Question

Two objects attract each other gravitationally with a force of 2.2×10⁻¹⁰ NN when they are 0.20 mm apart. Their total mass is 4.30 kgkg . Find their individual masses.

1. Mlarger

2. Msmaller

Answer 1

Suppose, the larger mass = M

and the smaller mass = m

Gravitational force of attraction between two masses M and 'm' situated at a distance 'd' is given as -

Fg = G * M * m ÷ d^2 -----------------------------------------------(i)

Now,

M + m = 4.30 kg

=> M = 4.3 - m

So,

M * m = 4.3 * m – m^2

Therefore, from equation (i) -

2.2 * 10^-10 = 6.67 * 10^-11 * (4.3 * m – m^2) ÷ 0.2^2

Multiply both sides by 0.2^2.

0.88 * 10^-11 = 6.67 * 10^-11 * (4.3 * m – m^2)

Divide both sides by 6.67 * 10^-11.

0.88/6.67 = 4.3 * m – m^2

m^2 – 4.3 * m + 0.132 = 0

m = [4.3 ± √(-4.3^2 – 4 * 1 * 0.132)] ÷ 2

m = [4.3 + √17.962] ÷ 2

= [4.3 + 4.238] / 2 = 4.269 kg

m = [4.3 – √17.962] ÷ 2

= [4.3 - 4.238]/2 = 0.031 kg

Let’s add these two numbers.

Sum = 4.269 + 0.031 = 4.30

This proves that the masses are correct.

Therefore,

(1) Larger mass = 4.269 kg

(2) Smaller mass = 0.031 kg (Answer)