Question

18 if the satellite is put into a new circular orbit of a smaller radius, how will the force of gravity and the speed of the satellite change?
a. both the gravitational force and the speed will decrease.
b. the gravitational force will decrease, and the speed will increase.
c. the gravitational force will increase, and the speed will decrease.
d. both the gravitational force and the speed will increase.

Explanation:

Step1: Gravitational force formula

The gravitational force \(F = G\frac{Mm}{r^{2}}\), where \(G\) is the gravitational constant, \(M\) is the mass of the central body (e.g., Earth), \(m\) is the mass of the satellite, and \(r\) is the radius of the orbit. When \(r\) decreases (new orbit of smaller radius), since \(F\propto\frac{1}{r^{2}}\), the gravitational force \(F\) increases.

Step2: Satellite speed formula

For a satellite in circular orbit, the centripetal force is provided by the gravitational force. So \(G\frac{Mm}{r^{2}}=m\frac{v^{2}}{r}\), solving for \(v\) gives \(v = \sqrt{\frac{GM}{r}}\). When \(r\) decreases, since \(v\propto\frac{1}{\sqrt{r}}\), the speed \(v\) increases.

Answer:

D. Both the gravitational force and the speed will increase.