A body of mass 0.4 kg is whirled in a vertical circle making 2 rev/sec. If the radius of the circle is 1.2 m, then tension in the string when the body is at the top of the circle, is

Options

A. 41.56N
B. 89.86N
C. 109.86
D. 115.86N

Explanation

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: Calculate the angular velocity (ω)
Given that the body makes 2 revolutions per second, we can calculate the angular velocity using the formula:
\[ \omega = 2\pi f \]
where \( f \) is the frequency in revolutions per second.

\[ \omega = 2\pi \times 2 = 4\pi \, \text{rad/sec} \]

### Step 2: Calculate the linear velocity (v)
The linear velocity can be calculated from the angular velocity using the formula:
\[ v = r \omega \]
where \( r \) is the radius of the circle.

\[ v = 1.2 \times 4\pi = 4.8\pi \, \text{m/s} \]

### Step 3: Calculate the centripetal force (Fc)
The centripetal force required to keep the body moving in a circle is given by:
\[ F_c = \frac{mv^2}{r} \]
Substituting the values:
- \( m = 0.4 \, \text{kg} \)
- \( v = 4.8\pi \, \text{m/s} \)
- \( r = 1.2 \, \text{m} \)

First, calculate \( v^2 \):
\[ v^2 = (4.8\pi)^2 = 23.04\pi^2 \]

Now, substituting into the centripetal force formula:
\[ F_c = \frac{0.4 \times 23.04\pi^2}{1.2} = \frac{9.216\pi^2}{1.2} = 7.68\pi^2 \, \text{N} \]

### Step 4: Calculate the gravitational force (Fg)
The gravitational force acting on the body is given by:
\[ F_g = mg \]
where \( g = 9.8 \, \text{m/s}^2 \).

\[ F_g = 0.4 \times 9.8 = 3.92 \, \text{N} \]

### Step 5: Calculate the tension in the string (T)
At the top of the circle, the tension in the string and the gravitational force both act downwards, providing the necessary centripetal force. Thus, we can write:
\[ F_c = T + F_g \]
Rearranging gives:
\[ T = F_c - F_g \]

Substituting the values we calculated:
\[ T = 7.68\pi^2 - 3.92 \]

Using \( \pi^2 \approx 9.87 \):
\[ T \approx 7.68 \times 9.87 - 3.92 \approx 75.91 - 3.92 \approx 71.99 \, \text{N} \]

### Final Answer
The tension in the string when the body is at the top of the circle is approximately \( 71.99 \, \text{N} \).

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