Circular disc of mass 2 kg and radius 1 m is rotating about an axis perpendicular to its plane and passing through its centre of mass with a rotational kinetic energy of 8 J. The angular momentum is (Js) is

Options

A. 8
B. 4
C. 2
D. 1

Explanation

To find the angular momentum of the circular disc, we can follow these steps:

### Step 1: Determine the Moment of Inertia (I)
The moment of inertia \( I \) for a circular disc rotating about an axis perpendicular to its plane and passing through its center of mass is given by the formula:
\[ I = \frac{1}{2} M R^2 \]
where \( M \) is the mass of the disc and \( R \) is the radius.

Given:
- Mass \( M = 2 \, \text{kg} \)
- Radius \( R = 1 \, \text{m} \)

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

### Step 2: Use the Rotational Kinetic Energy to Find Angular Velocity (ω)
The rotational kinetic energy \( K \) is given by the formula:
\[ K = \frac{1}{2} I \omega^2 \]
We know the rotational kinetic energy \( K = 8 \, \text{J} \) and we have calculated \( I = 1 \, \text{kg m}^2 \).

Substituting the values into the equation:
\[ 8 = \frac{1}{2} \times 1 \times \omega^2 \]
This simplifies to:
\[ 8 = \frac{1}{2} \omega^2 \]
Multiplying both sides by 2:
\[ 16 = \omega^2 \]
Taking the square root:
\[ \omega = \sqrt{16} = 4 \, \text{radians/second} \]

### Step 3: Calculate Angular Momentum (L)
The angular momentum \( L \) is given by the formula:
\[ L = I \omega \]
Substituting the values we found:
\[ L = 1 \times 4 = 4 \, \text{Js} \]

### Final Answer
The angular momentum of the circular disc is \( 4 \, \text{Js} \).

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