planetary motion

Question

For orbiting satellites, the mass of the satellite always cancels in our centripetal force equation. Use the equation to justify why it is the satellite’s mass, and not the mass of the central planet, that must cancel.

Answer 1

Suppose a satellite of mass m is orbiting a planet of mass M, along a circular path of radius r, with the planet at the center of the circle.  Set the gravitational force between the planet and satellite equal to the centripetal force on the satellite.  You can do this because the gravitational force keeps the satellite on the circular path.

 

GMm/r² = mv²/r

 

Note that m appears on both sides and therefore is canceled out.

 

GM/r² = v²/r = a_c

 

a_c = centripetal acceleration

 

Note that a_c depends on the mass of the planet, but not on the mass of the satellite.