A stone tied to a rope is rotated in a vertical circle with uniform speed. If the difference between the maximum and minimum tensions in the rope is 20 N, mass of the stone in kg is (take, g= 10 `ms^(-2)`)

Options

A. `0.75`
B. `1.0`
C. `1.5`
D. `0.5`

Explanation

To solve the problem, we need to find the mass of the stone based on the difference in maximum and minimum tensions in the rope when the stone is rotated in a vertical circle.

### Step-by-Step Solution:

1. **Understanding the Forces**:
- At the lowest point of the circle (point B), the tension (T_max) in the rope is at its maximum. The forces acting on the stone are:
- The gravitational force (weight) acting downwards: \( mg \)
- The centripetal force required to keep the stone moving in a circle, which is provided by the tension in the rope: \( T_{max} = mg + \frac{mv^2}{l} \)
- At the highest point of the circle (point A), the tension (T_min) in the rope is at its minimum. The forces acting on the stone are:
- The gravitational force (weight) acting downwards: \( mg \)
- The centripetal force is now provided by the difference between the tension and the weight: \( T_{min} = \frac{mv^2}{l} - mg \)

2. **Setting Up the Equations**:
- From the above points, we have:
- \( T_{max} = mg + \frac{mv^2}{l} \) (Equation 1)
- \( T_{min} = \frac{mv^2}{l} - mg \) (Equation 2)

3. **Finding the Difference in Tensions**:
- The difference between the maximum and minimum tensions is given as:
\[ \Delta T = T_{max} - T_{min} \]
- Substituting the expressions from Equations 1 and 2:
\[ \Delta T = \left( mg + \frac{mv^2}{l} \right) - \left( \frac{mv^2}{l} - mg \right) \]
- Simplifying this:
\[ \Delta T = mg + \frac{mv^2}{l} - \frac{mv^2}{l} + mg = 2mg \]

4. **Using the Given Information**:
- We know from the problem statement that the difference in tensions is \( \Delta T = 20 \, N \):
\[ 2mg = 20 \]
- Rearranging this gives:
\[ mg = 10 \]

5. **Finding the Mass of the Stone**:
- Given \( g = 10 \, m/s^2 \), we can find the mass \( m \):
\[ m = \frac{mg}{g} = \frac{10}{10} = 1 \, kg \]

### Final Answer:
The mass of the stone is \( 1 \, kg \).