A metre stick is pivoted about its centre. A piece of wax of mass 20 g travelling horizontally and perpendicular to it at 5 m/s strikes and adheres to one end of the stick so that the stick starts to rotate in a horizontal circle. Given the moment of inertia of the stick and wax about the pivot is 0.02 kg `m^(2)`, the initial angular velocity of the stick is :

Options

A. 1.58 rad/s
B. 2.24 rad/s
C. 2.50 rad/s
D. 5.00 rad/s

Answer

To solve the problem, we will apply the principle of conservation of angular momentum. Here are the steps to find the initial angular velocity of the stick after the wax adheres to it:

### Step 1: Understand the System
- We have a meter stick pivoted at its center.
- A piece of wax with a mass of 20 g (0.02 kg) is moving horizontally at a speed of 5 m/s and strikes one end of the stick.

### Step 2: Identify Initial Angular Momentum
- The initial angular momentum of the system consists only of the wax since the stick is at rest.
- The angular momentum \( L \) of the wax can be calculated using the formula:
\[ L = m \cdot v \cdot r \]
where:
- \( m = 0.02 \) kg (mass of the wax)
- \( v = 5 \) m/s (velocity of the wax)
- \( r = 0.5 \) m (distance from the pivot to the end of the stick)

### Step 3: Calculate Initial Angular Momentum
- Substitute the values into the formula:
\[ L = 0.02 \, \text{kg} \cdot 5 \, \text{m/s} \cdot 0.5 \, \text{m} \]
\[ L = 0.02 \cdot 5 \cdot 0.5 = 0.05 \, \text{kg m}^2/\text{s} \]

### Step 4: Identify Final Angular Momentum
- After the wax sticks to the end of the stick, the final angular momentum \( L_f \) of the system is given by:
\[ L_f = I \cdot \omega \]
where:
- \( I = 0.02 \, \text{kg m}^2 \) (moment of inertia of the stick and wax about the pivot)
- \( \omega \) is the angular velocity we need to find.

### Step 5: Apply Conservation of Angular Momentum
- According to the conservation of angular momentum:
\[ L_i = L_f \]
Therefore:
\[ 0.05 = 0.02 \cdot \omega \]

### Step 6: Solve for Angular Velocity \( \omega \)
- Rearranging the equation to find \( \omega \):
\[ \omega = \frac{0.05}{0.02} \]
\[ \omega = 2.5 \, \text{rad/s} \]

### Final Answer
The initial angular velocity of the stick after the wax adheres to it is \( \omega = 2.5 \, \text{rad/s} \).

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