The kinetic energy K of a rotating body depends on its moment of inertia I and its angular speed `omega`. Assuming the relation to be `K=kI^(alpha) omega^b` where k is a dimensionless constatnt, find a and b. Moment of inertia of a spere about its diameter is `2/5Mr^2`.

Explanation

To solve the problem, we need to find the values of \( a \) and \( b \) in the equation \( K = k I^a \omega^b \), where \( K \) is the kinetic energy, \( I \) is the moment of inertia, and \( \omega \) is the angular speed. We will use dimensional analysis to achieve this.

### Step-by-Step Solution:

1. **Identify the dimensions of kinetic energy \( K \)**:
The dimension of kinetic energy is given by:
\[ [K] = [\text{Energy}] = [\text{Mass}] \times [\text{Length}]^2 \times [\text{Time}]^{-2} = M^1 L^2 T^{-2} \]

2. **Identify the dimensions of moment of inertia \( I \)**:
The moment of inertia for a sphere about its diameter is given as:
\[ I = \frac{2}{5} M r^2 \]
Thus, the dimension of moment of inertia is:
\[ [I] = [\text{Mass}] \times [\text{Length}]^2 = M^1 L^2 \]

3. **Identify the dimensions of angular speed \( \omega \)**:
Angular speed \( \omega \) has the dimension of:
\[ [\omega] = [\text{Time}]^{-1} = T^{-1} \]

4. **Write the dimensions of the right-hand side of the equation**:
The right-hand side of the equation is \( K = k I^a \omega^b \). Since \( k \) is dimensionless, we can ignore its dimensions. Therefore, the dimensions can be expressed as:
\[ [K] = [I]^a \times [\omega]^b = (M^1 L^2)^a \times (T^{-1})^b = M^a L^{2a} T^{-b} \]

5. **Set up the dimensional equation**:
Now we equate the dimensions from both sides:
\[ M^1 L^2 T^{-2} = M^a L^{2a} T^{-b} \]

6. **Compare the powers of each dimension**:
- For mass \( M \):
\[ a = 1 \]
- For length \( L \):
\[ 2a = 2 \implies a = 1 \quad (\text{This confirms our previous result}) \]
- For time \( T \):
\[ -b = -2 \implies b = 2 \]

7. **Final values**:
From the comparisons, we find:
\[ a = 1, \quad b = 2 \]

### Conclusion:
The values of \( a \) and \( b \) are:
\[ \boxed{a = 1, \quad b = 2} \]