A circular disc has a mass of 1kg and radius 40 cm. It is rotating about an axis passing through its centre and perpendicular to its plane with a speed of 10rev/s. The work done in joules in stopping it would be-

Options

A. 4
B. 47.5
C. 79
D. 158

Answer

To solve the problem of calculating the work done in stopping a rotating circular disc, we can follow these steps:

### Step 1: Identify the given values
- Mass of the disc (m) = 1 kg
- Radius of the disc (r) = 40 cm = 0.4 m (conversion from cm to m)
- Speed of rotation (f) = 10 revolutions per second

### Step 2: Convert the speed of rotation to angular velocity
Angular velocity (ω) in radians per second can be calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the given frequency:
\[ \omega = 2\pi \times 10 = 20\pi \text{ rad/s} \]

### Step 3: Calculate the moment of inertia (I) of the disc
The moment of inertia for a circular disc rotating about an axis through its center is given by:
\[ I = \frac{1}{2} m r^2 \]
Substituting the values:
\[ I = \frac{1}{2} \times 1 \times (0.4)^2 = \frac{1}{2} \times 1 \times 0.16 = 0.08 \text{ kg m}^2 \]

### Step 4: Calculate the initial rotational kinetic energy (KE)
The rotational kinetic energy is given by:
\[ KE = \frac{1}{2} I \omega^2 \]
Substituting the values of I and ω:
\[ KE = \frac{1}{2} \times 0.08 \times (20\pi)^2 \]
Calculating \( (20\pi)^2 \):
\[ (20\pi)^2 = 400\pi^2 \]
Now substituting back:
\[ KE = \frac{1}{2} \times 0.08 \times 400\pi^2 = 0.04 \times 400\pi^2 = 16\pi^2 \text{ J} \]

### Step 5: Calculate the numerical value of the kinetic energy
Using the approximation \( \pi \approx 3.14 \):
\[ KE \approx 16 \times (3.14)^2 \approx 16 \times 9.8596 \approx 157.76 \text{ J} \]

### Step 6: Determine the work done to stop the disc
The work done (W) in stopping the disc is equal to the change in kinetic energy. Since we are stopping the disc, the final kinetic energy is 0:
\[ W = KE_{\text{initial}} - KE_{\text{final}} = 157.76 - 0 = 157.76 \text{ J} \]
Rounding this value gives approximately:
\[ W \approx 158 \text{ J} \]

### Final Answer
The work done in stopping the disc is approximately **158 joules**.

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