Question

A body of mass M is broken into two smaller parts each of mass m₁ and m₂ such that the gravitational force between them is the maximum. Then which of the following is the correct relation?
1. m₁ = m₂
2. m₂ = 0.75 m₁
3. M = 3m₁ 
4. M = m₁m₂

Answer 1

Correct Answer - Option 1 : m₁ = m₂

The correct answer is option 1) i.e. m1 = m2

CONCEPT:

• The universal law of gravitation: It states that every object in the universe attracts every other with a force varying directly as the product of their masses and inversely as the square of the distance between them.

This is given by:

\(F =\frac{GMm}{R^2}\)

Where F is the gravitational force, M and m are the masses of two objects, R is the distance between the centre of two objects, and G is the universal gravitational constant. 

CALCULATION:

Given that:

Mass of one part = m₁

Mass of the second part = m₂ = M - m₁

Let us assume the distance between their centres is R.

\(⇒ F =\frac{Gm_1m_2}{R^2} = \frac{Gm_1(M-m_1)}{R^2}\)

For this force to be maximum, \(\frac{dF}{dm_1} = 0\)

\(⇒ \frac{d F}{dm_1} =\frac{d( \frac{Gm_1(M-m_1)}{R^2})}{dm_1}\)

\(⇒ \frac{d F}{dm_1} =\frac{G(M - 2m_1)}{R^2}\)

⇒ M - 2m₁ = 0

⇒ m1 = M/2

∴ m₂ = M/2

Thus, the gravitational force is maximum when m1 = m2.