A simple pendulum of length l has a maximum angular displacement `theta`. The maximum kinetic energy of the bob of mass m will be

Options

A. `mgl(1-costheta)`
B. `mglcostheta`
C. `mgl sintheta`
D. None of these

Explanation

To find the maximum kinetic energy of a simple pendulum of length \( l \) with a maximum angular displacement \( \theta \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Energy Conservation Principle**:
The total mechanical energy in a pendulum system is conserved when only conservative forces (like gravity) are acting. This means that the sum of potential energy (PE) and kinetic energy (KE) at one position will equal the sum at another position.

2. **Identify the Positions**:
Let's denote the lowest point of the pendulum swing as position A (where kinetic energy is maximum) and the highest point of the swing as position B (where kinetic energy is zero).

3. **Potential Energy at Position B**:
At the highest point (position B), the pendulum has maximum potential energy and zero kinetic energy. The height \( h \) from the lowest point to the highest point can be calculated using the length of the pendulum \( l \) and the angle \( \theta \):
\[ h = l - l \cos \theta = l(1 - \cos \theta) \]

4. **Calculate Potential Energy at Position B**:
The potential energy at position B can be expressed as:
\[ PE_B = mgh = mg(l(1 - \cos \theta)) \]

5. **Kinetic Energy at Position A**:
At the lowest point (position A), all the potential energy has been converted into kinetic energy. Therefore, the maximum kinetic energy \( KE_{max} \) at position A is equal to the potential energy at position B:
\[ KE_{max} = PE_B = mg(l(1 - \cos \theta)) \]

6. **Final Expression for Maximum Kinetic Energy**:
Thus, the maximum kinetic energy of the bob of mass \( m \) is given by:
\[ KE_{max} = mg l (1 - \cos \theta) \]

### Conclusion:
The maximum kinetic energy of the bob of mass \( m \) in a simple pendulum of length \( l \) with a maximum angular displacement \( \theta \) is:
\[ KE_{max} = mg l (1 - \cos \theta) \]