A particle of mass m is tied to a string of length L and whirled into a horizontal plan. If tension in the string is T then the speed of the particle will be :

Options

A. `sqrt((Tl)/m)`
B. `sqrt((2Tl)/(m))`
C. `sqrt((3Tl)/m)`
D. `sqrt((T)/(ml))`

Answer

To find the speed of a particle of mass \( m \) that is tied to a string of length \( L \) and whirled in a horizontal plane with a tension \( T \) in the string, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Particle**:
The only force providing the necessary centripetal force to keep the particle moving in a circular path is the tension \( T \) in the string.

2. **Centripetal Force Equation**:
The centripetal force \( F_c \) required to keep the particle moving in a circle is given by the formula:
\[ F_c = \frac{mv^2}{R} \]
where \( v \) is the speed of the particle and \( R \) is the radius of the circular path. In this case, the radius \( R \) is equal to the length of the string \( L \).

3. **Relate Tension to Centripetal Force**:
Since the tension \( T \) in the string provides the centripetal force, we can set the tension equal to the centripetal force:
\[ T = \frac{mv^2}{L} \]

4. **Solve for Speed \( v \)**:
Rearranging the equation to solve for \( v \), we get:
\[ mv^2 = TL \]
\[ v^2 = \frac{TL}{m} \]
\[ v = \sqrt{\frac{TL}{m}} \]

5. **Final Expression for Speed**:
Thus, the speed of the particle is:
\[ v = \sqrt{\frac{TL}{m}} \]

### Final Answer:
The speed of the particle is \( v = \sqrt{\frac{TL}{m}} \).

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