# Chapter 5 - Problems - Page 187: 35

The second satellite takes 128 hours to orbit Jupiter.

#### Work Step by Step

When a satellite orbits a planet, the gravitational force provides the centripetal force to keep the satellite moving in a circle. Let $M_p$ be the mass of the planet and let $M_s$ be the mass of the satellite. We can find an expression for the angular speed of a satellite: $\frac{G~M_p~M_s}{R^2} = M_s~\omega^2~R$ $\omega = \sqrt{\frac{G~M_p}{R^3}}$ We can find an expression for a satellite's orbital period $P$: $P = \frac{2\pi}{\omega} = 2\pi~\sqrt{\frac{R^3}{G~M_p}}$ We can find an expression for the period $P_1$ of the first satellite: $P_1 = 2\pi~\sqrt{\frac{r^3}{G~M_p}} = 16~h$ We can find the period $P_2$ of the other satellite: $P_2 = 2\pi~\sqrt{\frac{(4.0~r)^3}{G~M_p}}$ $P_2 = 8\times 2\pi~\sqrt{\frac{r^3}{G~M_p}}$ $P_2 = 8\times P_1$ $P_2 = (8)(16~h)$ $P_2 = 128~h$ The second satellite takes 128 hours to orbit Jupiter.

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