Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition)

Published by Pearson
ISBN 10: 0133942651
ISBN 13: 978-0-13394-265-1

Chapter 8 - Dynamics II: Motion in a Plane - Exercises and Problems: 20

Answer

The ratio of the normal force to the gravitational force is 3.

Work Step by Step

The critical speed is the speed at which the centripetal acceleration is equal to $g$. We can find an expression for the critical speed as; $a_c = \frac{v^2}{r} = g$ $v = \sqrt{g~r}$ Note that at the top of the loop, the normal force from the track is directed straight down. We can use an equation for the centripetal force to find an expression for the normal force $F_N$. We can let the speed $v = 2\sqrt{g~r}$. Therefore; $\sum F = \frac{mv^2}{r}$ $F_N+mg = \frac{(m)(2\sqrt{g~r})^2}{r}$ $F_N+mg = 4~mg$ $F_N = 3~mg$ The ratio of the normal force to the gravitational force is 3.
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