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?

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Solutions of Savchenko Problems in Physics
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    <h3 id="back-link"><a href="/#1.3">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <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?
    </p>

    <h3>Solution</h3>
    <p>
        <p>We will divide the movement into vertical and in the plane of the cross section of the cylinder</p>

$x = vt \sin \alpha$

$$y = vt \cos \alpha - \frac{gt^2}{2}$$

We take into account that the displacement along $Ox$ is equal to the circumference, and along $Oy$ — 0:

$$vt \cos \alpha - \frac{gt^2}{2} = 0$$

$$vt \sin \alpha = 2\pi n R,\\,n\in \mathbb{N}$$ We find time $t$ as $$t = \frac{2\pi n R}{v \sin \alpha}$$ $$2v \cos \alpha = g\frac{2\pi n R}{v \sin \alpha}$$

From here we express $v$:

$$\fbox{$v=\sqrt{2\pi Rgn/\sin2\alpha}$}$$

    <h4>Answer</h4>
    <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.

    </p>


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