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, 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))`