Fields, gravitational fields and orbits – A-level Physics exam questions with mark scheme

1Core3 marks

Mars has a mass of 6.42 × 10$^{23}$ kg and a radius of 3.39 × 10$^6$ m.

(a) Calculate the gravitational field strength at the surface of Mars. [2]

(b) Calculate the gravitational potential at the surface of Mars. [1]

(a) $g = \frac{GM}{r^2} = \frac{6.67\times10^{-11} \times 6.42\times10^{23}}{(3.39\times10^6)^2}$(1)
(a) $g$ = 3.7 N kg$^{-1}$ (3.73) (allow m s$^{-2}$)(1)
(b) $V = -\frac{GM}{r}$ = −1.3 × 10$^7$ J kg$^{-1}$ (minus sign required)(1)
Answer: (a) 3.73 N kg$^{-1}$ (b) −1.26 × 10$^7$ J kg$^{-1}$
Examiner tip: Gravitational potential is always negative — the minus sign is part of the mark.

2Standard5 marks

A geostationary satellite orbits the Earth. The mass of the Earth is 5.97 × 10$^{24}$ kg.

(a) Show that the radius of a geostationary orbit is about 4.2 × 10$^7$ m. [3]

(b) State two features of a geostationary orbit, other than its radius. [2]

(a) $\frac{GMm}{r^2} = m\omega^2 r$ or $T^2 = \frac{4\pi^2}{GM}r^3$(1)
(a) $T$ = 24 h = 86 400 s used (allow 86 164 s, giving 4.21 × 10$^7$ m)(1)
(a) $r = \sqrt[3]{\frac{6.67\times10^{-11}\times5.97\times10^{24}\times86\,400^2}{4\pi^2}}$ = 4.22 × 10$^7$ m (at least 3 sf)(1)
(b) any one feature from: period of 24 hours (one day) / same angular speed as the Earth's rotation; orbits directly above the equator (in the equatorial plane); moves west to east, in the same direction as the Earth rotates; stays above the same point on the Earth's surface(1)
(b) a second, different feature from the list above ('period 24 h' and 'same angular speed as the Earth' count as the same feature; ignore uses such as 'used for communications/TV')(1)
Answer: (a) 4.22 × 10$^7$ m (b) any two of: period 24 h; above the equator; same direction as Earth's rotation; stays above the same point on the surface
Examiner tip: In a 'show that', give your answer to one more significant figure than the value quoted.

3Hard4 marks

A satellite of mass 500 kg is moved from the Earth's surface (radius 6.37 × 10$^6$ m) into a circular orbit of radius 7.00 × 10$^6$ m. The mass of the Earth is 5.97 × 10$^{24}$ kg.

(a) Calculate the increase in gravitational potential energy of the satellite. [2]

(b) Calculate the orbital speed of the satellite. [2]

(a) $\Delta E = GMm\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$ with correct substitution(1)
(a) $\Delta E$ = 2.8 × 10$^9$ J (2.81)(1)
(b) $\frac{GMm}{r^2} = \frac{mv^2}{r}$ so $v = \sqrt{\frac{GM}{r}}$(1)
(b) $v$ = 7.5 × 10$^3$ m s$^{-1}$ (7540)(1)
Answer: (a) 2.81 × 10$^9$ J (b) 7.54 × 10$^3$ m s$^{-1}$
Examiner tip: Use the change in potential between two radii — mgh is not valid here because g is not uniform over 630 km.

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