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| <h3 id="back-link"><a href="../../#1.3">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.3">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.3.15.$ A ball is launched along the inner surface of a smooth vertical cylinder of radius $R$ at an angle $\alpha$ to the vertical. What initial velocity must be given to it so that it returns to the starting point? | | $1.3.15.$ A ball is launched along the inner surface of a smooth vertical cylinder of radius $R$ at an angle $\alpha$ to the vertical. What initial velocity must be given to it so that it returns to the starting point? |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p>We will divide the movement into vertical and in the plane of the cross section of the cylinder</p> | | <p>We will divide the movement into vertical and in the plane of the cross section of the cylinder</p> |
| <p class="exp"> | | <p class="exp"> |
| $x = vt \sin \alpha$ | | $x = vt \sin \alpha$ |
| </p> | | </p> |
| <p class="exp"> | | <p class="exp"> |
| $$ y = vt \cos \alpha - \frac{gt^2}{2} $$ | | $$ y = vt \cos \alpha - \frac{gt^2}{2} $$ |
| </p> | | </p> |
| <p>We take into account that the displacement along $Ox$ is equal to the circumference, and along $Oy$ — 0:</p> | | <p>We take into account that the displacement along $Ox$ is equal to the circumference, and along $Oy$ — 0:</p> |
| | | |
| <p class="exp"> | | <p class="exp"> |
| $$ vt \cos \alpha - \frac{gt^2}{2} = 0 $$ | | $$ vt \cos \alpha - \frac{gt^2}{2} = 0 $$ |
| </p> | | </p> |
| $$vt \sin \alpha = 2\pi n R,\,n\in \mathbb{N}$$ | | $$vt \sin \alpha = 2\pi n R,\,n\in \mathbb{N}$$ |
| We find time $t$ as | | We find time $t$ as |
| $$ t = \frac{2\pi n R}{v \sin \alpha} $$ | | $$ t = \frac{2\pi n R}{v \sin \alpha} $$ |
| $$ 2v \cos \alpha = g\frac{2\pi n R}{v \sin \alpha} $$ | | $$ 2v \cos \alpha = g\frac{2\pi n R}{v \sin \alpha} $$ |
| <p>From here we express $v$: | | <p>From here we express $v$: |
| <p class="exp"> | | <p class="exp"> |
| $$\fbox{$v=\sqrt{2\pi Rgn/\sin2\alpha}$}$$ | | $$\fbox{$v=\sqrt{2\pi Rgn/\sin2\alpha}$}$$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $v=\sqrt{2\pi Rgn/\sin2\alpha}$, where $n$ is any natural number; when $\alpha = 0$, the speed can be any in absolute value. | | $v=\sqrt{2\pi Rgn/\sin2\alpha}$, where $n$ is any natural number; when $\alpha = 0$, the speed can be any in absolute value. |
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