The mass $M$ of a spherical planet may be assumed to be a point mass at the centre of the planet. (a) A stone, travelling at speed $v$, is in a circular orbit of radius $r$ about the planet, as illustrated in Fig. 1.1. Show that the speed $v$ is given by the expression $$ v = \sqrt{\left(\frac{GM}{r}\right)} $$ where $G$ is the gravitational constant. Explain your working. (b) A second stone, initially at rest at infinity, travels towards the planet, as illustrated in Fig.…

Physics 9702/42 — May/June 2014

Cambridge A-Level · A Level Structured Questions · worked solutions for every part, with the mark scheme

questions

100

marks

120

minutes

Topics Motion in a Circle · Gravitational Fields · Ideal Gases · Electric Fields · Nuclear Physics · Magnetic Fields · +8 more

Q1Motion in a CircleGravitational FieldsFree sample

The mass $M$ of a spherical planet may be assumed to be a point mass at the centre of the planet.

(a)

A stone, travelling at speed $v$ , is in a circular orbit of radius $r$ about the planet, as illustrated in Fig. 1.1.

Fig. 1.1

Show that the speed $v$ is given by the expression

v = \sqrt{\left(\frac{GM}{r}\right)}

where $G$ is the gravitational constant.
Explain your working.

(b)

A second stone, initially at rest at infinity, travels towards the planet, as illustrated in Fig. 1.2.

Fig. 1.2

The stone does not hit the surface of the planet.

(i)

Determine, in terms of the gravitational constant $G$ and the mass $M$ of the planet, the speed $V_0$ of the stone at a distance $x$ from the centre of the planet. Explain your working.
You may assume that the gravitational attraction on the stone is due only to the planet.

(ii)

Use your answer in (i) and the expression in (a) to explain whether this stone could enter a circular orbit about the planet.

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