A simple pendulum oscillates in a vertical plane. When it passes through the mean position, the tension in the string is `3` times the weight of the pendulum bob.what is the maximum displacement of the pendulum with respect to the vertical

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `90^(@)`

Explanation

To solve the problem, we will analyze the forces acting on the pendulum bob and use the principles of energy conservation and circular motion.

### Step-by-Step Solution:

1. **Understanding the Forces at Mean Position**:
At the mean position of the pendulum, two forces act on the bob: the gravitational force (weight) \( Mg \) acting downward and the tension \( T \) in the string acting upward. According to the problem, the tension is three times the weight of the bob:
\[ T = 3Mg \]

2. **Applying Newton's Second Law**:
At the mean position, the net force acting on the bob provides the centripetal force required for circular motion. The equation can be set up as follows:
\[ T - Mg = \frac{Mv^2}{L} \]
Substituting \( T = 3Mg \) into the equation gives:
\[ 3Mg - Mg = \frac{Mv^2}{L} \]
Simplifying this, we find:
\[ 2Mg = \frac{Mv^2}{L} \]

3. **Solving for Velocity**:
We can cancel \( M \) from both sides (assuming \( M \neq 0 \)):
\[ 2g = \frac{v^2}{L} \]
Rearranging gives us the expression for the velocity \( v \):
\[ v^2 = 2gL \quad \Rightarrow \quad v = \sqrt{2gL} \]

4. **Using Conservation of Energy**:
As the pendulum swings to its maximum height \( h \), all the kinetic energy will convert into potential energy. At the mean position, the kinetic energy is:
\[ KE = \frac{1}{2}Mv^2 = \frac{1}{2}M(2gL) = MgL \]
At the maximum height \( h \), the potential energy is:
\[ PE = Mgh \]
Setting the kinetic energy equal to the potential energy at maximum height gives:
\[ MgL = Mgh \]

5. **Solving for Maximum Height**:
Cancelling \( M \) from both sides (assuming \( M \neq 0 \)):
\[ gL = gh \quad \Rightarrow \quad h = L \]

6. **Finding Maximum Displacement**:
The maximum displacement \( \theta \) with respect to the vertical occurs when the pendulum reaches height \( h \). Since the pendulum swings to height \( h = L \), the angle \( \theta \) can be calculated using the geometry of the pendulum:
\[ \text{Maximum displacement} = L \quad \text{(which corresponds to an angle of 90 degrees)} \]

### Conclusion:
The maximum displacement of the pendulum with respect to the vertical is equal to the length of the string \( L \).