A simple pendulum of length `l` and mass `m` is initially at its lowest position. It is given the minimum horizontal speed necessary after to move in a circular path about the point of suspension. The tension in the string at the lowest positive of the bob is

Options

A. `3mg`
B. `4mg`
C. `5mg`
D. `6mg`

Explanation

To solve the problem of finding the tension in the string at the lowest point of a simple pendulum given the minimum horizontal speed necessary for it to move in a circular path, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the parameters**:
- Length of the pendulum: \( l \)
- Mass of the bob: \( m \)
- Gravitational acceleration: \( g \)

2. **Determine the minimum speed**:
The minimum horizontal speed \( v \) required for the bob to move in a circular path is given by the formula:
\[ v = \sqrt{g l} \]
This is the speed necessary to maintain circular motion at the top of the swing.

3. **Draw the free body diagram**:
At the lowest point of the swing, the forces acting on the bob are:
- The weight of the bob acting downwards: \( mg \)
- The tension in the string acting upwards: \( T \)

4. **Apply Newton's second law**:
At the lowest point, the net force acting on the bob provides the centripetal force required for circular motion. Therefore, we can write:
\[ T - mg = \frac{mv^2}{l} \]
where \( \frac{mv^2}{l} \) is the centripetal force.

5. **Substitute the expression for \( v \)**:
Substitute \( v = \sqrt{g l} \) into the centripetal force equation:
\[ T - mg = \frac{m(\sqrt{g l})^2}{l} \]
Simplifying this gives:
\[ T - mg = \frac{mg l}{l} = mg \]

6. **Solve for tension \( T \)**:
Rearranging the equation gives:
\[ T = mg + mg = 2mg \]

7. **Final result**:
The tension in the string at the lowest point of the bob is:
\[ T = 2mg \]