A stone of mass m has gravitational potential energy EP at a point X in a gravitational field.

Question 2

(a) Define gravitational potential at a point. [2]

(b) A stone of mass m has gravitational potential energy EP at a point X in a gravitational field.

The magnitude of the gravitational potential at X is ϕ.

State the relation between m, EP and ϕ. [1]

(c) An isolated spherical planet of radius R may be assumed to have all its mass concentrated at its centre. The gravitational potential at the surface of the planet is − 6.30 × 10⁷ J kg^-1.

A stone of mass 1.30 kg is travelling towards the planet such that its distance from the centre of the planet changes from 6R to 5R.

Calculate the change in gravitational potential energy of the stone. [4]

Reference: Past Exam Paper – June 2014 Paper 41 & 43 Q1

Solution:

(a) Gravitational potential at a point is defined as the work done in bringing unit mass from infinity (to the point).

(b)

{Gravitational potential energy =  – mass × gravitational potential

From the question, we are told that ϕ is the magnitude of the gravitational potential.

From the definition of gravitational potential (as given in part (a)), ϕ always has a negative value. So, we need to include the negative sign.}

E_p = - mϕ        

(c)

{Gravitational potential: ϕ = - GM / R           or – GM / x

The gravitational potential is inversely proportional to the distance x from the centre of the planet.}

Gravitational potential , ϕ ∝ 1 / x      

{Change in potential energy = mass of object × change in gravitational potential

ΔPE = m × Δϕ}

EITHER

{ ϕ ∝ 1 / x

ϕ = k / x          where k is a constant

At the surface, distance x = R and ϕ = (–) 6.30×10⁷ J kg^-1

Since we will finally be dealing with a ‘change’, we may neglect the negative sign

At x = 6R, potential ϕ = 6.30×10⁷ / 6             since ϕ is inversely proportional to x}

At a distance of 6R from the centre, potential = (6.3×10⁷) / 6 [= 1.05×10⁷ J kg^-1]

and at 5R from the centre, potential = (6.3×10⁷) / 5 [= 1.26×10⁷ J kg^-1]

{change in potential Δϕ = final potential – initial potential

Change in potential Δϕ = potential at 5R – potential at 6R

Change in potential energy = m Δϕ }

Change in energy = (1.26 – 1.05)×10⁷ × 1.3 = 2.7×10⁶ J

OR

Change in gravitational potential = (1/5 – 1/6) × 6.3×10⁷     

Change in energy =   (1/5 – 1/6) × (6.3×10⁷) × 1.3 = 2.7×10⁶ J