A body of mass 0.4 kg is whirled in a vertical circle making 2 rev/sec . If the radius of the circle is 2 m , then tension in the string when the body is at the top of the circle, is

Options

A. `41.56 N`
B. `89.86 N`
C. `109.86 N`
D. `122.2 N`

Answer

To find the tension in the string when the body is at the top of the vertical circle, we can follow these steps:

### Step 1: Identify the given values
- Mass of the body, \( m = 0.4 \, \text{kg} \)
- Radius of the circle, \( r = 2 \, \text{m} \)
- Frequency of revolution, \( f = 2 \, \text{rev/sec} \)

### Step 2: Calculate the angular velocity (\( \omega \))
The angular velocity in radians per second can be calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the given frequency:
\[ \omega = 2\pi \times 2 = 4\pi \, \text{rad/sec} \]

### Step 3: Calculate the linear velocity (\( v \))
The linear velocity can be calculated using the formula:
\[ v = \omega \times r \]
Substituting the values:
\[ v = 4\pi \times 2 = 8\pi \, \text{m/s} \]

### Step 4: Calculate the centripetal force (\( F_c \))
The centripetal force required to keep the body moving in a circle is given by:
\[ F_c = \frac{mv^2}{r} \]
Substituting the values:
\[ F_c = \frac{0.4 \times (8\pi)^2}{2} \]
Calculating \( (8\pi)^2 \):
\[ (8\pi)^2 = 64\pi^2 \]
Now substituting this back into the equation:
\[ F_c = \frac{0.4 \times 64\pi^2}{2} = 0.2 \times 64\pi^2 = 12.8\pi^2 \, \text{N} \]

### Step 5: Calculate the weight of the body (\( mg \))
The weight of the body is given by:
\[ mg = 0.4 \times 10 = 4 \, \text{N} \]

### Step 6: Apply the forces at the top of the circle
At the top of the circle, the tension (\( T \)) in the string and the weight of the body both act downwards. The centripetal force is provided by the sum of these two forces:
\[ F_c = T + mg \]
Rearranging for tension:
\[ T = F_c - mg \]
Substituting the values:
\[ T = 12.8\pi^2 - 4 \]

### Step 7: Calculate the numerical value of tension
Using \( \pi \approx 3.14 \):
\[ \pi^2 \approx 9.86 \]
Now substituting this into the equation:
\[ T = 12.8 \times 9.86 - 4 \]
Calculating:
\[ T \approx 126.08 - 4 = 122.08 \, \text{N} \]

### Final Answer
The tension in the string when the body is at the top of the circle is approximately:
\[ \boxed{122.08 \, \text{N}} \]