Question

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 $K=k{I}^a{\omega }^B$ where k is a dimensionless constant, find a and b. Moment of inertia of a sphere about its diameter is $\frac{2}{5}M{r}^2$

Answer 1

Kinetic energy of a rotating body is K = kI ^aω^b.

Dimensions of the quantities are [K] = [ML²T⁻²], [I] = [ML²] and [ω] = [T⁻¹].

Now, dimension of the right side are [I]^a = [ML²]^a and [ω]^b = [T⁻¹]^b.

According to the principal of homogeneity of dimension, we have:
[ML²T⁻²] = [ML²]^a [T⁻¹]^b

Equating the dimensions of both sides, we get:
2 = 2a
⇒ a = 1
And,
−2 = −b
⇒ b = 2