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 circular motion of a body of mass \( m \) that is being rotated in a horizontal circle of radius \( r \) with a velocity \( v \). The tension \( T \) in the rope is responsible for providing the necessary 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 keep it in that path. This force is known as the centripetal force.

2. **Formula for Centripetal Force**:
- The formula for the centripetal force \( F_c \) required to keep an object of mass \( m \) moving in a circle of radius \( r \) at a velocity \( v \) is given by:
\[ F_c = \frac{mv^2}{r} \]

3. **Identifying the Role of Tension**:
- In this scenario, the tension \( T \) in the rope acts as the centripetal force. Therefore, we can equate the tension in the rope to the centripetal force required for circular motion:
\[ T = F_c \]

4. **Final Equation**:
- Substituting the expression for centripetal force into the equation gives us:
\[ T = \frac{mv^2}{r} \]
- This means that the tension in the rope is equal to the centripetal force needed to keep the body moving in a circular path.

### Conclusion:
Thus, the force required for circular motion, which is provided by the tension in the rope, is given by:
\[ T = \frac{mv^2}{r} \]