University Physics with Modern Physics (14th Edition)

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

Chapter 7 - Potential Energy and Energy Conservation - Problems - Exercises - Page 229: 7.17

Answer

(a) (i) $U = 4\times U_0$ (ii) $U = 0.25\times U_0$ (b) (i) The spring should be compressed $\sqrt{2}~x_0$. (ii) The spring should be compressed $\frac{x_0}{\sqrt{2}}$.

Work Step by Step

$U_0 = \frac{1}{2}kx_0^2$ (a) (i) $U = \frac{1}{2}k(2~x_0)^2$ $U = 4\times \frac{1}{2}kx_0^2 = 4\times U_0$ (ii) $U = \frac{1}{2}k(\frac{x_0}{2})^2$ $U = \frac{1}{4}\times \frac{1}{2}kx_0^2 = 0.25\times U_0$ (b) (i) $2\times U_0 = 2\times \frac{1}{2}kx_0^2$ $2\times U_0 = \frac{1}{2}k(\sqrt{2}~x_0)^2$ The spring should be compressed $\sqrt{2}~x_0$. (i) $\frac{1}{2}\times U_0 = \frac{1}{2}\times \frac{1}{2}kx_0^2$ $\frac{1}{2}\times U_0 = \frac{1}{2}k(\frac{x_0}{\sqrt{2}})^2$ The spring should be compressed $\frac{x_0}{\sqrt{2}}$.
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