Phobos orbits mars in 27,553 s at a distance of 9.378 × 106 m. what is the mass of mars? 2.58 × 1011 kg 2.05 × 1023 kg 6.43 × 1023 kg 1.09 × 1030 kg

The Newton's law of gravitational force is
F = (GmM)/r²
where
F = force
G = gravitational constant = 6.67428 × 10¯¹¹ Nm²/kg²
m = mass of the first body
M = mass of the second body
r = distance between two bodies (or also called radius)

Since Phobos orbits around Mars, it has a centrifugal force. We recall the formula of centrifugal force:
F = mv²/r
where v = velocity
We know that the velocity is distance over time, that is v = d/t. Specifically, the distance here is the circumference C where we assume that Phobos travels around Mars in a circular path, so C = 2πr and the time is the period t.

Let
m = mass of Phobos
M = mass of Mars

So, plug in and arrange the quantities in terms of M:
mv² / r = (GmM) / r²
v² / r = (GM) / r²
(2πr / t)² / r = (GM) / r²
(4π²r) / t² = (GM) / r²
4π²r³ / t² = GM
M = (4π²r³) / (Gt²)
M = [4π² (9.378 × 10⁶)³] / [(6.67428 × 10¯¹¹) (27553)²]
M ≈ 6.43 × 10²³ kg (ANSWER)

There's no need to convert the units since we are provided SI units right away.

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