Question

4. Show from Newton's law of gravitation and Newton's second law of motion that the acceleration of a freely falling body does not depend on the mass of the body.​

Answer 1

TO PROVE:

Acceleration of a freely falling body does not depend on the mass of the body.

PROOF:

NEWTON's SECOND LAW:

Newton's second law states that the rate of change of momentum is directly propotional to the applied unbalanced force in the direction of force

$\begin{gathered}F \: \alpha \: \frac{ m(v - u) }{t} \\ \\ \frac{v - u}{t} = a \\ \\ F \: \alpha \: ma \\ \\ F= k \: m \: a\end{gathered}$

k is the constant of proportionality and is equal to one

F = ma

NEWTON'S LAW OF GRAVITATION:

Newton's law of gravitation states that every object in the universe attract every other object by a force which is directly proportional to the product of masses and inversely proportional to the square of the distance between them.

$\begin{gathered}F \: \alpha \: \frac{Mm}{ {d}^{2} } \\ \\ F = \frac{GMm}{ {d}^{2} }\end{gathered}$

$\tt \: G \: is \: universal \: gravitational \: constant \: and \: is \: equal \: to \: 6.673 × 10^{-11} Nm^2kg^{-2}10$

EQUATE BOTH THE F VALUES

ma = GMm / d²

a = GM / d²

Here 'a' means acceleration due to gravity(g).

We can notice that 'a' is independent of mass of the body(m).

Also we can notice that 'a' is dependent of mass of the other body(M).

We need to prove that acceleration of a freely falling body does not depend on the mass of the body.

As the acceleration of a freely falling body = GM / d². It is independent of its own mass.

HENCE PROVED.