Question

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

• A. $2\pi \sqrt{\frac{R^3}{GM}}$
• B. $\pi \sqrt{\frac{R^3}{GM}}$
• C. $\pi \sqrt{\frac{R}{GM}}$
• D. None of these

Answer 1

Answer: (A)

We have $[T]=[G{]}^a[M{]}^b[R{]}^c$
$[M{]}^0[L{]}^0[T{]}^1=[M{]}^{-a}[L{]}^{3a}[T{]}^{-2a}\times [M{]}^b\times [L{]}^c$
$=[M{]}^{b-a}[L{]}^{c+3a}[T{]}^{-2a}$
Comparing the exponents
$\text{ For }[T]:1=-2a$
$\Rightarrow a=-\frac{1}{2}$
$\text{ For }[M]:0=b-a$
$\Rightarrow b=a=-\frac{1}{2}$
$\text{ For }[L]:0=c+3a$
$\Rightarrow c=-3a=-\frac{1}{2}$
$\text{ Putting the values we get }$
$T\propto G^{-1/2}{M}^{-1/2}{R}^{3/2}\propto \sqrt{\frac{R^3}{GM}}$
$\text{ The actual expression is }T=2\pi \sqrt{\frac{R^3}{GM}}$