A small sphere of mass m is supended by a thread of length l. It is raised upto the height of suspension with thread fully stretched and released. Then, the maximum tension in thread will be

Options

A. mg
B. 2 mg
C. 3 mg
D. 6 mg

Explanation

To solve the problem of determining the maximum tension in the thread when a small sphere of mass \( m \) is suspended and released from a height equal to the length of the thread \( l \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- The sphere is raised to the height \( l \) (the length of the thread) and then released. At this point, the potential energy is at its maximum, and the kinetic energy is zero.

2. **Calculating Initial Potential Energy**:
- The potential energy (PE) at the height \( l \) is given by:
\[ PE = mgh = mgL \]
- Here, \( h = l \) is the height from which the sphere is released.

3. **Conservation of Energy**:
- As the sphere falls, the potential energy converts into kinetic energy (KE). At the lowest point, all the potential energy will have converted into kinetic energy.
- The kinetic energy at the lowest point is:
\[ KE = \frac{1}{2} mv^2 \]
- By conservation of energy:
\[ mgL = \frac{1}{2} mv^2 \]

4. **Solving for Velocity**:
- Rearranging the energy equation gives:
\[ v^2 = 2gL \]
- Thus, the velocity \( v \) at the lowest point is:
\[ v = \sqrt{2gL} \]

5. **Calculating the Forces at the Lowest Point**:
- At the lowest point, the tension \( T \) in the thread must counteract both the weight of the sphere and provide the centripetal force required for circular motion.
- The weight of the sphere is \( mg \).
- The centripetal force needed is given by:
\[ F_c = \frac{mv^2}{R} \]
- Here, \( R = L \) (the radius of the circular path).

6. **Substituting for Centripetal Force**:
- Substituting \( v^2 \):
\[ F_c = \frac{m(2gL)}{L} = 2mg \]

7. **Finding the Total Tension**:
- The total tension in the thread at the lowest point is:
\[ T = mg + F_c = mg + 2mg = 3mg \]

8. **Conclusion**:
- Therefore, the maximum tension in the thread when the sphere is at the lowest point is:
\[ T_{max} = 3mg \]

### Final Answer:
The maximum tension in the thread will be \( 3mg \).