$(a)$ Prove that if the Earth attracts two bodies placed at the same distance from the center of the Earth with equal force,then their masses will be the same.
$(b)$ Mathematically express the acceleration due to gravity experienced by a free-falling object.
$(c)$ Why is $G$ called a universal gravitational constant?

(N/A) Let the masses of two bodies be $m_{1}$ and $m_{2}$.
Let the mass of the Earth be $M$ and the distance from the center of the Earth be $d$.
According to Newton's law of universal gravitation,the force $F$ is given by $F = \frac{GMm}{d^{2}}$.
For the two bodies,$F_{1} = \frac{GMm_{1}}{d^{2}}$ and $F_{2} = \frac{GMm_{2}}{d^{2}}$.
Given that $F_{1} = F_{2}$,we have $\frac{GMm_{1}}{d^{2}} = \frac{GMm_{2}}{d^{2}}$.
By canceling the common terms $\frac{GM}{d^{2}}$ from both sides,we get $m_{1} = m_{2}$.
$(b)$ The acceleration due to gravity $g$ for an object of mass $m$ at the surface of the Earth (radius $R$) is given by $g = \frac{GM}{R^{2}}$.
$(c)$ $G$ is called a universal gravitational constant because its value remains the same throughout the entire universe,regardless of the location or the nature of the interacting bodies.