<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="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?">
<meta property="og:title" content="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?">
<meta property="og:image" content="img/logo.png">
<meta property="og:description" content="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?">
<title>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?</title>
$1.3.15.$ 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>
@@ -57,20 +57,20 @@
<p>
<p>We will divide the movement into vertical and in the plane of the cross section of the cylinder</p>
<p class="exp">
−
$x = vt \sin\alpha$
+
$x = vt \sin\alpha$
</p>
<p class="exp">
−
$$ y = vt \cos\alpha - \frac{gt^2}{2}$$
+
$$ y = vt \cos\alpha - \frac{gt^2}{2}$$
</p>
<p>We take into account that the displacement along $Ox$ is equal to the circumference, and along $Oy$ — 0:</p>
<p class="exp">
−
$$ vt \cos\alpha - \frac{gt^2}{2} = 0 $$
+
$$ vt \cos\alpha - \frac{gt^2}{2} = 0 $$
</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
−
$$ t = \frac{2\pi n R}{v \sin\alpha}$$
−
$$ 2v \cos\alpha = g\frac{2\pi n R}{v \sin\alpha}$$
+
$$ t = \frac{2\pi n R}{v \sin\alpha}$$
+
$$ 2v \cos\alpha = g\frac{2\pi n R}{v \sin\alpha}$$
<p>From here we express $v$:
<p class="exp">
$$\fbox{$v=\sqrt{2\pi Rgn/\sin2\alpha}$}$$
</p>
</p>
<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.
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="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?">
<meta name="description" content="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?">
<meta property="og:title" content="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?">
<meta property="og:title" content="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?">
<meta property="og:image" content="img/logo.png">
<meta property="og:image" content="img/logo.png">
<meta property="og:description" content="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?">
<meta property="og:description" content="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?">
<title>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?</title>
<title>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?</title>
$1.3.15.$ 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>
@@ -57,20 +57,20 @@
<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.
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
