A horizontal trough to the left of the bottom line is bent on a cylindrical surface of radius r, and to the right, on a surface of radius R. Determine the ratio of the largest deflections to the left and to the right during small oscillations of the body in this trough.

Savchenko Solutions

Solutions of Savchenko Problems in Physics
knowledge must be free

    <h3 id="back-link"><a href="/#3.1">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $3.1.7.$ A horizontal trough to the left of the bottom line is bent on a cylindrical surface of radius $r$, and to the right, on a surface of radius $R$. Determine the ratio of the largest deflections to the left and to the right during small oscillations of the body in this trough.

    <h3>Solution</h3>
    <p>
        According to the law of conservation of body energy, the maximum potential energy will be conserved in both cases

$$E_{p1}=E_{p2}$$
And therefore the height $h$ will rise relative to the lower point of the trajectory to which the body will rise
will be equal

We will use the expression $\Delta U$ for small oscillations obtained in 3.1.5 $$\Delta U = \frac{mgx_1^2}{2r}=\frac{mgx_2^2}{2R}$$ Whence it follows that $$\boxed{\frac{x_2}{x_1}=\sqrt{\frac{R}{r}}}$$

    </p>

    <h4>Answer</h4>
    <p>
        $$\frac{x_2}{x_1}=\sqrt{\frac{R}{r}}$$
    </p>
	<p style="text-align: right; font-style: italic; font-size: 14;">
      Dzikan Mikita<br>
    </p>


<footer class="row container">
  <br>
    <p>
        <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small>
    </p>
    <p>
        <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>
    </p>
</footer>