A body of mass 2 kg is rotating on a circular path of radius 0.5 m, with an angular velocity of 20 rad/s. If the radius of the path is doubled, then the new angular velocity will be

Options

A. 5 rad/sec
B. 2.5 rad/sec
C. 10 rad/sec
D. 8 rad/sec

Explanation

To solve the problem, we will use the principle of conservation of angular momentum. The angular momentum of a rotating body is given by the formula:

\[ L = I \omega \]

where \(L\) is the angular momentum, \(I\) is the moment of inertia, and \(\omega\) is the angular velocity.

### Step-by-Step Solution:

1. **Identify the initial conditions**:
- Mass of the body, \(m = 2 \, \text{kg}\)
- Initial radius, \(r_1 = 0.5 \, \text{m}\)
- Initial angular velocity, \(\omega_1 = 20 \, \text{rad/s}\)

2. **Calculate the initial moment of inertia**:
For a point mass rotating in a circular path, the moment of inertia \(I\) is given by:
\[ I_1 = m r_1^2 \]
Substituting the values:
\[ I_1 = 2 \, \text{kg} \times (0.5 \, \text{m})^2 = 2 \times 0.25 = 0.5 \, \text{kg m}^2 \]

3. **Calculate the initial angular momentum**:
Using the formula for angular momentum:
\[ L_1 = I_1 \omega_1 = 0.5 \, \text{kg m}^2 \times 20 \, \text{rad/s} = 10 \, \text{kg m}^2/\text{s} \]

4. **Identify the new conditions**:
- New radius, \(r_2 = 2r_1 = 2 \times 0.5 \, \text{m} = 1.0 \, \text{m}\)

5. **Calculate the new moment of inertia**:
\[ I_2 = m r_2^2 = 2 \, \text{kg} \times (1.0 \, \text{m})^2 = 2 \times 1 = 2 \, \text{kg m}^2 \]

6. **Apply conservation of angular momentum**:
Since there is no external torque acting on the system, the angular momentum before and after must be equal:
\[ L_1 = L_2 \]
Therefore:
\[ I_1 \omega_1 = I_2 \omega_2 \]
Substituting the known values:
\[ 10 \, \text{kg m}^2/\text{s} = 2 \, \text{kg m}^2 \times \omega_2 \]

7. **Solve for the new angular velocity \(\omega_2\)**:
\[ \omega_2 = \frac{10 \, \text{kg m}^2/\text{s}}{2 \, \text{kg m}^2} = 5 \, \text{rad/s} \]

### Final Answer:
The new angular velocity \(\omega_2\) when the radius is doubled is \(5 \, \text{rad/s}\).

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