Set 7 Problem number 10

A small object orbits a planet at a distance of 20000 kilometers from the center of the planet with a period of 87 minutes. What is the mass of the planet?

Solution

We know that the centripetal force required to hold the object in a circular orbit is provided by the gravitational force between the object and the planet. We write this fact as the equation

condition for circular orbit: mv ^ 2 / r = G m M / r ^ 2

where M, m and r are the massof the planet and of the satellite and the radius of the orbit about the planet's center.

Solving for the planet mass M, we obtain

planet mass = M = v ^ 2 * r / G.

We know r, and G is the universal gravitational constant, so if we can find the velocity v of the orbit object, we can find the mass M of the planet.

To find v, we need only know a distance the object travels and the time required.

We know the time required for an orbit, so if we can find the distance traveled in an orbit we will have what we require.
The radius of the circular orbit is 20000 km, so its circumference is

circumference = distance = `ds = 2 `pi ( 20000 km).

Dividing this by the 87 minutes = 5220 seconds required for an orbit, we obtain orbital velocity

v = 24060 m/s.

Substituting into the expression M = v ^ 2 r / G, we find that the massof the planet is

Planet mas = M = ( 24060 m/s) ^ 2( 20000 m) / (6.67*10^-11 N m ^ 2/kg ^ 2)= 173.5 * 10^24 kilograms.

Generalized Solution

To find planetary mass from the orbital radius and period of a small satellite we solve the orbital condition m v^2 / r = G M m / r^2 for M to obtain

M = v^2 r / G,

then determine v from the orbital circumference and period. We obtain

v = 2 `pi r / T,

where T is the period of the orbit.

Substituting this expression for v into M = v^2 r / G we obtain

M = (2 `pi r / T) ^ 2 r / G = 2 `pi / G * (r^3 / T^2).