A disc of mass 1 kg and radius 0.1 m is rotating with angular velocity 20 rad/s. What is angular velocity (in rad/s) if a mass of 0.5 kg is put on the circumference of the disc ?

Options

A. 10
B. 20
C. 40
D. 30

Answer

To solve the problem step by step, we will follow the principles of rotational motion and the conservation of angular momentum.

### Step 1: Identify the initial parameters
- Mass of the disc (m_disk) = 1 kg
- Radius of the disc (r) = 0.1 m
- Initial angular velocity (ω₁) = 20 rad/s

### Step 2: Calculate the moment of inertia of the disc
The moment of inertia (I) for a solid disc is given by the formula:
\[ I = \frac{1}{2} m r^2 \]

Substituting the values:
\[ I_1 = \frac{1}{2} \times 1 \, \text{kg} \times (0.1 \, \text{m})^2 \]
\[ I_1 = \frac{1}{2} \times 1 \times 0.01 \]
\[ I_1 = 0.005 \, \text{kg m}^2 \]

### Step 3: Calculate the new moment of inertia when a mass is added
When a mass (m_particle) of 0.5 kg is placed on the circumference of the disc, the new moment of inertia (I₂) can be calculated using:
\[ I_2 = I_1 + m_{\text{particle}} \cdot r^2 \]

Substituting the values:
\[ I_2 = 0.005 \, \text{kg m}^2 + 0.5 \, \text{kg} \cdot (0.1 \, \text{m})^2 \]
\[ I_2 = 0.005 + 0.5 \cdot 0.01 \]
\[ I_2 = 0.005 + 0.005 \]
\[ I_2 = 0.01 \, \text{kg m}^2 \]

### Step 4: Apply the conservation of angular momentum
According to the conservation of angular momentum:
\[ L_1 = L_2 \]
Where:
\[ L_1 = I_1 \cdot \omega_1 \]
\[ L_2 = I_2 \cdot \omega_2 \]

Setting them equal gives:
\[ I_1 \cdot \omega_1 = I_2 \cdot \omega_2 \]

### Step 5: Solve for the new angular velocity (ω₂)
Rearranging the equation to find ω₂:
\[ \omega_2 = \frac{I_1 \cdot \omega_1}{I_2} \]

Substituting the known values:
\[ \omega_2 = \frac{0.005 \, \text{kg m}^2 \cdot 20 \, \text{rad/s}}{0.01 \, \text{kg m}^2} \]
\[ \omega_2 = \frac{0.1}{0.01} \]
\[ \omega_2 = 10 \, \text{rad/s} \]

### Final Answer
The new angular velocity (ω₂) after adding the mass is **10 rad/s**.

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