A particle of mass m is tied to light string and rotated with a speed v along a circular path of radius r. If T=tension in the string and `mg=` gravitational force on the particle then actual forces acting on the particle are

Options

A. mg and T only
B. mg T and on additional force of `(mv^(2))/(r )` directed inwards.
C. mg, T and an additional force of `(mv^(2))/(r )` directed outwards.
D. only a force `(mv^(2))/(r )` directed outwards

Answer

To solve the problem, we need to analyze the forces acting on a particle of mass \( m \) that is tied to a light string and rotated in a circular path with a speed \( v \) and radius \( r \).

### Step-by-step Solution:

1. **Identify the Forces Acting on the Particle:**
- The particle experiences two main forces:
- The gravitational force acting downward, which is \( mg \).
- The tension \( T \) in the string acting upward.

2. **Draw the Free Body Diagram:**
- Draw the particle and indicate the forces:
- An arrow pointing downwards labeled \( mg \).
- An arrow pointing upwards labeled \( T \).

3. **Understand Circular Motion:**
- For an object moving in a circular path, a centripetal force is required to keep it in motion. This force is directed towards the center of the circular path.
- The required centripetal force \( F_c \) can be expressed as:
\[ F_c = \frac{mv^2}{r} \]

4. **Resolve the Tension Force:**
- The tension \( T \) in the string can be resolved into two components:
- A vertical component \( T \cos \theta \) that balances the gravitational force \( mg \).
- A horizontal component \( T \sin \theta \) that provides the necessary centripetal force.
- Therefore, we can write:
\[ T \cos \theta = mg \quad \text{(1)} \]
\[ T \sin \theta = \frac{mv^2}{r} \quad \text{(2)} \]

5. **Equilibrium in the Vertical Direction:**
- Since the particle is in equilibrium in the vertical direction, the upward force (tension component) must equal the downward force (weight):
\[ T \cos \theta = mg \]

6. **Centripetal Force Requirement:**
- The horizontal component of tension provides the centripetal force necessary for circular motion:
\[ T \sin \theta = \frac{mv^2}{r} \]

7. **Conclusion:**
- The actual forces acting on the particle are the gravitational force \( mg \) and the tension \( T \) in the string. Thus, the answer to the question is:
\[ \text{The actual forces acting on the particle are } mg \text{ and } T \text{ only.} \]