Question

1. A satellite orbits a planet. The distance between the satellite and the center of the planet is r. The time it takes the satellite to complete one orbit is T.

a. Find the speed of the satellite in its orbit, in terms of the quantities given above. Do not use the law of gravity in part a.!

b. Find the acceleration of the satellite in its orbit, in terms of the quantities given above. Do not use the law of gravity in part b.!

c. Use Newton’s second law and the gravitational force law to write the acceleration of the satellite, in terms of its mass m, the planet’s mass M, the distance r and the gravitational constant G.

d. Set the expressions for the acceleration you found in b. and c. equal. From this equation, find T in terms of r, G and M. Note: the answer will not depend on any other quantities.

Answer 1

Radius of orbit = r

Time period of orbit = T

Speed of satellite = V

The distance covered by the satellite in time period T is equal to the circumference of the orbit.

Acceleration of satellite = a

The satellite is in circular motion and will have acceleration equal to centripetal acceleration

Mass of satellite = m

Mass of planet = M

Gravitational constant = G

Gravitational force acting on satellite = GMm/r²

a) Speed of satellite in its orbit

  

b) Acceleration of satellite in its orbit

  

c) Acceleration of satellite from gravitational force

  

d) Time in terms of r, M and m