Question

You are on a planet where the moon's orbit is approximately 9 weeks (i.e. twice the period of earth's moon), but the distance to the moon is also twice as much. The mass of the planet is ____ times that of earth.

Answer 1

First Lets see on what factors Time Period depends on while in a orbit.

Then we know that when a satellite(of mass m) orbits around the planet(of mass M) of, say, radius "R" then there exists the gravitational attractive force between them given by

  

where G= Gravitational Constant

This force is exactly balanced by the centripetal force acting on the satellite when it moving with velocity"v".

or   

Then Time Period of the motion of satellite around planet is

Squaring both sides,

  

  

....................................(1)

Then First consider the case of Earth,

Radius of orbit of moon around Earth= R_e

Time Period of moon's orbit around Earth= T_e

Mass of Earth= M_e

then accroding to equation 1, we'll have

.....................................(2)

Now Considering the orbit of other moon around some planet.

Given: Time period of that moon around planet, T_p= 2T_e

        Radius of that moon from planet, R_p= 2R_e

then using equation 1,

Mass of that planet will be

or   

  

Using equation 1 in above,

  (ANS)

Thus   Mass of planet= 2(Mass of Earth) (ANS)