A-LevelPhysicsGravitational fields9702/42 · Oct/Nov 2017 · Q110 Marks

Answer all the questions in the spaces provided. (b) The planet Jupiter and one of its moons, Io, may be considered to be uniform spheres that are isolated in space. Jupiter has radius R and mean density ρ. Io has mass m and is in a circular orbit about Jupiter with radius nR, as illustrated in Fig. 1.1. [Figure 1.1] The time for Io to complete one orbit of Jupiter is T. (c) (i) The radius R of Jupiter is 7.15×10⁴km and the distance between the centres of Jupiter and Io is 4.32 × 10⁵ km. The period T of the orbit of Io is 42.5 hours.

(a)State Newton's law of gravitation.[2]
(b)Show that the time T is related to the mean density ρ of Jupiter by the expression ρT² = 3πη³ / G where G is the gravitational constant.[4]
(c)(i)Calculate the mean density ρ of Jupiter. ρ = .............................. kg m⁻³[3]
(c)(ii)The Earth has a mean density of 5.5×10³ kg m⁻³. It is said to be a planet made of rock. By reference to your answer in (i), comment on the possible composition of Jupiter.[1]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

[1] (a) force proportional to product of masses and inversely proportional to square of separation
[1] (a) idea of force between point masses
[1] (b) mass of Jupiter (M) = (4/3)πR³ρ
[1] (b) ω = 2π/Τ or v = 2πnR/T
[1] (b) (m)ω²x = GM(m)/x² or (m)v²/x = GM(m)/x²
[1] (b) substitution and correct algebra leading to ρT² = 3πη³/G
[1] (c)(i) n = (4.32 × 10⁵) / (7.15 × 10⁴) or n = 6.04
[1] (c)(i) ρx (42.5 × 3600)² = (3π × 6.04³)/(6.67 × 10⁻¹¹)
[1] (c)(i) ρ = 1.33 × 10³kgm⁻³
[1] (c)(ii) Jupiter likely to be a gas/liquid (at high pressure) [allow other sensible suggestions]

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