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 12 - Rotation of a Rigid Body - Exercises and Problems - Page 349: 33

Answer

$0.375\;\rm J$

Work Step by Step

Since the can is filled with uniform dense food, its center of mass is at its geometric center. Now we have two motions of the can while rolling on the floor, the translational motion ($v_{cm}=1$ m/s) and the rotational motion. This means that the can's kinetic energy is given by $$K_{tot}=K_{translational }+K_{rotational}$$ $$K_{tot}=\frac{1}{2}mv_{cm}^2+\frac{1}{2}I\omega^2$$ where $\omega$ is the can's angular velocity. We know that the linear speed is given by $v=R\omega$, Hence $\omega=v/R$. $$K_{tot}=\frac{1}{2}mv_{cm}^2+\frac{1}{2}I\left[\dfrac{v_{cm}}{R}\right]^2$$ $$K_{tot}=\frac{1}{2}mv_{cm}^2+\frac{1}{2}I \dfrac{v_{cm}^2}{R^2} $$ Recall that the moment of inertia of a disk or a cylinder is $I=\frac{1}{2}mR^2$, and the can is a solid cylinder. Thus, $$K_{tot}=\frac{1}{2}mv_{cm}^2+\frac{1}{2}\left(\frac{1}{2}mR^2\right)\dfrac{v_{cm}^2}{R^2} $$ $$K_{tot}=\frac{1}{2}mv_{cm}^2+\frac{1}{4} m \color{red}{\bf\not}R^2 \dfrac{v_{cm}^2}{ \color{red}{\bf\not}R^2} $$ $$K_{tot}=\frac{1}{2}mv_{cm}^2+\frac{1}{4} mv_{cm}^2= \frac{3}{4} mv_{cm}^2 $$ Plugging the given; $$K_{tot}= \frac{3}{4} (0.500)(1.0^2)=\color{red}{\bf 0.375}\;\rm J $$
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