(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.

[ 1 ]

(i)

Show that the numerical value of $\alpha$ is $1.35 \times 10^{20} \mathrm{~kg}^{2} \mathrm{C}^{-2}$ .

[ 1 ]

(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 $6.4 \times 10^{6} \mathrm{~m}$ .

[ 2 ]

(i)

In practice, the Earth is not an isolated sphere because it is orbited by the Moon, as illustrated in Fig. 1.2.
initial path
of meteorite

Fig. 1.2 (not to scale)

The initial path of the meteorite is also shown.
Suggest two changes to the motion of the meteorite caused by the Moon.
1.
2.

[ 2 ]

[Maximum number: 2]

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 $6.4 \times 10^{6} \mathrm{~m}$ .

[ 2 ]

(i)

In practice, the Earth is not an isolated sphere because it is orbited by the Moon, as illustrated in Fig. 1.2.
initial path
of meteorite

Fig. 1.2 (not to scale)

The initial path of the meteorite is also shown.
Suggest two changes to the motion of the meteorite caused by the Moon.

[ 2 ]

[Maximum number: 8]

gel and soft tissue,

\alpha=

(a)

The Earth and the Moon may be considered to be spheres that are isolated in space with their masses concentrated at their centres.
The masses of the Earth and the Moon are $6.00 \times 10^{24} \mathrm{~kg}$ and $7.40 \times 10^{22} \mathrm{~kg}$ respectively.
The radius of the Earth is $R_{\mathrm{E}}$ and the separation of the centres of the Earth and the Moon is $60 R_{\mathrm{E}}$ , as illustrated in Fig. 1.2.

Fig. 1.2 (not to scale)

[ 8 ]

(i)

Explain why there is a point between the Earth and the Moon at which the gravitational field strength is zero.

[ 2 ]

(ii)

Determine the distance, in terms of $R_{\mathrm{E}}$ , from the centre of the Earth at which the gravitational field strength is zero.

distance =

[ 3 ]

(iii)

On the axes of Fig. 1.3, sketch a graph to show the variation of the gravitational field strength with position between the surface of the Earth and the surface of the Moon.

[ 3 ]

[Maximum number: 2]

A planet of mass m is in a circular orbit of radius r about the Sun of mass M, as illustrated in Fig. 1.1.

Fig. 1.1

The magnitude of the angular velocity and the period of revolution of the planet about the Sun are $\omega$ and T respectively.

(a)

Data for the planets Venus and Neptune are given in Fig. 1.2.

Fig. 1.2

Assume that the orbits of both planets are circular.

[ 2 ]

(i)

Use the expression in (b) to calculate the value of T for Neptune.
T= years

[ 2 ]

(a)

Newton's law of gravitation applies to point masses.

[ 3 ]

(i)

State Newton's law of gravitation.

[ 2 ]

(ii)

Explain why, although the planets and the Sun are not point masses, the law also applies to planets orbiting the Sun.

[ 1 ]

(a)

Two protons are isolated in space. Their centres are separated by a distance R. Each proton may be considered to be a point mass with point charge. Determine the magnitude of the ratio
> force between protons due to electric field force between protons due to gravitational field

ratio =

[ 3 ]

[Maximum number: 4]

the mass of Mars,
mass = kg

(a)

A moon is in a circular orbit of radius r about a planet. The angular speed of the moon in its orbit is $\omega$ . The planet and its moon may be considered to be point masses that are isolated in space.

Show that r and $\omega$ are related by the expression

r^{3} \omega^{2}=\text { constant. }

Explain your working.

[ 3 ]

(b)

Phobos and Deimos are moons that are in circular orbits about the planet Mars.

Data for Phobos and Deimos are shown in Fig. 1.1.

Fig. 1.1

[ 1 ]

(i)

Use data from Fig. 1.1 to determine

(ii)

The period of rotation of Mars about its axis is 24.6 hours.

Deimos is in an equatorial orbit, orbiting in the same direction as the spin of Mars about its axis.

Use your answer in (i) to comment on the orbit of Deimos.

[ 1 ]

(a)

State Newton's law of gravitation.

[ 2 ]

(b)

The Earth and the Moon may be considered to be isolated in space with their masses concentrated at their centres.
The orbit of the Moon around the Earth is circular with a radius of $3.84 \times 10^{5} \mathrm{~km}$ . The period of the orbit is 27.3 days.

Show that

[ 2 ]

(i)

the mass of the Earth is $6.0 \times 10^{24} \mathrm{~kg}$ .

[ 2 ]

(c)

The mass of the Moon is $7.4 \times 10^{22} \mathrm{~kg}$ .

(i)

Using data from (b), determine the gravitational force between the Earth and the Moon.

force =

[Maximum number: 6]

kinetic energy,
change in kinetic energy = J
2. gravitational potential energy.
change in potential energy = J

(a)

State Newton's law of gravitation.

[ 2 ]

(b)

A satellite of mass m is in a circular orbit of radius r about a planet of mass M. For this planet, the product G M is $4.00 \times 10^{14} \mathrm{Nm}^{2} \mathrm{~kg}^{-1}$ , where G is the gravitational constant.
The planet may be assumed to be isolated in space.

[ 4 ]

(i)

By considering the gravitational force on the satellite and the centripetal force, show that the kinetic energy $E_{\mathrm{K}}$ of the satellite is given by the expression

E_{\mathrm{K}}=\frac{G M m}{2 r} .

[ 2 ]

(ii)

The satellite has mass 620 kg and is initially in a circular orbit of radius $7.34 \times 10^{6} \mathrm{~m}$ , as illustrated in Fig. 1.1.

Fig. 1.1 (not to scale)

Resistive forces cause the satellite to move into a new orbit of radius $7.30 \times 10^{6} \mathrm{~m}$ .

Determine, for the satellite, the change in

(iii)

Use your answers in (ii) to explain whether the linear speed of the satellite increases, decreases or remains unchanged when the radius of the orbit decreases.

[ 2 ]