Problem: A small object orbits a planet at a distance of 2.000 * 10 ^ 4 kilometers from the center of the planet. Each orbit requires 33 minutes. What is the mass of the planet?
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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 mass of 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 it 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 2.000 * 10 ^ 4 km, so its circumference is
circumference = distance = `ds = 2 `pi ( 2.000 * 10 ^ 4 km).
Dividing this by the 33 minutes = 1980 seconds required for an orbit, we obtain orbital velocity
v = 634.3 m/s.
Substituting into the expression M = v ^ 2 r / G, we find that the mass of the planet is
M = ( 634.3 m/s) ^ 2( 2.000 * 10 ^ 4 m) / (6.67*10^-11 N m ^ 2/kg ^ 2) = .001206 * 10^24 kilograms.
Generalized Response: 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).