Find the value of a and b in the following cases :
(a) The velocity `v` of the ball falling freely under gravity is proportional to `g^(a) h^(b)`, where `g` is the acceleration due to gravity, `h` is the height from which the ball is dropped.
(b) The kinetic energy `K` of a rotating body is proportional to `I^(a) omega^(b)` where `I` is the moment if inertia and `omega` is the angular speed.
(c ) The time-period `T` of spring pendulum is proportiona to `m^(a) k^(b)`, where `m` is the mass of block attached to the spring and `k` is the spring constant.
The speed of sound `v` in a gaseous medium is proportional to `P^(a) rho^(b)`, where `P` is the pressure and `rho` is the density of medium.

Explanation

(a) `v prop g^(a) h^(b)`
`[LT^(-1)] prop [LT^(-2)]^(a) [L]^(b)`
`L^(1) T^(-1) prop L(a + b) T^(-2a)`
Comparing powers of `L` and `T`
`a + b = 1`
`-2 a = - 1`
`a = (1)/(2), b = (1)/(2)`
`K prop I^(a) omega^(b)`
`[ML^(2) T^(-2)] prop [ML^(2)]^(a) [T^(-1)]^(b)`
`M^(1) L^(2) T^(-2) prop M^(a) L^(2a) T^(-b)`
Comparing powers of `M` and `T`
`a = 1`
`-b = - 2`
`a = 1, b= 2`
(c ) `T prop m^(a) k^(b)`
`[T] prop [M]^(a) [MT^(-2)]^(b)`
`M^(0) T^(-1) prop M^(a + b) T^(-2b)`
Comparing power of `M` and `T`
`a + b = 0`
`-2 b = 1`
`a = (1)/(2), b = - (1)/(2)`
(d) `v prop P^(a) rho^(b)`
`[LT^(-1)] prop [ML^(-1) T^(-2)]^(a) [ML^(-3)]^(b)`
`M^(0) L^(1) T^(-1) prop M^(a + b) L^(-a - 3b) T^(-2a)`
Comparing powers of `M` and `T`
`a + b = 0`
`- 2a = - 1`
`a = (1)/(2), b = - (1)/(2)`