Two discs of moment of inertia `9kgm^2` and `3kgm^2`were rotating with angular velocity 6 rad/sec and 10 rad/sec respectively in same direction. They are brought together gently to move with same angular velocity. The loss of kinetic energy in Joules is______.

Explanation

To solve the problem, we need to find the loss of kinetic energy when two discs with different moments of inertia and angular velocities are brought together to rotate with the same angular velocity. Here’s a step-by-step solution:

### Step 1: Calculate the Initial Angular Momentum
The angular momentum \( L \) of a rotating object is given by the formula:
\[ L = I \cdot \omega \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

For the first disc:
- Moment of inertia \( I_1 = 9 \, \text{kgm}^2 \)
- Angular velocity \( \omega_1 = 6 \, \text{rad/s} \)

Calculating the angular momentum of the first disc:
\[ L_1 = I_1 \cdot \omega_1 = 9 \cdot 6 = 54 \, \text{kgm}^2/\text{s} \]

For the second disc:
- Moment of inertia \( I_2 = 3 \, \text{kgm}^2 \)
- Angular velocity \( \omega_2 = 10 \, \text{rad/s} \)

Calculating the angular momentum of the second disc:
\[ L_2 = I_2 \cdot \omega_2 = 3 \cdot 10 = 30 \, \text{kgm}^2/\text{s} \]

### Step 2: Total Initial Angular Momentum
The total initial angular momentum \( L_{\text{initial}} \) is the sum of the angular momenta of both discs:
\[ L_{\text{initial}} = L_1 + L_2 = 54 + 30 = 84 \, \text{kgm}^2/\text{s} \]

### Step 3: Calculate the Final Angular Velocity
Since no external torque is applied, angular momentum is conserved. Let \( \omega_f \) be the final angular velocity when both discs are rotating together.

The total moment of inertia when they are combined:
\[ I_{\text{total}} = I_1 + I_2 = 9 + 3 = 12 \, \text{kgm}^2 \]

Using conservation of angular momentum:
\[ L_{\text{initial}} = I_{\text{total}} \cdot \omega_f \]
\[ 84 = 12 \cdot \omega_f \]
Solving for \( \omega_f \):
\[ \omega_f = \frac{84}{12} = 7 \, \text{rad/s} \]

### Step 4: Calculate Initial Kinetic Energy
The initial kinetic energy \( KE_{\text{initial}} \) is the sum of the kinetic energies of both discs:
\[ KE_{\text{initial}} = \frac{1}{2} I_1 \omega_1^2 + \frac{1}{2} I_2 \omega_2^2 \]
Calculating:
\[ KE_{\text{initial}} = \frac{1}{2} \cdot 9 \cdot 6^2 + \frac{1}{2} \cdot 3 \cdot 10^2 \]
\[ = \frac{1}{2} \cdot 9 \cdot 36 + \frac{1}{2} \cdot 3 \cdot 100 \]
\[ = 162 + 150 = 312 \, \text{J} \]

### Step 5: Calculate Final Kinetic Energy
The final kinetic energy \( KE_{\text{final}} \) when both discs rotate together at \( \omega_f \):
\[ KE_{\text{final}} = \frac{1}{2} I_{\text{total}} \omega_f^2 \]
Calculating:
\[ KE_{\text{final}} = \frac{1}{2} \cdot 12 \cdot 7^2 \]
\[ = \frac{1}{2} \cdot 12 \cdot 49 = 294 \, \text{J} \]

### Step 6: Calculate the Loss of Kinetic Energy
The loss of kinetic energy is given by:
\[ \text{Loss} = KE_{\text{initial}} - KE_{\text{final}} \]
Calculating:
\[ \text{Loss} = 312 - 294 = 18 \, \text{J} \]

### Final Answer
The loss of kinetic energy is **18 Joules**.