Prove that if the earth attracts two bodies placed at same distance from the centre of the earth with the same force, then their masses are equal.

Let P and Q be the two bodies,

the mass of body P = m₁

And the mass of body Q = m₂

As per the universal law of gravitation, the force of attraction between the earth and the body P is given by,

FP = {𝐺 × 𝑀_𝑒 × 𝑀_1 \over 𝑅^2}            … (1)

Where, R is the distance of the body from the centre of the earth.

Similarly, the force of attraction between the earth and the body Q is given by

FQ = {𝐺 × 𝑀_𝑒 × 𝑀_2 \over 𝑅^2}            … (2)

Since, the two forces, i.e., FP and FQ are equal, thus from (1) and (2),

{𝐺 × 𝑀_𝑒 × 𝑀_1 \over 𝑅^2}  =  {𝐺 × 𝑀_𝑒 × 𝑀_2 \over 𝑅^2}

⟹ m_{1 =} m₂