A `1kg` stone at the end of `1m` long string is whirled in a vertical circle at a constant speed of `4m//s`. The tension in the string is `6N`, when the stone is at `(g=10m//s^(2))`

Options

A. (a) At the top of the circle
B. (b) At the bottom of the circle
C. (c) Half way down
D. (d) None of the above

Explanation

To solve the problem, we need to analyze the forces acting on the stone when it is at the top of the vertical circle. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Forces Acting on the Stone
When the stone is at the top of the vertical circle, two main forces act on it:
- The gravitational force (weight) acting downwards: \( W = mg \)
- The centripetal force required to keep the stone moving in a circle, which is provided by the tension in the string and the weight of the stone.

### Step 2: Calculate the Weight of the Stone
Given:
- Mass of the stone, \( m = 1 \, \text{kg} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

The weight of the stone is calculated as:
\[ W = mg = 1 \, \text{kg} \times 10 \, \text{m/s}^2 = 10 \, \text{N} \]

### Step 3: Calculate the Centripetal Force
The centripetal force required to keep the stone moving in a circle is given by:
\[ F_c = \frac{mv^2}{r} \]
Where:
- \( v = 4 \, \text{m/s} \) (the speed of the stone)
- \( r = 1 \, \text{m} \) (the radius of the circle)

Substituting the values:
\[ F_c = \frac{1 \, \text{kg} \times (4 \, \text{m/s})^2}{1 \, \text{m}} = \frac{1 \times 16}{1} = 16 \, \text{N} \]

### Step 4: Analyze the Forces at the Top of the Circle
At the top of the circle, the tension in the string \( T \) and the weight of the stone \( W \) together provide the necessary centripetal force:
\[ F_c = T + W \]
Given that the tension \( T = 6 \, \text{N} \) when the stone is at the top, we can substitute the known values:
\[ 16 \, \text{N} = T + 10 \, \text{N} \]
Substituting \( T = 6 \, \text{N} \):
\[ 16 \, \text{N} = 6 \, \text{N} + 10 \, \text{N} \]
This confirms that the forces are balanced correctly.

### Step 5: Conclusion
The tension in the string is 6 N when the stone is at the top of the vertical circle.

### Final Answer
The tension in the string is 6 N when the stone is at the top of the circle.

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