CAIE A-Level Physics 13.4 Gravitational Potential

Practise defining and calculating gravitational potential and potential energy, interpreting signs and relating potential, work, energy changes and escape.

Exam points

define and calculate gravitational potential ϕ = −GM/r and interpret its sign and reference
derive and calculate gravitational potential energy EP = −GMm/r
relate potential, potential energy, work and energy changes in a gravitational field

Covered syllabus points

13.4.1—Gravitational potential at a point as the work done per unit mass in 13.4.2—Φ = –GM / r for the gravitational potential in the field due to a point mass 13.4.3—The concept of gravitational potential leads to the gravitational potential

[Maximum number: 5]

(a)

Define gravitational potential.

[ 2 ]

(b)

The Earth E and the Moon M can both be considered as isolated point masses at their centres. The mass of the Earth is $5.98 \times 10^{24} \mathrm{~kg}$ and the mass of the Moon is $7.35 \times 10^{22} \mathrm{~kg}$ . The Earth and the Moon are separated by a distance of $3.84 \times 10^{8} \mathrm{~m}$ , as shown in Fig. 2.1.

Fig. 2.1 (not to scale)

P is a point, on the line joining the centres of E and M, where the resultant gravitational field strength is zero. Point P is at a distance x from the centre of the Earth.

[ 3 ]

(i)

Calculate the gravitational potential $\phi$ at point P .

\phi=\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . . \mathrm{Jkg}^{-1}[3]

[ 3 ]

$\begin{aligned} \phi(\text { Earth })=(-) 6.67 \times 10^{-11} \times\left(5.98 \times 10^{24} / 3.5 \times 10^{8}\right) \text { and } \phi(\text { Moon })=(-) 6.67 \times 10^{-11} \times\left(7.35 \times 10^{22} / 0.38 \times 10^{8}\right) \end{aligned}$

C1

$\phi=(-) 6.67 \times 10^{-11} \times\left[\left(5.98 \times 10^{24} / 3.5 \times 10^{8}\right)+\left(7.35 \times 10^{22} / 0.38 \times 10^{8}\right)\right]$

C1

$=-1.3 \times 10^{6} \mathrm{~J} \mathrm{~kg}^{-1}$

A1

[Maximum number: 8]

(a)

Define gravitational potential at a point.

[ 2 ]

(b)

An isolated solid sphere of radius r may be assumed to have its mass M concentrated at its centre. The magnitude of the gravitational potential at the surface of the sphere is $\phi$ .

On Fig. 1.1, show the variation of the gravitational potential with distance d from the centre of the sphere for values of d from d=r to d=4 r.

Fig. 1.1

[ 3 ]

curve from r to 4 r, with gradient of decreasing magnitude and starting at ( $r, \pm \phi$ )

B1

line passing through ( $2 r, \pm 0.5 \phi$ ) and ( $4 r, \pm 0.25 \phi$ )

B1

line showing potential is negative throughout

B1

(c)

The sphere in (b) is a planet with radius r of $6.4 \times 10^{6} \mathrm{~m}$ and mass M of $6.0 \times 10^{24} \mathrm{~kg}$ . The planet has no atmosphere.

A rock of mass $3.4 \times 10^{3} \mathrm{~kg}$ moves directly towards the planet. Its distance from the centre of the planet changes from 4 r to 3 r.

[ 3 ]

(i)

Calculate the change in gravitational potential energy of the rock.
change =

[ 3 ]

gravitational potential energy = (-) GMm / R

C1

change $=\left(6.67 \times 10^{-11} \times 6.0 \times 10^{24} \times 3.4 \times 10^{3}\right) /\left(6.4 \times 10^{6}\right) \times[1 / 3-1 / 4]$

C1

$=1.8 \times 10^{10} \mathrm{~J}$

A1

All question bank results loaded

Covered syllabus points

13.4 Gravitational potential question 1

Question (a)

Question (b)

Question (i)

13.4 Gravitational potential question 2

Question (a)

Question (b)

Question (c)

Question (i)