Question

Derive an expression to show that acceleration due to gravity is independent of the mass of the object.

Answer 1

Let the mass of the body be m.

From the second law of motion, the force on a body is the product of its mass and acceleration i.e.; $F=ma$.

The acceleration of falling objects due to the gravitational force of earth is equal to the acceleration due to gravity(g).

Thus, the force(F) on the falling objects is given by, F = mg..........(i)

This force is equal to the gravitational force between the body and the earth.

From the universal law of gravitation the force of attraction between the body and the earth will be,

$F=G\frac{Mm}{r^2}$..............................(ii)

Where, G=Universal Gravitational constant

M=mass of earth

m=mass of body

r=distance between the body and the centre of earth

From eq. (i) and eq. (ii) as both the forces are equal so we have,

$mg=G\frac{Mm}{r^2}$

$g=G\frac{M}{r^2}$.

Thus $g$ is independent of the mass ($m$) of the body.