(a)

(i)

Explain, with reference to gravitational field lines, why the gravitational field near the surface of the Earth is approximately constant for small changes in height.

[ 2 ]

(b)

A large isolated uniform sphere has mass M and radius R.

Point P lies on a straight line passing through the centre of the sphere, at a variable displacement x from the centre, as shown in Fig. 1.1.

Fig. 1.1

Fig. 1.2 shows the variation with x of the gravitational field g at point P due to the sphere for the values of x for which P is inside the sphere.

Fig. 1.2

The magnitude of the gravitational field at the surface of the sphere is Y.

[ 7 ]

(i)

Determine an expression for Y in terms of M and R. Identify any other symbols that you use.

[ 2 ]

(ii)

Explain why, at the surface of the sphere, g always has the opposite sign to x.

[ 2 ]

(iii)

Complete Fig. 1.2 to show the variation of g with x for values of x, up to $\pm 3 R$ , for which point P is outside the sphere.

[ 3 ]

(a)

An isolated uniform conducting sphere has mass M and charge Q.

The gravitational field strength at the surface of the sphere is g.
The electric field strength at the surface of the sphere is E.

[ 3 ]

(i)

Show that

\frac{M}{Q}=\alpha \frac{g}{E}

where $\alpha$ is a constant.

[ 3 ]

(a)

Artemis is a spherical planet that may be assumed to be isolated in space. The variation with distance x from the centre of Artemis of the gravitational potential $\phi$ is shown in Fig. 1.1.

Fig. 1.1

(i)

Calculate the gravitational field strength g on the surface of Artemis.

(a)

(i)

Use Newton's law of gravitation to show that the gravitational field strength g at a distance r away from a point mass M is given by

g=\frac{G M}{r^{2}} .

[ 2 ]

(b)

The Earth has a mass of $5.98 \times 10^{24} \mathrm{~kg}$ and a radius of $6.37 \times 10^{6} \mathrm{~m}$ .

The Moon has a mass of $7.35 \times 10^{22} \mathrm{~kg}$ and a radius of $1.74 \times 10^{6} \mathrm{~m}$ .
The Earth and the Moon can both be considered as point masses at their centres. Their centres are a distance of $3.84 \times 10^{8} \mathrm{~m}$ apart.

[ 1 ]

(i)

Show that the gravitational field strength at the surface of the Moon due to the mass of the Moon is $1.62 \mathrm{~N} \mathrm{~kg}^{-1}$ .

[ 1 ]

(a)

(i)

Use Newton's law of gravitation to show that the gravitational field strength g at a distance r away from a point mass M is given by

g=\frac{G M}{r^{2}} .

[ 2 ]

(b)

The Earth has a mass of $5.98 \times 10^{24} \mathrm{~kg}$ and a radius of $6.37 \times 10^{6} \mathrm{~m}$ .

The Moon has a mass of $7.35 \times 10^{22} \mathrm{~kg}$ and a radius of $1.74 \times 10^{6} \mathrm{~m}$ .
The Earth and the Moon can both be considered as point masses at their centres. Their centres are a distance of $3.84 \times 10^{8} \mathrm{~m}$ apart.

[ 1 ]

(i)

Show that the gravitational field strength at the surface of the Moon due to the mass of the Moon is $1.62 \mathrm{~N} \mathrm{~kg}^{-1}$ .

[ 1 ]

[Maximum number: 2]

An isolated spherical planet has a diameter of $6.8 \times 10^{6} \mathrm{~m}$ . Its mass of $6.4 \times 10^{23} \mathrm{~kg}$ may be assumed to be a point mass at the centre of the planet.

(a)

Show that the gravitational field strength at the surface of the planet is $3.7 \mathrm{Nkg}^{-1}$ .

[ 2 ]

[Maximum number: 2]

An isolated spherical planet has a diameter of $6.8 \times 10^{6} \mathrm{~m}$ . Its mass of $6.4 \times 10^{23} \mathrm{~kg}$ may be assumed to be a point mass at the centre of the planet.

(a)

Show that the gravitational field strength at the surface of the planet is $3.7 \mathrm{Nkg}^{-1}$ .

[ 2 ]

(a)

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

State and explain why the change in gravitational potential energy $\Delta E_{\mathrm{P}}$ of the mass is given by the expression

\Delta E_{\mathrm{p}}=m g h

where g is the acceleration of free fall.

[ 4 ]

[Maximum number: 2]

gel and soft tissue,

\alpha=

(a)

An isolated star has radius R. The mass of the star may be considered to be a point mass at the centre of the star.
The gravitational field strength at the surface of the star is $g_{\mathrm{s}}$ .
On Fig. 1.1, sketch a graph to show the variation of the gravitational field strength of the star with distance from its centre. You should consider distances in the range R to 4 R.

Fig. 1.1

[ 2 ]

[Maximum number: 3]

2.

(a)

The Earth may be considered to be an isolated sphere of radius R with its mass concentrated at its centre.
The variation of the gravitational potential $\phi$ with distance x from the centre of the Earth is shown in Fig. 1.1.

Fig. 1.1

The radius R of the Earth is $6.4 \times 10^{6} \mathrm{~m}$ .

[ 3 ]

(i)

By considering the gravitational potential at the Earth's surface, determine a value for the mass of the Earth.

[ 3 ]