A small stone of mass 50 g is rotated in a vertical circle of radius 40 cm. What is the minimum tension in the string at the lowest point?

Options

A. `6`N
B. `2`N
C. `3`N
D. `1.5`N

Answer

To solve the problem of finding the minimum tension in the string at the lowest point of a vertical circle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Mass of the stone, \( m = 50 \, \text{g} = 0.05 \, \text{kg} \) (convert grams to kilograms).
- Radius of the circle, \( r = 40 \, \text{cm} = 0.4 \, \text{m} \) (convert centimeters to meters).
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \).

2. **Understand the Forces at the Lowest Point:**
At the lowest point of the vertical circle, two forces act on the stone:
- The gravitational force acting downwards: \( F_g = mg \).
- The tension in the string acting upwards: \( T \).

3. **Apply Newton's Second Law:**
At the lowest point, the net force is directed upwards, and it must provide the centripetal force required to keep the stone moving in a circle. Therefore, we can write:
\[ T - mg = \frac{mv^2}{r} \]
where \( v \) is the velocity of the stone at the lowest point.

4. **Determine the Minimum Velocity:**
The minimum velocity at the lowest point of the circle can be derived from the condition that the stone must have enough speed to maintain circular motion. The minimum speed occurs when the stone is just about to lose contact with the string, which is at the top of the circle. The minimum speed \( v_{\text{min}} \) can be derived as:
\[ v_{\text{min}} = \sqrt{g r} \]
However, at the lowest point, we consider \( v_{\text{min}}^2 = 5gr \) (as derived from the centripetal force requirement).

5. **Substituting for Tension:**
Now substituting \( v_{\text{min}}^2 \) into the tension equation:
\[ T = mg + \frac{m(5gr)}{r} \]
Simplifying this gives:
\[ T = mg + 5mg = 6mg \]

6. **Calculate the Tension:**
Now substituting the values of \( m \) and \( g \):
\[ T = 6 \times 0.05 \, \text{kg} \times 10 \, \text{m/s}^2 \]
\[ T = 6 \times 0.5 = 3 \, \text{N} \]

### Final Answer:
The minimum tension in the string at the lowest point is \( T = 3 \, \text{N} \).

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