A particle is moving in a circular path in the vertical plane. It is attached at one end of a string of length l whose other end is fixed. The velocity at lowest point is u. The tension in the string is T and acceleration of the particle is a at any position then `T. a' is positive at the lowest point for

Options

A. `u=sqrt(2gl)`
B. `u=sqrt(5gl)`
C. `u=sqrt(7gl)`
D. any value of `u`

Explanation

To solve the problem step by step, we will analyze the motion of the particle in a vertical circular path and the relationship between tension (T) and acceleration (a) at the lowest point of the circular motion.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- A particle is attached to a string of length \( l \) and moves in a vertical circular path.
- The particle has a velocity \( u \) at the lowest point of the circular path.

2. **Forces Acting on the Particle**:
- At the lowest point, the forces acting on the particle are:
- The tension \( T \) in the string acting upwards.
- The weight \( mg \) of the particle acting downwards (where \( m \) is the mass of the particle and \( g \) is the acceleration due to gravity).

3. **Applying Newton's Second Law**:
- At the lowest point, the net force acting on the particle provides the centripetal force required for circular motion.
- The equation can be written as:
\[ T - mg = \frac{mv^2}{l} \]
- Rearranging gives:
\[ T = mg + \frac{mv^2}{l} \]

4. **Acceleration of the Particle**:
- The acceleration \( a \) of the particle at the lowest point is the centripetal acceleration, which can be expressed as:
\[ a = \frac{u^2}{l} \]

5. **Dot Product of Tension and Acceleration**:
- The dot product \( T \cdot a \) is given by:
\[ T \cdot a = T \cdot a \cdot \cos(\theta) \]
- At the lowest point, both tension \( T \) and acceleration \( a \) act in the same direction (upwards), making the angle \( \theta = 0^\circ \).
- Therefore, \( \cos(0^\circ) = 1 \):
\[ T \cdot a = T \cdot a \]

6. **Conclusion on the Sign of \( T \cdot a \)**:
- Since both \( T \) and \( a \) are positive quantities at the lowest point, their product \( T \cdot a \) is also positive.
- Hence, we conclude that \( T \cdot a \) is positive at the lowest point for any value of \( u \).

### Final Answer:
The dot product \( T \cdot a \) is positive at the lowest point for any value of \( u \).