Question

Planet Lex is four times as massive as the Earth and has a radius four times larger than the Earth's radius. Note: You do not have to express your answers in scientific notation.

Mass of the Earth = 5.97 x 10²⁴ kg

Radius of the Earth = 6.37 x 10⁶ m

a.

What is the value of the acceleration due to gravity (in m/s²) on the surface of this planet? Round off your answer up to two decimal digits.

b.

At what altitude, h, above the surface of the Earth (in meter) is the value of the acceleration due to gravity equal to that on the surface of Planet Lex? Round off your answer in two decimal digits.

c.

If a GPS satellite of mass 2000 kg is to be placed in a circular orbit at this altitude, h, from the Earth's surface, find the value of tangential speed (in m/s). Round off your answer up to two decimal digits.

d.

Find the period of rotation (in seconds) of the GPS satellite at this altitude, h, above the surface of Earth.

e.

Find the magnitude of the centripetal force (in Newton) exerted by the Earth on the GPS satellite at this altitude.

Answer 1

The acceleration due to gravity of a spherical planet of mass M, at a distance r > R from the center of the planet is given by
  
Now at the surface of the planet, r = R, where, R is the radius of the planet.
So, at the surface of the planet, the acceleration due to gravity is given by
  
Now for earth the acceleration due to gravity at the surface of the earth is given by
   

So, the acceleration due to gravity at the surface of the planet Lex is
  



b)

Now at an altude h above the surface of the earth, the distance from the center is r = R_e + h. So, at this distance the acceleration due to gravity is
  

Now if
   
then, we have
  


c)

From the equation of motion of an object of mass m around a circular orbit of radius r, we have
  

Now for r = R_e + h = R_e + R_e = 2R_e, we have
   

So, for the given values, we have
   


  Period of rotation is
  

e)

  The magnitude of the centripetal force at this altude is