Question

Show that if a satellite orbits very near the surface of a planet, with the period T, the density of thhe planet =3pi/GT^2

Answer 1

Let, the mass of the planet is M and the mass of the satellite is m and so, the equation of motion of the satellite around the planet near the surface of the planet ( radius of the orbit is approximately equal to the radius of the planet ) is
     
  
And from the definition of the time period of the satellite, we write
   
  
And so, we get

  
And from the definition of the volume of a solid sphere and the mass in terms of density, we write
   
And so, comparing these two expressions, we get