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= kIᵃωᵇ where k is a dimensionless constant, find a and b. Moment of inertia of a sphere about its diameter is $\frac{2}{5}$ Mr² .
Concept of Physics - 1 , HC VERMA , Chapter "Introduction to Physics".

Answer 1

Hello Dear.

Given K= kIᵃωᵇ -------→ (i)

where k is the kinetic energy of any rotating body also a dimensionless constant.
So,
Dimension of Kinetic energy, K= [ML²T⁻²]

Dimension of Moment of Inertia , Iᵃ = [ML²]ᵃ

Dimension of angular speed , ωᵇ = [T⁻¹]ᵇ

as we studied the principle of homogeneity of dimension,
Now put dimension in eq (i)
[ML²T⁻²] = [ML²]ᵃ × [T⁻¹]ᵇ
Equating the Exponents of the similar Quantities.
2=2a ; -b=-2    ∵ (L²=L²ᵃ & T⁻²=T⁻ᵇ)

Hence, a=1 ; b= 2


Hope it Helps.

Answer 2

Heya user☺☺

Hope it will help☺☺