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

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A body of mass M is broken into two smaller parts each of mass m1 and msuch 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

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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

⇒ m= M/2

∴ m= M/2

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