A stone of mass 0.2 kg is tied to one end of a string of length 80 cm. Holding the other and, the stone is whiled into a vertical circle. The minimum speed of the stone and tension at the lowest point of circular path so that it just comletes the circle are

Options

A. `5ms^(-1),10N`
B. `6.32ms^(-1),12N`
C. `6.32ms^(-1),10N`
D. `2ms^(-1),10N`

Answer

To solve the problem of finding the minimum speed of the stone at the lowest point of the circular path and the tension in the string at that point, we will follow these steps:

### Step 1: Identify the parameters
- Mass of the stone, \( m = 0.2 \, \text{kg} \)
- Length of the string (radius of the circle), \( r = 80 \, \text{cm} = 0.8 \, \text{m} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

### Step 2: Determine the minimum speed at the lowest point
To ensure that the stone completes the circular motion, we need to find the minimum speed at the lowest point. At this point, the forces acting on the stone are:
- Tension \( T \) acting upwards
- Weight \( mg \) acting downwards

The centripetal force required to keep the stone moving in a circle is provided by the net force acting towards the center of the circle. Therefore, we can write the equation for the forces at the lowest point:

\[ T - mg = \frac{mv^2}{r} \]

At the minimum speed required to complete the circle, the tension \( T \) will be zero at the top of the circle. Therefore, we can derive the minimum speed \( v \) at the lowest point using the following relation:

\[ v = \sqrt{5gr} \]

### Step 3: Substitute the values to find the minimum speed
Substituting the values into the equation:

\[ v = \sqrt{5 \cdot 10 \cdot 0.8} \]

Calculating this gives:

\[ v = \sqrt{40} \approx 6.32 \, \text{m/s} \]

### Step 4: Calculate the tension at the lowest point
Now, we need to calculate the tension \( T \) at the lowest point. Using the rearranged force equation:

\[ T = \frac{mv^2}{r} + mg \]

Substituting \( v^2 = 5gr \):

\[ T = \frac{m(5gr)}{r} + mg \]

This simplifies to:

\[ T = 5mg + mg = 6mg \]

### Step 5: Substitute the values to find the tension
Now substituting the values:

\[ T = 6 \cdot 0.2 \cdot 10 \]

Calculating this gives:

\[ T = 12 \, \text{N} \]

### Final Answers
- Minimum speed at the lowest point: \( 6.32 \, \text{m/s} \)
- Tension at the lowest point: \( 12 \, \text{N} \)