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?

The correct expression for the time T it takes the satellite to make one complete revolution around the moon is .

We can find the period T (the time it takes the satellite to make one complete revolution around the moon) from the gravitational force:

   (1)

Where:

G: is the gravitational constant = 6.67x10⁻¹¹ Nm²/kg²

R: is the distance between the satellite and the center of the moon

m: is the satellite's mass

M: is the moon's mass

The gravitational force is also equal to the centripetal force:

  (2)

The centripetal acceleration () is equal to the tangential velocity (v):

  (3)

And from the tangential velocity we can find the period:

  (4)

Where:

ω: is the angular speed = 2π/T

By entering equations (4) and (3) into (2), we have:

  (5)

By equating (5) and (1), we get:

Therefore, the expression for the time T is .

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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 ))