College Physics (4th Edition)

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

Chapter 5 - Problems - Page 189: 67


The spacecraft's orbital period is $57,600~seconds$

Work Step by Step

When a spacecraft orbits a planet, the gravitational force provides the centripetal force to keep the spacecraft moving in a circle. Let $M_p$ be the mass of the planet and let $M_s$ be the mass of the spacecraft. We can find the angular speed of a spacecraft: $\frac{G~M_p~M_s}{(3.0~R_J)^2} = M_s~\omega^2~(3.0~R_J)$ $\frac{G~M_p}{R_J^2} = 27~\omega^2~R_J$ $\omega^2 = (\frac{1}{27~R_J})(\frac{G~M_p}{R_J^2})$ $\omega = \sqrt{(\frac{1}{27~R_J})(\frac{G~M_p}{R_J^2})}$ $\omega = \sqrt{(\frac{1}{(27)(7.15\times 10^7~m)})(23~N/kg)}$ $\omega = 1.09\times 10^{-4}~rad/s$ We can find the spacecraft's orbital period $P$: $P = \frac{2\pi}{\omega}$ $P = \frac{2\pi}{1.09\times 10^{-4}~rad/s}$ $P = 57,600~s$ The spacecraft's orbital period is $57,600~seconds$
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