The string of a pendulum is horizontal. The mass of the bob is m. Now the string is released. The tension in the string in the lowest position is -

Options

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

Explanation

To solve the problem of finding the tension in the string of a pendulum at its lowest position after being released from a horizontal position, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- The pendulum consists of a string of length \( l \) with a bob of mass \( m \) attached to it.
- The string is initially horizontal and then released.

2. **Identify Forces at the Lowest Point**:
- At the lowest point (let's call it point B), the forces acting on the bob are:
- The gravitational force \( mg \) acting downward.
- The tension \( T \) in the string acting upward.

3. **Use Conservation of Mechanical Energy**:
- Since the pendulum is released from rest, we can use the conservation of mechanical energy to find the speed of the bob at the lowest point.
- At the initial position (point A), the potential energy (PE) is maximum and kinetic energy (KE) is zero:
\[ PE_A = mgh = mgL \quad (\text{since height } h = L) \]
- At the lowest point (point B), the potential energy is zero and the kinetic energy is maximum:
\[ PE_B = 0, \quad KE_B = \frac{1}{2} mv^2 \]

4. **Set Up the Energy Conservation Equation**:
- According to the conservation of energy:
\[ PE_A + KE_A = PE_B + KE_B \]
\[ mgL + 0 = 0 + \frac{1}{2} mv^2 \]
- Simplifying gives:
\[ mgL = \frac{1}{2} mv^2 \]

5. **Solve for the Speed \( v \)**:
- Cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ gL = \frac{1}{2} v^2 \]
- Rearranging gives:
\[ v^2 = 2gL \]

6. **Calculate the Tension at the Lowest Point**:
- At the lowest point, we can apply Newton's second law:
\[ T - mg = \frac{mv^2}{l} \]
- Substitute \( v^2 \) from the previous step:
\[ T - mg = \frac{m(2gL)}{l} \]
- Rearranging gives:
\[ T = mg + \frac{2mgL}{l} \]
- Since \( L = l \), we have:
\[ T = mg + 2mg = 3mg \]

### Final Answer:
The tension in the string at the lowest position is \( T = 3mg \).

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