Question

centripetal force is required for a body to move on a circular track. What provides this force to the moon to move around the earth? write an expression for it.

Answer 1

Gravitational force of attraction provides centripetal force to the Moon to move around the Earth.
Centripetal acceleration is given as, $a=\frac{v^2}{r}=r{\omega }^2$
Where v is linear velocity, $\omega$ is angular velocity and r is the radius of the circular path.
Using Newton's second law of motion as,
$$\begin{gathered} F=m\times a \\ Therefore,Centripetalforce\left(F_c\right)=mas\times centripetalacceleration\left(a_c\right) \\ or,F_c=\frac{m{v}^2}{r}=mr{\omega }^2............\left(i\right) \end{gathered}$$

Gravitational force of attraction for the moon orbiting the earth will be

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

From equation (i) and (ii).

$$\begin{gathered} \frac{GmM}{r^2}=\frac{m{v}^2}{r} \\ or,v=\sqrt{\frac{2GM}{r}} \end{gathered}$$

This expression can be used to find the orbital speed of the Moon.

Also, equating the expression (i) and (ii) for the force on the moon and substituting the relationship for a circular orbit that
$v=\frac{2\pi r}{T}$.

$$\begin{gathered} \frac{GmM}{r^2}=\frac{m{v}^2}{r}=\frac{4{\pi }^{2}mr}{T^2} \\ or,M=\frac{4{\pi }^2{r}^3}{G{T}^2} \end{gathered}$$

Thus a value for mass of the Earth can be calculated by knowing the period and radius of the Moon's orbit.