physics gravitational force

Question

Satellite A of mass m orbits a planet of mass M at a distance of R from the center of the planet. The speed of the satellite is v.

(a) What is the speed of a satellite B whose mass is 3 m? The distance from the center of the planet to the satellite B is R and the mass of the planet is M. Answer in terms of v.

(b) Satellite B of mass m has orbital radius R. Satellite B orbits a planet of mass 2 M. What is the speed of the satellite B? Answer in terms of v.

(c) Satellite B of mass m orbits a planet of mass M with speed 2 v. What is the radius of the satellite B’s orbit? Answer in terms of R.

(d) How fast should satellite B of mass m orbit a planet of mass M so that its orbital radius is 2 R? Answer in terms of v.

(e) Satellite B has mass m and orbits a planet of mass M. What should satellite B’s orbital radius be so that its period is 8 times that of satellite A? Answer in terms of R.

Answer 1

(a)

v_B = √(GM/R) = v

Note the speed depends only on the radius of the orbit and the mass of the planet, but not on the mass of the satellite.

(b)

v_B = √[G(2M)/R] = √2 √(GM/R) = √2 v

(c)

2v = √(GM/R_B) ⇒

R_B = GM / (4v²) = GM / [4(GM/R)] = R/4

(d)

v_B = √[GM / (2R)] = (1/√2) v

(e)

Use Kepler's 3rd law.

R_B³ / T_B² = R_A³ / T_A² = R³ / T_A²

R_B³ / (8T_A)² = R³ / T_A²

R_B³ / 64 = R³

R_B = 4R