Shi Qi 4 Mar 2018 - 2

Question

In two widely-separated planetary systems whose suns have masses S₁ and S₂, planet P₁ of mass M₁ and planet P₂ of mass M₂ are observed to have circular orbits of equal radii respectively. If P₁ completes and orbit in half the time by P₂, it may be deduced that

A) S₁ = S₂ and M₁ = 0.25 M₂

B) S₁ = 4 S₂ only

C) S₁ = 4 S₂ and M₁ = M₂

D) S₁ = 0.25 S₂ only

E) S₁ = 0.25 S₂ and M₁ = M₂

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Answer

For the system with S₁ and P₁, the gravitational force on P₁ provides the centripetal force.

Similarly for the system with S₂ and P₂,

Since the period of P₁ is half of P₁,

Since the periods do not depend on the masses of the planets, we can only deduce that S₁ = 4 S₂.

Hence, answer is B.

Shi Qi 4 Mar 2018 - 1

Question

At a point outside the Earth and a distance of x from its centre, the Earth’s gravitational field strength is 5 N kg^-1. At the Earth’s surface, the field strength is about 10 N kg^-1. Which of the following gives an approximate value for the radius of the Earth?

A) x/5

B) x/2√2

C) x√2

D) x/√2

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Answer

At point x, the gravitational field strength is given by the formula:

At the surface of the Earth, the gravitational field strength is given by the formula:

Since g_s ≈ 2 g_x,

Hence, the answer is D.