Question

It is known that time of revolution T of a satellite around the Earth depends on the universal gravitational constant G, the mass of Earth M,and the radius of circular orbit R. Obtain an expression for T using dimensional analysis.

Answer 1

Step 1: Given that:

Time of revolution of the satellite around earth = T

Depends on = The universal gravitational constant G, the mass of Earth M, and the radius of circular orbit R.

Step 2: Determining the expression for T:

Let, the time of revolution of the satellite around the earth(T) depends on power $a$ of The universal gravitational constant G, power $b$ of the mass of the earth and power $c$ of the radius of circular orbit R.

That is; $T\propto G^a{M}^b{R}^c$

Now,

$T\propto G^a{M}^b{R}^c$

$T=k{G}^a{M}^b{R}^c$.........(1)

Writing the dimensions of the quantities on both sides, we get;

$\left[T\right]={\left[M^{-1}{L}^3{T}^{-2}\right]}^a{\left[M\right]}^b{\left[L\right]}^c$

$\left[T\right]=\left[M^{-a+b}{L}^{3a+c}{T}^{-2a}\right]$

Comparing the powers of similar quantities on both sides, we get

$-a+b=0$

$3a+c=0$

$-2a=1$

Solving these equations, we get;

$a=\frac{-1}{2}$

$b=\frac{-1}{2}$

$c=\frac{3}{2}$

Putting these values in equation (1), we get;

$T=k{G}^{\frac{-1}{2}}{M}^{{}^{\frac{-1}{2}}}{R}^{{}^{\frac{3}{2}}}$

$T=k\frac{R^{{}^{\frac{3}{2}}}}{G^{\frac{1}{2}}{M}^{{}^{\frac{1}{2}}}}$

$T=k\frac{\sqrt{R^3}}{\sqrt{GM}}$

$T=k\sqrt{\frac{R^3}{GM}}$

Thus,

The expression for the time of revolution of the satellite around the earth is

$T=k\sqrt{\frac{R^3}{GM}}$ .