A body of mass `m` is tied with rope and rotated along horizontal circle of radius r. If T is the tension in the rope and v is the velocity of body and an instant the force required for circular motion is

Options

A. `T + (mv^2)/r`
B. `T - (mv^2)/r`
C. ` (mv^2)/r`
D. `(mv^2)/(4r)`

Answer

To solve the problem, we need to determine the force required for a body of mass `m` to move in a circular path of radius `r` at a velocity `v`. This force is known as the centripetal force.

### Step-by-Step Solution:

1. **Understanding Circular Motion**:
- When an object moves in a circular path, it requires a force directed towards the center of the circle to maintain that circular motion. This force is called the centripetal force.

2. **Centripetal Force Formula**:
- The formula for centripetal force \( F_c \) is given by:
\[ F_c = \frac{mv^2}{r} \]
where:
- \( m \) is the mass of the body,
- \( v \) is the velocity of the body,
- \( r \) is the radius of the circular path.

3. **Identifying the Role of Tension**:
- In this scenario, the tension \( T \) in the rope provides the necessary centripetal force to keep the body moving in a circular path. Therefore, we can equate the tension to the centripetal force:
\[ T = \frac{mv^2}{r} \]

4. **Conclusion**:
- Thus, the force required for circular motion, which is the centripetal force, is equal to the tension in the rope:
\[ T = \frac{mv^2}{r} \]

### Final Answer:
The force required for circular motion is \( \frac{mv^2}{r} \).

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