A simple pendulum of length `l` has maximum angular displacement `theta` . Then maximum kinetic energy of a bob of mass `m` is

Options

A. `(1)/(2)mgl`
B. `(1)/(2)mglcostheta`
C. `mgl(1-costheta)`
D. `(1)/(2)mglsintheta`

Explanation

To find the maximum kinetic energy of a bob of mass `m` in a simple pendulum of length `l` with a maximum angular displacement `theta`, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Pendulum's Motion**:
- A simple pendulum swings back and forth. The maximum angular displacement is denoted as `theta`. At the extreme positions, the pendulum has maximum potential energy and zero kinetic energy.

2. **Identify the Positions**:
- At the extreme position (maximum displacement), the bob is momentarily at rest, so the kinetic energy (KE) is zero, and the potential energy (PE) is at its maximum.
- At the mean position (the lowest point of the swing), the kinetic energy is maximum, and the potential energy is zero.

3. **Calculate the Height (h)**:
- The height `h` of the bob at the extreme position can be calculated using trigonometry. If we consider the vertical line from the pivot to the lowest point of the swing:
- The length of the pendulum is `l`.
- The vertical distance from the pivot to the bob at the extreme position is `l * cos(theta)`.
- Therefore, the height `h` from the lowest point to the extreme position is:
\[ h = l - l \cos(\theta) = l(1 - \cos(\theta)) \]

4. **Calculate the Potential Energy (PE)**:
- The potential energy at the extreme position is given by:
\[ PE = mgh = mg \cdot h = mg \cdot l(1 - \cos(\theta)) = mgl(1 - \cos(\theta)) \]

5. **Apply Conservation of Energy**:
- Since energy is conserved in the absence of friction, the total mechanical energy at the extreme position (which is all potential energy) is equal to the total mechanical energy at the mean position (which is all kinetic energy).
- Therefore, we have:
\[ KE_{\text{max}} = PE_{\text{max}} = mgl(1 - \cos(\theta)) \]

6. **Conclusion**:
- The maximum kinetic energy of the bob is:
\[ KE_{\text{max}} = mgl(1 - \cos(\theta)) \]

### Final Answer:
The maximum kinetic energy of the bob of mass `m` is:
\[ KE_{\text{max}} = mgl(1 - \cos(\theta)) \]