Question

A satellite of mass m orbits a moon of mass M in uniform circular motion with a constant tangential speed of v. The satellite orbits at a distance R from the center of the moon. Write down the correct expression for the time T it takes the satellite to make one complete revolution around the moon?

Answer 1

The time T for a satellite to complete one orbit around a moon is found by dividing the circumference of the orbit (2πR) by the constant tangential speed v.

Step-by-step explanation:

To find the period T of a satellite orbiting a moon, we can use Kepler's third law which relates the time for one orbit to the radius of the orbit. In this case, we are given that the satellite orbits at a distance R from the center of the moon with a constant speed v. To find the time T it takes the satellite to make one complete revolution, we first need to calculate the circumference of the orbit using 2πR, where π is approximately 3.1415926. Then we divide the circumference by the constant tangential speed v to find T, which is the time for one complete orbit.

The correct expression to find T is:

T = πR / v

Answer 2

The gravitational force exerted by the moon on the satellite is such that

F = G M m / R ² = m a → a = G M / R ²

where a is the satellite's centripetal acceleration, given by

a = v ² / R

The satellite travels a distance of 2πR about the moon in complete revolution in time T, so that its tangential speed is such that

v = 2πR / T → a = 4π ² R / T ²

Substitute this into the first equation and solve for T :

4π ² R / T ² = G M / R ²

4π ² R ³ = G M T ²

T ² = 4π ² R ³ / (G M )

T = √(4π ² R ³ / (G M ))

T = 2πR √(R / (G M ))