Question

-Mathematically prove that acceleration due to the gravity of a body is independent of its mass.

Answer 1

Hint: acceleration due to gravity is denoted by g. When a body is dropped from a certain height then it falls freely under the influence of gravitational force acted by earth on the body and the acceleration achieved by the body is g.

Complete step by step answer:
Let us take the mass of the body be m, and let the body falls freely under the influence of gravity, then as per Newton’s second law the force on the body is \[F=mg\]
This force is on account of gravitational force which is given by, \[F=\dfrac{GMm}{{{r}^{2}}}\]where M is the mass of the earth and r is the distance between the body and earth’s surface.
Equating the two forces we get,
$
mg=\dfrac{GMm}{{{r}^{2}}} \\
\implies g=\dfrac{GM}{{{r}^{2}}} \\
$
Thus, it is clear that the value of g is independent of the mass of the body. It is dependent on the mass of the earth and the distance between the two bodies.
Hence, proved.

Additional Information:
The value of g changes from pole to equator.

Note:
The universal law of gravitation holds everywhere. It is due to the gravitational force of the earth which holds all the bodies. If a body is allowed to fall freely from a certain height then it falls under the influence of gravity and the acceleration of the body is g whose value is 9.8 \[m/{{s}^{2}}\]. The value of acceleration due to gravity changes with the height and the depth. The angular velocity of rotation of earth affects the value of g.