Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson
ISBN 10: 0321740904
ISBN 13: 978-0-32174-090-8

Chapter 14 - Oscillations - Exercises and Problems - Page 405: 57

Answer

$f=\dfrac{1}{2\pi }\sqrt{\dfrac{5g}{7R}}$

Work Step by Step

First, we need to find the moment of inertia of the sphere around the pivot point. $$I_{pivot}=I_{cm}+Md^2$$ where $d=R$, and the moment of inertia of the sphere is $\frac{2}{5}MR^2$. $$I_{pivot}=\frac{2}{5}MR^2+MR^2$$ $$I_{pivot}=\frac{7}{5}MR^2\tag 1$$ Now we need to find the frequency of our physical pendulum and we know that the angular frequency of such one is given by $$\omega=2\pi f=\sqrt{\dfrac{mgl}{I}}$$ where $m$ is the mass of the system, and $l$ is the distance from the pivot point to the center of mass of the system. So, in our case, $$ 2\pi f=\sqrt{\dfrac{MgR}{I}}$$ Plugging from (1); $$ 2\pi f=\sqrt{\dfrac{ \color{red}{\bf\not} Mg \color{red}{\bf\not} R}{\frac{7}{5} \color{red}{\bf\not} MR^{ \color{red}{\bf\not} 2}}}$$ $$ \boxed{f=\dfrac{1}{2\pi }\sqrt{\dfrac{5g}{7R}}}$$
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