Question

State Newton’s law of gravitation. The gravitational force between two objects is F.
What will be the value of gravitational force between them, if masses of both objects
are doubled and the distance between them is halved ?
Calculate the magnitude of gravitational force between the Earth and a 1 kg object on
its surface. (Mass of the Earth is 6 × 1024 kg and radius of the Earth is 6.4 × 106 m.
Take G = 6.7 × 10–11 Nm2 kg–2)

Answer 1

$\fbox{\bold{\mathsf{Newton's\:law\:of\:gravitation:}}}$

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

Question 1:

The gravitational force between two objects is F. What will be the value of gravitational force between them, if masses of both objects are doubled and the distance between them is halved ?

Answer 1:

Let product of the masses be M and m and the distance between them be r.

Given that,

Gravitational force = F

One object's mass = M

Other object's mass = m

Distance between them = $\frac{r}{2}$

To find,

$\mathsf{Gravitational\:force_{new}}$ = ?

We know that,

F = $\frac{GMm}{r^2}$

F = G $\dfrac{2M \times 2m}{(\frac{1}{2})R^2}$

F = G $\dfrac{2M \times 2m}{\frac{1}{4}R^2}$

F = G $\frac{2M \times 2m}{R^2} \times \frac{4}{1}$

F = G $\frac{4Mm}{R^2} \times \frac{4}{1}$

F = G $\frac{Mm}{R^2} \times 16$

Therefore, the force will increase 16 times.

Question 2:

Calculate the magnitude of gravitational force between the Earth and a 1 kg object on its surface.

Answer 2:

Given that,

G = $6.7 \times 10^{-11}$ Nm^2kg

Mass of Earth (m1) = $6 \times 10^{24}$ kg

Radius of Earth (r) = $6.4 \times 10^6$

Mass of object (m2) = 1 kg

To find,

Gravitational force = ?

We know that,

F = $\frac{GMm}{r^2}$

F = $\frac{6.7 \times 10^{-11} \times 6 \times 10^{24} \times 1}{6.4 \times 10^6}$

F = $\frac{6.7 \times 6 \times 1 \times 10^{-11 + 24}}{6.4 \times 10^6}$

F = $\frac{6.7 \times 6 \times 10^{13}}{6.4 \times 10^6}$

F = $\frac{6.7 \times 6 \times 10^{13-6}}{6.4}$

F = $\frac{6.7 \times 6 \times 10^7}{6.4}$

F = $\frac{40.2 \times 10^7}{6.4}$

F = $\frac{402 \times 10^7}{64}$

F = $\frac{\cancel{402} \times 10^7}{\cancel{64}}$

F = $6.28 \times 10^7$ (approx)

$\boxed{\bold{F = 0.628 \times 10^6\:N}}$