If angular momentum of a body increased by 200% its kinetic energy will increase by :

Options

A. `400%`
B. `800%`
C. `200%`
D. `100%`

Answer

To solve the problem of how much the kinetic energy of a body increases when its angular momentum is increased by 200%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**:
The kinetic energy (KE) of a rotating body is related to its angular momentum (L) by the formula:
\[ KE = \frac{L^2}{2I} \]
where \(I\) is the moment of inertia. For a given body, \(I\) remains constant.

2. **Express the Increase in Angular Momentum**:
If the angular momentum increases by 200%, it means the new angular momentum \(L_2\) can be expressed as:
\[ L_2 = L_1 + 2L_1 = 3L_1 \]
where \(L_1\) is the initial angular momentum.

3. **Calculate the Ratio of Angular Momenta**:
The ratio of the initial angular momentum to the new angular momentum is:
\[ \frac{L_1}{L_2} = \frac{L_1}{3L_1} = \frac{1}{3} \]

4. **Relate Kinetic Energies**:
Since kinetic energy is proportional to the square of angular momentum, we can write:
\[ \frac{KE_1}{KE_2} = \left(\frac{L_1}{L_2}\right)^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]

5. **Calculate the Ratio of Kinetic Energies**:
This implies:
\[ \frac{KE_2}{KE_1} = 9 \]
Therefore, the new kinetic energy \(KE_2\) is 9 times the initial kinetic energy \(KE_1\).

6. **Determine the Increase in Kinetic Energy**:
The increase in kinetic energy can be calculated as:
\[ \text{Increase} = KE_2 - KE_1 = 9KE_1 - KE_1 = 8KE_1 \]

7. **Calculate the Percentage Increase**:
The percentage increase in kinetic energy is given by:
\[ \text{Percentage Increase} = \frac{\text{Increase}}{KE_1} \times 100\% = \frac{8KE_1}{KE_1} \times 100\% = 800\% \]

### Final Answer:
The kinetic energy will increase by **800%**.

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