Question

A satellite of mass m is in a circular orbit of radius R2 around a spherical planet of radius R1 made of a material with density p. ( R2 is measured from the center of the planet, not its surface.) Use G for the universal gravitational constant. Part A Find the kinetic energy of this satellite, K.Part B Find U, the gravitational potential energy of the satellite. Take the gravitational potential energy to be zero for an object infinitely far away from the planet.

Answer 1

Answer:

a)      K = 2/3 π G m ρ R₁³ / R₂ ,  b) U = - G m M / r

Explanation:

The law of universal gravitation is

     F = G m M / r²

Part A

Let's use Newton's second law

     F = m a

The acceleration is centripetal

     a = v² / R₂

     

      G m M / R₂² = m v² / R₂

      v² = G M / R₂

They give us the density of the planet

    ρ = M / V

    V = 4/3 π R₁³

    M =   ρ V

    M =   ρ 4/3 π R₁³

    v² = 4/3 π G  ρ R₁³ / R₂

    K = ½ m v²

    K = ½ m (4/3 π G ρ R₁³ / R₂)

    K = 2/3 π G m ρ R₁³ / R₂

Part B

Potential energy and strength are related

     F = - dU / dr

     ∫ dU = - ∫ F. dr

The force was directed towards the center and the vector r outwards therefore there is an angle of 180º between the two cos 180 = -1

    U- U₀ = G m M ∫ dr / r²

    U - U₀ = G m M (- r⁻¹)

We evaluate for

    U - U₀ = -G m M (1 / $r_{f}$ -  1 /$r_{i}$)

They indicate that for ri = ∞     U₀ = 0

    U = - G m M / r