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, the radius of the circular orbit R. Obtain an expression for T using dimensional analysis.

Explanation

We have `[T]=[G]^(@) [M]^(@) [R]^(@)`
`[M]^(0)[L]^(0)[T]^(1)=[M]^(-a)[L]^(3a)[T]^(-2a)xx[M]^(b-a)[L]^(c+3a)[T]^(-2a)`
Comparing the exponents
For `[T]: 1=-2arArra=-(1)/(2)` For `[M]: 0-b-arArrb=a=-(1)/(2)`
For `[L]: 0=c+3arArrc=-3a=(3)/(2)`
Putting the values we get `Tprop G^(-1//2)M^(-1//2)R^(3//2)propsqrt((R^(3))/(GM))`
The actual expression is `T=2pisqrt((R^(3))/(GM))`