Essential University Physics: Volume 1 (4th Edition)

Published by Pearson
ISBN 10: 0-134-98855-8
ISBN 13: 978-0-13498-855-9

Chapter 10 - Exercises and Problems - Page 194: 72

Answer

$(a)\space P = (\frac{M}{2}+m)Va$ $(b)\space 1.1\%$ $(c)\space 0.28\%$

Work Step by Step

Please see the image below. (a) Let's apply equation $V=u +at$ $\rightarrow V=u+at$ $V=0+at$ $V=at-(1)$ Let's apply the equation of conservation of kinetic energy. Total kinetic energy = Linear kinetic energy + Rotational kinetic energy $E=\frac{1}{2}MV^{2}+\frac{1}{2}I\omega^{2}\times4-(2)$ $E=\frac{1}{2}MV^{2}+2I\omega^{2}$ For a solid disk, $I=\frac{1}{2}MR^{2}-(3)$ $(3)=\gt (2)$ $E=\frac{1}{2}MV^{2}+\frac{1}{2}mR^{2}\times\frac{V^{2}}{R^{2}}=V^{2}(\frac{M}{2}+m)-(3)$ Rate of change in kinetic energy = E - 0 = E Power (P) $=\frac{E}{t}$ $(3)=\gt P=\frac{V^{2}}{t}(\frac{M}{2}+m)=V\times \frac{V}{t}(\frac{M}{2}+m)$ $(1)=\gt P=Va(\frac{M}{2}+m)$ (b) Reduce the mass by 10 kg from non-rolling parts, $P_{1}=Va[\frac{(M-10)}{2}+m]=Va(\frac{M}{2}+m)-10Va$ $P_{1}=P-10Va$ Percentage of power decrease $= \frac{P-P_{1}}{P}\times 100\%$ $\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space=\frac{10Va}{Va(\frac{M}{2}+m)\times}\times100\%$ $\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space=\frac{1000}{\frac{1780}{2}+15.8}=1.1\%$ (c) Reduce the mass by 10 kg from rolling parts. $P_{2}=Va[\frac{M}{2}+(m-2.5)]$ $P_{2}=Va[\frac{M}{2}+m]-2.5Va$ $2.5Va=P-P_{2}$ Percentage of power decrease $= \frac{P-P_{2}}{P}\times 100\%$ $\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space=\frac{2.5Va}{Va(\frac{M}{2}+m)\times}\times100\%$ $\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space=\frac{250}{\frac{1780}{2}+15.8}=0.28\%$
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