Question

1. Weight of a body on the surfaces of two
planets is the same. If their densities are d1 and d2, then the ratio of their radius

1)d1/d2

2)d2/d1

3)d1^2/d2^2

4)d2^2/d1^2​

Answer 1

Given that a body kept on the surface of two planets experiences same weight. This just implies that the acceleration due to gravity, g, is same for the two planets.

Let m be the mass of the body kept. So its weight is mg. Since m is a constant, and if this mg is same for the two planets, then g will also be a constant.

We know that g = GM / R². Since LHS is constant, hence the RHS.

Since G is a constant, we have,

M1 / (R1)² = M2 / (R2)²

where M1 and R1 are the mass and radius of first planet respectively. Similarly, M2 and R2 are those of second one.

But density of first planet,

d1 = M1 / V1

=> M1 = d1 · V1

Similarly, M2 = d2 · V2.

So,

d1 · V1 / (R1)² = d2 · V2 / (R2)²

Well, since the planets are spherical in shape (assumed if not), volume of first planet,

V1 = 4 π (R1)³ / 3

Similarly, V2 = 4 π (R2)³ / 3.

So,

4 π d1 (R1)³ / 3 (R1)² = 4 π d2 (R2)³ / 3 (R2)²

4 π d1 · R1 / 3 = 4 π d2 · R2 / 3

d1 · R1 = d2 · R2

R1 / R2 = d2 / d1

Hence option (2) is the answer.