CAIE A-Level Physics A2 13.2 Gravitational Force Between Point Masses Questions

Practise applying Newton’s gravitational law to point masses and uniform spheres, analysing circular or geostationary orbits and calculating orbital quantities.

Exam points

apply Newton’s law of gravitation to point masses and uniform spheres
analyse circular and geostationary orbits by equating gravitational and centripetal effects
calculate orbital quantities and interpret geostationary conditions

Covered syllabus points

13.2.1—That, for a point outside a uniform sphere, the mass of the sphere may be 13.2.2—Newton’s law of gravitation F = Gm1m2 / r2 for the force between two point 13.2.3—Circular orbits in gravitational fields by relating the gravitational force 13.2.4—A satellite in a geostationary orbit remains at the same point above the

[Maximum number: 1]

The nearest star to the Sun is Proxima Centauri.

This star has a mass of $2.5 \times 10^{29} \mathrm{~kg}$ and is a distance of $4.0 \times 10^{13} \mathrm{~km}$ from the Sun. The Sun has a mass of $2.0 \times 10^{30} \mathrm{~kg}$ .

State why Proxima Centauri may be assumed to be a point mass when viewed from the Sun.

radius/diameter/size (of Proxima Centauri) << /is much less than $4.0 \times 10^{13} \mathrm{~km} /$ separation (of Sun and star)
or
(because) it is a uniform sphere

[Maximum number: 9]

(a)

State the equation for the gravitational force F between two point masses $m_{1}$ and $m_{2}$ that are separated by a distance r. State the meaning of any other symbols you use.

[ 2 ]

$F=\left(G m_{1} m_{2}\right) / r^{2}$
where G is the gravitational constant

(b)

A satellite is in a circular orbit of radius R around a planet of mass M.

Show that the period T of the orbit is given by

T^{2}=k R^{3}

where k is a constant that depends on the value of M. Explain your reasoning.

[ 3 ]

gravitational force provides the centripetal force
$m R \omega^{2}=G M m / R^{2} \text { and } \omega=2 \pi / T$

or
$m v^{2} / R=G M m / R^{2} \text { and } v=2 \pi R / T$

or
$4 \pi^{2} m R / T^{2}=G M m / R^{2}$
correct completion of algebra to get $T^{2}=\left(4 \pi^{2} / G M\right) R^{3}$ , with identification of ( $4 \pi^{2} / G M$ ) as k

(c)

A satellite is in a circular orbit around the Earth with a period of 24 hours. The mass of the Earth is $6.0 \times 10^{24} \mathrm{~kg}$ .

[ 4 ]

(i)

Calculate the radius of the orbit.
radius = m

[ 2 ]

$(24 \times 3600)^{2}=\left(4 \pi^{2} \times R^{3}\right) /\left(6.67 \times 10^{-11} \times 6.0 \times 10^{24}\right)$ $R=4.2 \times 10^{7} \mathrm{~m}$

(ii)

State the two other conditions that must be met for the orbit to be geostationary.

1
2

[ 2 ]

[Maximum number: 5]

(a)

A planet may be assumed to be a uniform sphere. It has gravitational potential $\phi$ at distance r from the centre of the planet.

The variation with $\frac{1}{r}$ of $\phi$ is shown in Fig. 1.1.

Fig. 1.1

[ 5 ]

(i)

Show that the mass of the planet is $8.8 \times 10^{25} \mathrm{~kg}$ .

[ 2 ]

M = (-) gradient / G
e.g. $M=\left(1.76 \times 10^{8}\right) /\left(3.0 \times 10^{-8} \times 6.67 \times 10^{-11}\right)=8.8 \times 10^{25} \mathrm{~kg}$

(ii)

The period of rotation of the planet is 0.72 Earth days.

A satellite in orbit around the planet remains above the same point on the surface of the planet.

Use the mass of the planet in (b)(i) to determine the radius R of the orbit of the satellite.

[ 3 ]

either $G M m / r^{2}=m r \omega^{2} \quad$ and $\omega=2 \pi / T$

or $G M m / r^{2}=m v^{2} / r$ and $v=2 \pi r / T$

or $G M m / r^{2}=4 \pi^{2} m r / T^{2}$ $R^{3}=6.67 \times 10^{-11} \times 8.8 \times 10^{25} \times(0.72 \times 24 \times 60 \times 60)^{2} / 4 \pi^{2}$ $R=8.3 \times 10^{7} \mathrm{~m}$

[Maximum number: 2]

State the equation for the gravitational force F between two point masses $m_{1}$ and $m_{2}$ that are separated by a distance r. State the meaning of any other symbols you use.

$F=\left(G m_{1} m_{2}\right) / r^{2}$
where G is the gravitational constant

[Maximum number: 3]

A planet may be assumed to be a uniform sphere. It has gravitational potential $\phi$ at distance r from the centre of the planet.

The variation with $\frac{1}{r}$ of $\phi$ is shown in Fig. 1.1.

Fig. 1.1

[Maximum number: 2]

Another satellite is in a circular orbit around the Earth with the same orbital radius and period as the satellite in (c).

Despite having the same orbital period, the orbit of this satellite is not geostationary.

Suggest two ways in which the orbit of this satellite could be different from the orbit of the satellite in (c).

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2

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Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)

Question 3

Question (a)

Question (i)

Question (ii)

Question 4

Question 5

Question 6