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 find the minimum tension in the string at the lowest point of the vertical circle, we can follow these steps:

### Step 1: Understand the Forces Acting on the Stone
At the lowest point of the vertical circle, two forces act on the stone:
1. The gravitational force (weight) acting downwards, which is given by \( mg \).
2. The tension in the string (T) acting upwards.

### Step 2: Write the Equation of Motion
Using Newton's second law, we can write the equation of motion for the stone at the lowest point:
\[ T - mg = \frac{mv^2}{r} \]
Where:
- \( T \) is the tension in the string,
- \( mg \) is the weight of the stone,
- \( v \) is the velocity of the stone at the lowest point,
- \( r \) is the radius of the circle.

### Step 3: Determine the Minimum Velocity
To maintain circular motion, the stone must have a minimum velocity at the lowest point. This minimum velocity can be derived from the condition that the stone must have enough speed to complete the vertical circle. The minimum velocity \( v_{\text{min}} \) at the lowest point is given by:
\[ v_{\text{min}} = \sqrt{5gr} \]
Where \( g \) is the acceleration due to gravity.

### Step 4: Substitute the Minimum Velocity into the Equation
Substituting \( v_{\text{min}} \) into the equation of motion gives:
\[ T - mg = \frac{m(\sqrt{5gr})^2}{r} \]
This simplifies to:
\[ T - mg = \frac{m(5gr)}{r} \]
\[ T - mg = 5mg \]

### Step 5: Solve for Tension
Now, rearranging the equation to solve for \( T \):
\[ T = mg + 5mg = 6mg \]

### Step 6: Substitute Values
Now we can substitute the values:
- Mass \( m = 50 \, \text{g} = 0.05 \, \text{kg} \)
- Acceleration due to gravity \( g = 10 \, \text{m/s}^2 \)

Calculating \( T \):
\[ T = 6 \times 0.05 \, \text{kg} \times 10 \, \text{m/s}^2 \]
\[ T = 6 \times 0.5 = 3 \, \text{N} \]

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

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