A simple pendulum of length 1m has bob of mass 100 g. It is displaced through an angle of `60^@` from the vertcal and then released. The kinetic energy of bob when it passes through the mean position is

Options

A. `0.25J`
B. `0.5J`
C. `1.0J`
D. `1.4J`

Explanation

To solve the problem of finding the kinetic energy of the bob of a simple pendulum when it passes through the mean position, we can follow these steps:

### Step 1: Understand the System
We have a simple pendulum with:
- Length \( L = 1 \, \text{m} \)
- Mass of the bob \( m = 100 \, \text{g} = 0.1 \, \text{kg} \)
- The bob is displaced at an angle of \( 60^\circ \) from the vertical.

### Step 2: Determine the Height Change
When the pendulum is displaced to an angle of \( 60^\circ \), we need to find the vertical height \( h \) that the bob rises. The height can be calculated using the formula:

\[ h = L - L \cos(\theta) \]

Substituting the values:

\[ h = 1 - 1 \cos(60^\circ) \]

Since \( \cos(60^\circ) = \frac{1}{2} \):

\[ h = 1 - 1 \times \frac{1}{2} = 1 - 0.5 = 0.5 \, \text{m} \]

### Step 3: Calculate the Potential Energy at the Displaced Position
The potential energy (PE) at the height \( h \) is given by:

\[ PE = mgh \]

Substituting the values:

\[ PE = 0.1 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 0.5 \, \text{m} \]

Calculating this gives:

\[ PE = 0.1 \times 9.8 \times 0.5 = 0.49 \, \text{J} \]

### Step 4: Apply the Conservation of Energy Principle
When the bob is released, all of the potential energy will convert into kinetic energy (KE) at the mean position (lowest point). Therefore, we have:

\[ KE = PE \]

Thus, the kinetic energy at the mean position is:

\[ KE = 0.49 \, \text{J} \]

### Step 5: Conclusion
The kinetic energy of the bob when it passes through the mean position is approximately:

\[ \text{Kinetic Energy} = 0.49 \, \text{J} \]