A disc of mass 2 kg and radius 0.2 m is rotating with angular velocity `30 "rad s"^(-1)`. What is angular velocity, if a mass of 0.25 kg is put on periphery of the disc ?

Options

A. 24 rad `s^(-1)`
B. 36 rad `s^(-1)`
C. 15 rad `s^(-1)`
D. 26 rad `s^(-1)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Calculate the Moment of Inertia of the Disc
The moment of inertia \( I \) of a disc is given by the formula:
\[ I = \frac{1}{2} m r^2 \]
where \( m \) is the mass of the disc and \( r \) is its radius.

Given:
- Mass of the disc, \( m = 2 \, \text{kg} \)
- Radius of the disc, \( r = 0.2 \, \text{m} \)

Substituting the values:
\[ I = \frac{1}{2} \times 2 \times (0.2)^2 = \frac{1}{2} \times 2 \times 0.04 = 0.04 \, \text{kg m}^2 \]

### Step 2: Calculate the Moment of Inertia of the Added Mass
The moment of inertia \( I_m \) of a point mass \( m \) at a distance \( r \) from the axis of rotation is given by:
\[ I_m = m r^2 \]
where \( m = 0.25 \, \text{kg} \) (the mass added at the periphery) and \( r = 0.2 \, \text{m} \).

Substituting the values:
\[ I_m = 0.25 \times (0.2)^2 = 0.25 \times 0.04 = 0.01 \, \text{kg m}^2 \]

### Step 3: Calculate the New Moment of Inertia
The new moment of inertia \( I' \) after adding the mass is:
\[ I' = I + I_m = 0.04 + 0.01 = 0.05 \, \text{kg m}^2 \]

### Step 4: Apply Conservation of Angular Momentum
Since no external torque is acting on the system, the angular momentum before adding the mass must equal the angular momentum after adding the mass.

The initial angular momentum \( L_1 \) is:
\[ L_1 = I \omega_1 = 0.04 \times 30 = 1.2 \, \text{kg m}^2/\text{s} \]
where \( \omega_1 = 30 \, \text{rad/s} \).

The final angular momentum \( L_2 \) is:
\[ L_2 = I' \omega_2 \]
where \( \omega_2 \) is the new angular velocity we want to find.

Setting \( L_1 = L_2 \):
\[ 1.2 = 0.05 \omega_2 \]

### Step 5: Solve for the New Angular Velocity
Rearranging the equation to find \( \omega_2 \):
\[ \omega_2 = \frac{1.2}{0.05} = 24 \, \text{rad/s} \]

### Final Answer
The new angular velocity \( \omega_2 \) after adding the mass is:
\[ \omega_2 = 24 \, \text{rad/s} \]

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