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 1: Determine the height change
When the pendulum bob is at its maximum angular displacement (point A), it is at a height above its lowest point (point B). The height `h` that the bob descends when it swings down can be calculated as follows:

\[ h = l - l \cos \theta \]

This simplifies to:

\[ h = l(1 - \cos \theta) \]

### Step 2: Use energy conservation to find maximum velocity
At the maximum height (point A), the bob has potential energy and no kinetic energy (since it is momentarily at rest). As it swings down to the lowest point (point B), all of the potential energy converts into kinetic energy.

Using the conservation of mechanical energy:

\[ \text{Potential Energy at A} = \text{Kinetic Energy at B} \]

The potential energy at point A is given by:

\[ PE = mgh = mg \cdot h = mg \cdot l(1 - \cos \theta) \]

At point B, the kinetic energy (KE) is given by:

\[ KE = \frac{1}{2} mv^2 \]

Setting the potential energy equal to the kinetic energy:

\[ mg \cdot l(1 - \cos \theta) = \frac{1}{2} mv^2 \]

### Step 3: Solve for maximum velocity
We can cancel `m` from both sides (assuming `m` is not zero):

\[ g \cdot l(1 - \cos \theta) = \frac{1}{2} v^2 \]

Multiplying both sides by 2:

\[ 2g \cdot l(1 - \cos \theta) = v^2 \]

Taking the square root gives us the maximum velocity:

\[ v = \sqrt{2g \cdot l(1 - \cos \theta)} \]

### Step 4: Calculate maximum kinetic energy
Now we can substitute `v` back into the kinetic energy formula:

\[ KE_{max} = \frac{1}{2} m v^2 \]

Substituting for `v^2`:

\[ KE_{max} = \frac{1}{2} m (2g \cdot l(1 - \cos \theta)) \]

This simplifies to:

\[ KE_{max} = mg \cdot l(1 - \cos \theta) \]

### Conclusion
Thus, the maximum kinetic energy of the bob is:

\[ KE_{max} = mg \cdot l(1 - \cos \theta) \]