The mass of a simple pendulum bob is 100 gm. The length of the pendulum is 1 m. The bob is drawn aside from the equilibrium position so that the string makes an angle of `60^(@)` with the vertical and let go. The kinetic energy of the bob while crossing its equilibrium position will be

Options

A. `0.49 J`
B. `0.94 J`
C. `1 J`
D. `1.2 J`

Explanation

To find the kinetic energy of the pendulum bob while crossing its equilibrium position, we can use the principle of conservation of mechanical energy. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Mass of the pendulum bob (m) = 100 gm = 0.1 kg (since 1 gm = 0.001 kg)
- Length of the pendulum (L) = 1 m
- Angle with the vertical (θ) = 60°

### Step 2: Calculate the height (h) of the bob when displaced
When the bob is displaced to an angle of 60°, we can find the height (h) it is raised from the lowest point (equilibrium position). The vertical height can be calculated using the formula:
\[ h = L - L \cos(θ) \]
Substituting the values:
\[ h = 1 - 1 \cos(60°) \]
Since \(\cos(60°) = 0.5\):
\[ h = 1 - 1 \times 0.5 = 1 - 0.5 = 0.5 \, \text{m} \]

### Step 3: Calculate the potential energy (PE) at the height
The potential energy at the height (h) can be calculated using the formula:
\[ PE = mgh \]
Where \( g \) (acceleration due to gravity) is approximately \( 9.8 \, \text{m/s}^2 \).
Substituting the values:
\[ PE = 0.1 \times 9.8 \times 0.5 \]
\[ PE = 0.1 \times 4.9 = 0.49 \, \text{J} \]

### Step 4: Apply the conservation of energy principle
At the highest point (when the bob is displaced), all the energy is potential energy. At the equilibrium position, all this potential energy will convert into kinetic energy (KE). Therefore:
\[ KE = PE \]
Thus:
\[ KE = 0.49 \, \text{J} \]

### Conclusion
The kinetic energy of the bob while crossing its equilibrium position is **0.49 J**.

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