College Physics (4th Edition)

Published by McGraw-Hill Education
ISBN 10: 0073512141
ISBN 13: 978-0-07351-214-3

Chapter 8 - Problems - Page 316: 68


The minimum starting height is $~3 r$

Work Step by Step

To find the minimum height $h$ where the cylinder will not lose contact with the track, we can assume that the normal force is zero and the gravitational force provides the centripetal force to keep the cylinder moving in a circle. We can find the speed at the top of the loop: $\frac{Mv^2}{r} = Mg$ $v = \sqrt{gr}$ At the top of the loop, the cylinder also has rotational kinetic energy. We can use conservation of energy to find the minimum starting height $h$: $Mgh = Mgh_2+\frac{1}{2}Mv^2+ \frac{1}{2}I~\omega^2$ $Mgh = Mgh_2+\frac{1}{2}Mv^2+ \frac{1}{2}(MR^2)(\frac{v}{R})^2$ $Mgh = Mgh_2+\frac{1}{2}Mv^2+ \frac{1}{2}Mv^2$ $Mgh = Mgh_2+Mv^2$ $h = h_2+\frac{v^2}{g}$ $h = 2r+\frac{(\sqrt{gr})^2}{g}$ $h = 2r+\frac{gr}{g}$ $h = 2r+r$ $h = 3 r$ The minimum starting height is $~3 r$.
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