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 25Mr2

Answer 1

Value of a = 1 and b = 2.

Explanation:

Diameter = 2/5 Mr²

            K = KIawb

Where k = kinetic energy of moving body  and

           K = dimensionless constant

Dimensions of left side are :

           K = [ M L² $T^-^2$ ]

Dimensions of right side are :

          Ia = [ML² ]a  

        Wb = [ $T^-^1$ ]b

According to the principle of homogeneity of dimension  :

[ M L² $T^-^2$ ] = [ M L² $T^-^2$ ] [ $T^-^1$ ]b

Equating the dimension of both sides,

               2 = 2a  

And,       -2 = -b

                a = 1  ; b = 2

Therefore, the value of "a" and "b" is "a = 1" and "b = 2".

Answer 2

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(a)=1

(b) =2