The kinetic energy K of a rotating body depends on its moment
of inertia I and its angular speed ω. Assuming the relation to be K =
kIa ωb where k is a dimensionless constant, find a and b. Moment of
inertia of a sphere about its diameter is Mr2.
K = k Ia ωb
Dimensions of K = [ML2T-2]
Dimensions of I = [ML2]
Dimensions of ω = [T-1]
Dimensionally
[ML2T-2] = [ML2] a [T-1] b
= [ ML2aT-b]
From here 2a = 2 and -2 = -b
So a = 1 and b = 2.