A-LevelPhysicsGravitational fields9702/43 · May/June 2018 · Q19 Marks

State Newton's law of gravitation.

(a)State Newton's law of gravitation.[2]
(b)(i)A distant star is orbited by several planets. Each planet has a circular orbit with a different radius. Each planet orbits at constant speed. Explain whether the planets are in equilibrium.[1]
(b)(ii)1.The radius of the orbit of a planet is R and the orbital period is T. Data for some of the planets are given in Fig. 1.1. [Figure 1.1] The relationship between R and T is given by the expression R³ = kT². Show that the constant k is given by the expression k = GM / 4π² where G is the gravitational constant and M is the mass of the star.[3]
(b)(ii)2.Use data from Fig. 1.1 for the three planets and the expression for k to calculate the mass M of the star.[3]

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Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

[B1] force proportional to product of masses and inversely proportional to square of separation
[B1] idea of force between point masses
[B1] velocity changes/direction of motion changes/there is an acceleration/there is a resultant force so not in equilibrium
[C1] gravitational force equals/is centripetal force
[M1] GMm/R² = mRω² and ω=2π/T or Gm/R² = mv²/R and v=2πr/T or GMm/R² = mR (2π/T)²
[A1] convincing algebra leading to k = GM/4π²
[C1] correct use of R³/T² for one planet (c gives 3.54 × 10²¹; e and g both give 3.56 × 10²¹)
[A1] 3.5(5) × 10²¹ = (6.67 × 10⁻¹¹ × M) / 4π²
[B1] M = 2.1 x 10³³ kg
[B1] two or three values of R³/T² correctly calculated and used in a valid way to find a value for M based on more than one k

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