University Physics with Modern Physics (14th Edition)

Published by Pearson
ISBN 10: 0321973615
ISBN 13: 978-0-32197-361-0

Chapter 9 - Rotation of Rigid Bodies - Problems - Discussion Questions - Page 294: Q9.14

Answer

If the hollow sphere is to be modified so that the kinetic energy triples while the mass stays the same, the radius should be increased by a factor of $\sqrt{3}$.

Work Step by Step

The moment of inertia $I=\frac{2}{3}MR^2$ for a hollow sphere of radius R, and the kinetic energy is $KE=\frac{1}{2}I\omega^2=\frac{1}{3} MR^2\omega^2$. If the sphere is to be modified so that the kinetic energy triples while the mass stays the same, the radius should be increased by a factor of $\sqrt{3}$. $$KE_{old}=\frac{1}{3} MR_{old}^2\omega^2$$ $$KE_{new}=\frac{1}{3} M(R_{old}\sqrt3)^2\omega^2=3 KE_{old}$$
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