If the orbital velocity of a planet is given by `v = G^(a) M^(b) R^(c )` then

Options

A. `a = (1)/(2), b = (1)/(2), c = (1)/(2)`
B. `a = (1)/(2), b = (1)/(2), c = - (1)/(2)`
C. `a = (1)/(2), b = - (1)/(2), c = (1)/(2)`
D. `a = (1)/(3), b = - (1)/(3), c = - (1)/(3)`

Answer

B

Explanation

(2) `v = G^(a) M^(b) R^(c )`
`LT^(-1) = [M^(-1) L^(3) T^(-2)]^(a) [M]^(b) [L]^(c )`
`M^(0) L^(1) T^(-1) = M^(-a + b) L^(3a + c) T^(-2a)`
Comparing power of `M, L` and `T`
`-a + b = 0`
`3 a + c = 1`
`-2a = - 1 implies a = (1)/(2), b = (1)/(2), c = -(1)/(2)`
OR
Orbital velocity `v = sqrt((GM)/(R )), v = G^(a) M^(b) R^(c )`