Chapter 12, Problem 32RE

If a particle of mass m has uniform circular motion (see Exercise 31), then the acceleration vector $\text{a}\left(t\right)$ is called the centripetal acceleration. According to Newton's second law, this acceleration must be produced by some force $\text{F}\left(t\right),$ called the centripetal force, that is related to $\text{a}\left(t\right)$ by the equation $\text{F}\left(t\right)=ma\left(t\right).$ If this force is not present, then the particle cannot undergo uniform circular motion.

(a) Show that the direction of the centripetal force varies with time but that it has constant magnitude F given by $F=\frac{m{\upsilon }^2}{R}$

(b) An astronaut with a mass of $m=60\text{ kg}$ orbits the Earth at an altitude of $h=3200\text{ km}$ with a constant speed of $\upsilon =6.43\text{ km/s}.$ Find her centripetal acceleration assuming that the radius of the Earth is 6440 km.

(c) What centripetal gravitational force in newtons does the Earth exert on the astronaut?