application/x-tex">v = r\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mord mathnormal" style="margin-right:0.0359em;">Ï</span></span></span></span>, resulting in a different ratio of the two forms of kinetic energy:</p> 168 169<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><msub><mi>E</mi><mtext>t</mtext></msub><msub><mi>E</mi><mtext>r</mtext></msub></mfrac><mo>=</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mfrac><msup><mi>r</mi><mn>2</mn></msup><msup><mi>R</mi><mn>2</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\frac{E_\text{t}}{E_\text{r}} = \frac{5}{2}\frac{r^2}{R^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1963em;vertical-align:-0.836em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">r</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0576em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">t</span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0077em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">â</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> 170 171<p>The ratio is tuned by the rail distance <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span>. If <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span></span></span></span> is zero, we get <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>=</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r=R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span></span></span></span> and are back to the behavior of a marble running on a flat surface. But what happens if <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi><mo>
171â</mo><mn>2</mn><mi>R</mi></mrow><annotation encoding="application/x-tex">w \to 2R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">â</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span></span></span></span>?</p> 172 173<h2 id="slow-tracks">Slow tracks</h2> 174 175<p>As the rail distance becomes larger and larger, the effective radius <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">r</span></span></span></span> becomes smaller and smaller. In terms of energy, most energy now goes into rotational energy. Visually, the marble only moves a small distance per rotation, so it has to spin really fast to achieve the same velocity as before.</p> 176 177<p>We can use this to create slow tracks. If we support the marble with different rail distances, such that its center of mass remains at the same height (so, wider rails need to be higher to support the marble), we still have the same total energy - the energy that was potential energy when we let go of the marble at the top of our marble run. But with different rail distances, the same energy is distributed across translational and rotational energy differently, leading to different velocities.</p> 178 179<p>If we assume no slippage and no friction, the total energy remains the same. So, we could have a fast-spinning marble that barely moves forward on a wide rail distance, and then have it suddenly speed up when moving onto close rails where it is almost rolling on a surface.</p> 180 181<figure> 182 183<img src="/assets/resized/images/2026-08-14/1280/rail_velocity_plot.png" style="max-width: min(1500px, 100%)" alt="Plot of the velocity as a function of the rail distance. It is a curve that starts parallel to the x axis and then slowly descents until it rather quickly drops to zero when the rail distance approaches the width of the marble of 14mm." srcset=" /assets/resized/images/2026-08-14/640/rail_velocity_plot.png 640w, /assets/resized/images/2026-08-14/768/rail_velocity_plot.png 768w, /assets/resized/images/2026-08-14/1024/rail_velocity_plot.png 1024w, /assets/resized/images/2026-08-14/1280/rail_velocity_plot.png 1280w, /assets/images/2026-08-14/rail_velocity_plot.png 1500w" sizes="(max-width: 60rem) 100vw, 60rem" /> 184 185<figcaption>Marble velocity as a function of rail distance after a drop of 10mm.</figcaption> 186</figure> 187 188<h2 id="friction-ruining-the-day">Friction ruining the day</h2> 189 190<p>At the limit <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi><mo>â</mo><mn>2</mn><mi>R</mi></mrow><annotation encoding="application/x-tex">w \to 2R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0269em;">w</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">â</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0077em;">R</span></span></span></span>, the marble no longer moves, but rotates in position. Of course, at that point it is clear that we will not see that in reality. The problem is of course friction. Our intuition already expects that the marble will just get stuck between the rails. And the reason is again a geometrical one. When the marble is rolling on the top of a surface, it experiences rolling resistance, which is proportional to the normal force acting on the marble. When it is more or less stuck between the rails, not only can it be disputed if this is still rolling resistance or rather sliding friction (which is a stronger opposing force), but also the surface now has to apply a much stronger normal force to counter gravity as it has to do so at an angle. Friction becomes really strong and our marble comes to a halt.</p> 191 192<p>So, yeah, do not expect too much from those slow tracks. I did not widen the tracks enough to get more than a subtle effect, because otherwise we would lose our precious kinetic energy to friction.</p> 193 194<figure> 195
196<video autoplay="" loop="" muted="" playsinline=""> 197 <source src="/assets/images/2026-08-14/marble-loop.mp4" type="video/mp4" /> 198</video> 199 200<figcaption>The slowdown during the wider rail distance on the yellow track is subtle, but noticable.</figcaption> 201</figure> 202 203<p>Hope you enjoyed the little physics lesson - or just the marble run. See you for the next project.</p> 204 205<div class="footnotes" role="doc-endnotes"> 206 <ol> 207 <li id="fn:printables"> 208 <p>My favorite. Well, have not heard anything bad about printables.com. Please donât tell me about it. <a href="#fnref:printables" class="reversefootnote" role="doc-backlink">↩</a></p> 209 </li> 210 <li id="fn:thingiverse"> 211 <p>The old big one. If you want to learn how to lose being the dominant platform, have a look at thingiverse. Still plenty of reach. <a href="#fnref:thingiverse" class="reversefootnote" role="doc-backlink">↩</a></p> 212 </li> 213 <li id="fn:makerworld"> 214 <p>How I hate myself for including them. I bought a Bambulab printer and wish I had not. No, the machine is fine (if you ignore that I got one of the first that do not <a href="https://www.reddit.com/r/BambuLab/comments/1q1git2/new_a1_without_the_failing_ntc_thermistor/" target="_blank">spontaneously burn and die</a>), but that company goes against everything I believe in. If you follow this blog, you certainly know that I love open source, open platforms and open APIs. Since I got the printer, I could witness how Bambulab closes down their APIs, sues (or threatens to sue?) open source developers, violates open source licenses and does everything they can to build a walled garden. Oh, and that platform is full of AI slop and stolen models that users upload because of the incentive of free filament. <a href="#fnref:makerworld" class="reversefootnote" role="doc-backlink">↩</a></p> 215 </li> 216 </ol> 217</div> 218 219 220 <footer> 221 <div id="social"> 222 <a class="social" href="https://youtube.com/c/thereoughtabe" title="youtube"><img src="/assets/images/social/youtube.svg" alt="youtube.com" /></a> 223 <a class="social" rel="me" href="https://mastodon.social/@DiConX" title="mastodon"><img src="/assets/images/social/mastodon.svg" alt="mastodon.social" /></a> 224 <a class="social" href="https://bsky.app/profile/there.oughta.be" title="bluesky"><img src="/assets/images/social/bsky.svg" alt="bsky.app" /></a> 225<!-- <a class="social" href="https://twitter.com/diconx" title="twitter"><img src="/assets/images/social/twitter.svg" alt="twitter.com" /></a> --> 226 <a class="social" href="https://www.instagram.com/there.oughta.be/" title="instagram"><img src="/assets/images/social/instagram.svg" alt="instagram.com" /></a> 227 <a class="social" href="https://www.threads.net/@there.oughta.be" title="threads"><img src="/assets/images/social/threads.svg" alt="threads.net" /></a> 228 <a class="social" href="https://www.reddit.com/r/thereoughtabe/" title="reddit"><img src="/assets/images/social/reddit.svg" alt="reddit.com" /></a> 229 <a class="social" href="https://github.com/Staacks" title="github"><img src="/assets/images/social/github.svg" alt="github.com" /></a> 230 <a class="social" href="/feed.xml" title="RSS-Feed"><img src="/assets/images/social/rss.svg" alt="RSS-Feed" /></a> 231 </div> 232 <div class="split"> 233 <a id="coffee" href="https://buymeacoffee.com/there.oughta.be"><img src="/assets/images/social/buymeacoffee.svg" alt="buymeacoffee.com" />Buy me a coffee.</a> 234 <a id="privacy" href="/an/about/page">Impressum / Privacy Policy</a> 235 </div> 236</footer> 237 238 </body> 239</html>
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