A rotating body's kinetic energy K relies on its moment of inertia I and its angular speed . Suppose the relation is K = kI^a ^b , where k is a dimensionless constant. Determine a and b . The moment of inertia of a sphere about its diameter is given as 2 5 Mr^2 .

Step-by-step solution

Using dimensional analysis, the dimensions of the given quantities are: [K] = [M L^2 T⁻²] [I] = [M L^2] [ ] = [T⁻¹] Substituting these into the given relation K = k I^a ^b : [M L^2 T⁻²] = [M L^2]^a [T⁻¹]^b [M L^2 T⁻²] = [M^a L^ 2a T^ -b ] Comparing the powers of M , L , and T on both sides: For M : a = 1 For T : -b = -2 b = 2 Therefore, a = 1 and b = 2 .