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