Orbital Radius?

Question

Planet W has a radius of 8.2*10^7m and a gravitational accelaeration of 27m/s^2. Determine the orbital radius of a satellite that orbits this planet in 6.50 days.

Answer 1

First, you need to find the mass of the planet. This is done using Newton's Law of Gravity and the parameters in your first sentence:

g = G M / r²

where g is the local acceleration of gravity, G is the Newton Gravity Constant, M the mass of the planet, and r the distance from the center of the planet to the surface. Solve this equation for M.

Now, the force required to hold a mass m in a circular orbit o radius r and angular velocity ω is

F = m r ω²

This force can only come from gravity...

F = G M m / r²

where again r is the distance from the center of the planet.

Now put them together...

G M m / r² = m r ω²
G M / r² = r ω²
GM = r³ ω²

Your angular velocity is 2π / ( ( 6.50 days )( 24 hours / day ) ( 3600 seconds / hour ) ). You already know G and M; solve for the distance r.