(a) Define gravitational potential at a point. (b) Gravitational potential φ at distance r from

Q# 693: (Past Exam Paper – June 2012 Paper 41 & 43 Q1) [Gravitation]

(a) Define gravitational potential at a point.

(b) Gravitational potential φ at distance r from point mass M is given by expression

φ = – GM / r

where G is the gravitational constant.

Explain significance of the negative sign in this expression.

(c) A spherical planet may be assumed to be an isolated point mass with its mass concentrated at its centre. Small mass m is moving near to, and normal to, surface of the planet. Mass moves away from the planet through a short distance h.

State and explain why change in gravitational potential energy ΔE_P of the mass is given by the expression

ΔE_P = mgh

where g is the acceleration of free fall.

(d) Planet in (c) has mass M and diameter 6.8 × 10³ km. The product GM for this planet is 4.3 × 10¹³ N m² kg^–1.

A rock, initially at rest a long distance from planet, accelerates towards the planet. Assuming that the planet has negligible atmosphere, calculate speed of the rock as it hits the surface of the planet.

Solution 693:

(a) The gravitational potential at a point is the work done in bringing unit mass from infinity (to the point).

(b) The gravitational force is (always) attractive.

EITHER as the distance r decreases, the object/mass/body does work

OR work is done by masses as they come together

(c)

EITHER

Force on the mass = mg

(where g is the acceleration of free fall /gravitational field strength)

Gravitational field strength, g = GM / r²

if r is at h {that is, the distance of the mass m is at a distance h from the point mass M – h is a distance close to the surface.}, g is constant

{This is the same as the acceleration of free fall on Earth. Close to the surface of Earth, g is about 9.81ms^-2. However, as we go further away from the Earth into the space, the value changes. For g to be constant, we should be close to the surface of the Earth.

As stated in the question above, h is a short distance from the surface of the planet. So, the distance r should also be a short distance from the surface of the planet for g to be constant. If r is at h, then it is close to the surface of the planet.}

ΔE_P = force × distance moved = mgh

{The following formula relates the gravitational potential energy to the gravitational potential. This is obvious from the definition of the gravitational potential at a point.}

Change in gravitational potential energy, ΔE_P = mΔφ

{Consider the distance of the small mass changing from r₁ to r₂.

1/r₁ – 1/r₂= (r₂ – r₁) / r₁r₂}

ΔE_P = mΔφ = GMm(1/r₁ – 1/r₂) = GMm(r₂ – r₁) / r₁r₂

if r₂ ≈ r₁ (r₂ is approximately equal to r₁ – that h is small), then (r₂ – r₁) = h and r₁r₂ = r² (since r₁ is assumed to be approximately equal to r₂, the product is either r₁² or r₂². Let call it r².)

{ΔE_P = GMm h / r²}

let g = GM / r²

ΔE_P = mgh

(d)

{From the conservation of energy, the potential energy is converted to kinetic energy.}

½ mv² = mΔφ

{Distance r is the radius, not the diameter. So, the diameter should be halved.}

v² = 2 × GM/r = (2 × 4.3 × 10¹³) / (3.4 × 10⁶)

Speed v = 5.0 × 10³ m s^–1