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| <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.5.10.$ On the inner surface of a fixed cylinder of radius $2r$, a wheel of radius $r$ rolls without slipping. Find the path of the wheel rim point. | | $1.5.10.$ On the inner surface of a fixed cylinder of radius $2r$, a wheel of radius $r$ rolls without slipping. Find the path of the wheel rim point. |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.5.10/statement.png" | | <img src="https://savchenkosolutions.com/1/1.5.10/statement.png" |
| loading="lazy" width="150" /> | | loading="lazy" width="150" /> |
| <figcaption> | | <figcaption> |
| For problem $1.5.10$ | | For problem $1.5.10$ |
| </figcaption> | | </figcaption> |
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| <p> | | <p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
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| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| <figcaption> | | <figcaption> |
| Construction of a hypocycloid at $r= \frac{R}{2}$ | | Construction of a hypocycloid at $r= \frac{R}{2}$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| This trajectory is the so-called <a href="https://en.wikipedia.org/wiki/Hypocycloid" target="_blank">Hypocycloid</a><br><br> | | This trajectory is the so-called <a href="https://en.wikipedia.org/wiki/Hypocycloid" target="_blank">Hypocycloid</a><br><br> |
| Given by the parametric equation: | | Given by the parametric equation: |
| </p> | | </p> |
| <p class="exp"> | | <p class="exp"> |
| $$ \begin{gathered}\begin{cases}x=r(k-1)\left(\cos t+\frac{\cos((k-1)t)}{k-1}\right)\\y=r(k-1)\left(\sin t-\frac{\sin((k-1)t)}{k-1}\right)&\end{cases}\end{gathered} $$ | | $$ \begin{gathered}\begin{cases}x=r(k-1)\left(\cos t+\frac{\cos((k-1)t)}{k-1}\right)\\y=r(k-1)\left(\sin t-\frac{\sin((k-1)t)}{k-1}\right)&\end{cases}\end{gathered} $$ |
| </p> | | </p> |
| <p> | | <p> |
| For $r= R/2$, this trajectory lies on the diameter | | For $r= R/2$, this trajectory lies on the diameter |
| </p> | | </p> |
| <h3> Proof: </h3> | | <h3> Proof: </h3> |
| <p>Let the circles initially touch at point $A$, lying on the $Ox$ axis, where point $O$ is the center of the large circle. The coordinates of point $A$ are $ (kr, 0) $, where $ k = \frac{R}{r} $. Let us consider how the coordinates of point $A$, tied to the rolling circle ($A$ goes to $ A' $), change. Let the small circle roll so that its center moves from point $ C' $ to point $ C'' $ and rotates relative to point $ O $ by angle $ t $.</p> | | <p>Let the circles initially touch at point $A$, lying on the $Ox$ axis, where point $O$ is the center of the large circle. The coordinates of point $A$ are $ (kr, 0) $, where $ k = \frac{R}{r} $. Let us consider how the coordinates of point $A$, tied to the rolling circle ($A$ goes to $ A' $), change. Let the small circle roll so that its center moves from point $ C' $ to point $ C'' $ and rotates relative to point $ O $ by angle $ t $.</p> |
| <p>First, it can be shown that the rotation of the small circle relative to its center (i.e. the angle between $ CA $ and $ C' A' $) is equal to $ t - kt = -(k-1)t $. Second, the coordinates of point $ C' $ will be as follows: $ (cos(t)(k-1)r, sin(t)(k-1)r) $. Then, knowing where the center of the rolling circle will go and at what angle it has turned relative to this center, we can write down the coordinates of point $ A' $:</p> | | <p>First, it can be shown that the rotation of the small circle relative to its center (i.e. the angle between $ CA $ and $ C' A' $) is equal to $ t - kt = -(k-1)t $. Second, the coordinates of point $ C' $ will be as follows: $ (cos(t)(k-1)r, sin(t)(k-1)r) $. Then, knowing where the center of the rolling circle will go and at what angle it has turned relative to this center, we can write down the coordinates of point $ A' $:</p> |
| <p class="exp"> | | <p class="exp"> |
| $ \begin{cases} | | $ \begin{cases} |
| x = cos(t)(k-1)r + cos((k-1)t)r \\ | | x = cos(t)(k-1)r + cos((k-1)t)r \\ |
| y = sin(t)(k-1)r - sin((k-1)t)r | | y = sin(t)(k-1)r - sin((k-1)t)r |
| \end{cases} $ | | \end{cases} $ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| The trajectory of the wheel rim point passes along the cylinder diameter. | | The trajectory of the wheel rim point passes along the cylinder diameter. |
| </p> | | </p> |
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