A dumb bell consists of two identicasl small balls offmss 1/2 kg each connected to the ends of a 50 cm long light rod. The dumb bell is rotating about a fixed axis through the centre of the rod and perpendicular to it t an angular speed of 10 rad/s. An impulsive force of average magnitude 5.0 N acts on one of the masses in the direction of its velocity for 0.10s. Find the new angular velocity of the system.

Explanation

To solve the problem step by step, we will follow the principles of rotational dynamics.

### Step 1: Identify the given data
- Mass of each ball, \( m = 0.5 \, \text{kg} \)
- Length of the rod, \( L = 0.5 \, \text{m} \) (which means the distance from the center to each mass, \( r = \frac{L}{2} = 0.25 \, \text{m} \))
- Initial angular velocity, \( \omega_i = 10 \, \text{rad/s} \)
- Impulsive force, \( F = 5 \, \text{N} \)
- Time duration of the force, \( t = 0.1 \, \text{s} \)

### Step 2: Calculate the moment of inertia \( I \) of the dumbbell
The moment of inertia \( I \) for the dumbbell about the axis of rotation is given by:
\[ I = 2 \cdot m \cdot r^2 \]
Substituting the values:
\[ I = 2 \cdot 0.5 \cdot (0.25)^2 = 2 \cdot 0.5 \cdot 0.0625 = 0.0625 \, \text{kg m}^2 \]

### Step 3: Calculate the torque \( \tau \) due to the impulsive force
The torque \( \tau \) is given by:
\[ \tau = F \cdot r \]
Substituting the values:
\[ \tau = 5 \cdot 0.25 = 1.25 \, \text{N m} \]

### Step 4: Calculate the angular acceleration \( \alpha \)
Using the relation between torque and angular acceleration:
\[ \tau = I \cdot \alpha \]
We can rearrange this to find \( \alpha \):
\[ \alpha = \frac{\tau}{I} \]
Substituting the values:
\[ \alpha = \frac{1.25}{0.0625} = 20 \, \text{rad/s}^2 \]

### Step 5: Calculate the final angular velocity \( \omega_f \)
Using the equation of motion for angular velocity:
\[ \omega_f = \omega_i + \alpha t \]
Substituting the values:
\[ \omega_f = 10 + 20 \cdot 0.1 = 10 + 2 = 12 \, \text{rad/s} \]

### Final Answer
The new angular velocity of the system is \( \omega_f = 12 \, \text{rad/s} \).

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