Physics, 06.01.2021 18:30 bcarri4073

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?

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The correct expression for the time T it takes the satellite to make one complete revolution around the moon is $T = 2\pi R\sqrt{\frac{R}{GM}}$ .

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

$F = \frac{GmM}{R^{2}}$ (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:

$F = ma_{c}$ (2)

The centripetal acceleration ( $a_{c}$ ) is equal to the tangential velocity (v):

$a_{c} = \frac{v^{2}}{R}$ (3)

And from the tangential velocity we can find the period:

$v = \omega R = \frac{2\pi R}{T}$ (4)

Where:

ω: is the angular speed = 2π/T

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

$F = m\frac{v^{2}}{R} = m\frac{(\frac{2\pi R}{T})^{2}}{R} = \frac{mR(2\pi)^{2}}{T^{2}}$ (5)

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

$\frac{mR(2\pi)^{2}}{T^{2}} = \frac{GmM}{R^{2}}$

$T^{2} = \frac{R^{3}(2\pi)^{2})}{GM}$

$T = \sqrt{\frac{R^{3}(2\pi)^{2})}{GM}}$

$T = 2\pi R\sqrt{\frac{R}{GM}}$

Therefore, the expression for the time T is $T = 2\pi R\sqrt{\frac{R}{GM}}$ .

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Answer from: Thrillseeker886

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 ))

A satellite of mass m orbits a moon of mass M in uniform circular motion with a constant tangential

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