Question

The distance between two objects is reduced to half. So, the force of gravitation between the two objects become
A. Twice of the initial force of gravitation
B. Four times of the initial force of gravitation
C. One-fourth of the initial force of gravitation
D. Half of the initial force of gravitation

Answer 1

Gravitational force is dependent on mass and distance between two objects. As the distance between the objects is decreased, we will find the effect of that change on the force. We will compare the two forces using $F = \dfrac{{G\,M\,m}}{{{r^2}}}$formula.

Complete step by step answer:
Gravitational force: It is defined as the force that is directly proportional to the product of the two masses and inversely proportional to the square of distances between them.
$F \propto {m_1}\,{m_2}$
$F\dfrac{1}{ \propto }\,\,{r^2}$
$F \propto \dfrac{{M\,m}}{{{r^2}}}$
Proportionality constant is removed by gravitational constant (G) ,
$F = \,\dfrac{{G\,M\,m}}{{{r^2}}}$ …(1)
Value of $G = 6.67\, \times \,{10^{ - 11}}\,{m^3}k{g^{ - 1}}{s^{ - 2}}$
It is universal gravitational constant
Let us assume gravitational force (${F_1}$) is acting between two objects of masses ${m_1}$and ${m_2}$kg which are separated by a distance of ‘x’ metre.
From (1) we get;
${F_1} = \,\dfrac{{G{m_1}{m_2}}}{{{r_1}^2}}$ …(2)
Distance between the two objects is reduced to half ${r_2} = \,\,\dfrac{{{r_1}}}{2}$. Mass will remain the same as ${m_1}$and ${m_2}$.
New force will be ${F_2} = \dfrac{{G\,{m_1}\,{m_2}}}{{r_2^2}}$ …(3)
From (2) and (3),
${F_2} = \dfrac{{G{m_1}{m_2}}}{{\left[ {\dfrac{{{r_1}}}{2}} \right]}}$
${F_2} = \dfrac{{4\,G\,{m_1}\,{m_2}}}{{r_1^2}}$
${F_2}\, = \,4{F_1}$
Therefore, the new force obtained is four times the original force. So option B is the correct solution.

Note: The Gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. If gravitational force does not follow inverse square law or, then the new force will be twice the initial force of gravitation.