The string of pendulum of length l is displaced through `90^(@)` from the vertical and released. Then the minimum strength of the string in order to withstand the tension, as the pendulum passes through the mean position is

Options

A. mg
B. 2mg
C. 3mg
D. 4mg

Answer

To find the minimum strength of the string required to withstand the tension as the pendulum passes through the mean position, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The pendulum is displaced through \(90^\circ\) from the vertical and then released. We need to find the tension in the string when the pendulum reaches the mean position (the lowest point).

2. **Applying Work-Energy Theorem**:
- According to the work-energy theorem, the work done by gravity is equal to the change in kinetic energy.
- When the pendulum is at the highest point (point A), its velocity is zero, and when it reaches the mean position (point B), it has maximum kinetic energy.

3. **Calculating Work Done by Gravity**:
- The work done by gravity when the pendulum moves from point A to point B can be calculated as:
\[ \text{Work done by gravity} = mgh \]
- Here, \(h\) is the vertical height fallen, which is equal to the length of the pendulum \(L\) (since it falls a distance \(L\) when moving from the highest point to the lowest point).
- Therefore, the work done by gravity is:
\[ W = mgL \]

4. **Change in Kinetic Energy**:
- The change in kinetic energy from point A to point B is given by:
\[ \Delta KE = \frac{1}{2} mv^2 - 0 = \frac{1}{2} mv^2 \]

5. **Setting Work Done Equal to Change in Kinetic Energy**:
- From the work-energy principle:
\[ mgL = \frac{1}{2} mv^2 \]
- Canceling \(m\) from both sides (assuming \(m \neq 0\)):
\[ gL = \frac{1}{2} v^2 \]
- Rearranging gives:
\[ v^2 = 2gL \]

6. **Finding Tension at the Mean Position**:
- At the mean position (point B), the forces acting on the pendulum bob are:
- Tension \(T_B\) acting upwards.
- Weight \(mg\) acting downwards.
- The net force towards the center (centripetal force) is given by:
\[ T_B - mg = \frac{mv^2}{L} \]
- Substituting \(v^2\) from the previous calculation:
\[ T_B - mg = \frac{m(2gL)}{L} \]
- Simplifying gives:
\[ T_B - mg = 2mg \]
- Therefore:
\[ T_B = 2mg + mg = 3mg \]

7. **Conclusion**:
- The minimum strength of the string required to withstand the tension as the pendulum passes through the mean position is:
\[ T_B = 3mg \]