One end of a string of length 1 m is tied to a body of mass `0.5` kg. It is whirled in a vertical circle with angular velocity 4 rad `s^(-1)`. Find the tension in the string when the body is at the lower most point of its motion. (take, g = 10 `ms^(-1)`)

Options

A. 3 N
B. 5 N
C. 8 N
D. 13 N

Explanation

To find the tension in the string when the body is at the lower most point of its motion, we can follow these steps:

### Step 1: Identify the forces acting on the body
At the lowest point of the vertical circle, two forces act on the body:
1. The gravitational force (weight) acting downward, \( F_g = mg \)
2. The tension in the string acting upward, \( T \)

### Step 2: Write the equation of motion
At the lowest point, the net force acting on the body is the difference between the tension in the string and the weight of the body. According to Newton's second law, this net force is also equal to the centripetal force required to keep the body moving in a circle.

The equation can be written as:
\[ T - mg = m \omega^2 r \]

Where:
- \( T \) = tension in the string
- \( m \) = mass of the body = 0.5 kg
- \( g \) = acceleration due to gravity = 10 m/s²
- \( \omega \) = angular velocity = 4 rad/s
- \( r \) = radius of the circular path = length of the string = 1 m

### Step 3: Substitute the known values into the equation
Now, we can substitute the values into the equation:
\[ T - (0.5 \, \text{kg} \cdot 10 \, \text{m/s}^2) = 0.5 \, \text{kg} \cdot (4 \, \text{rad/s})^2 \cdot 1 \, \text{m} \]

### Step 4: Calculate the gravitational force
Calculate the weight of the body:
\[ mg = 0.5 \cdot 10 = 5 \, \text{N} \]

### Step 5: Calculate the centripetal force
Calculate the centripetal force:
\[ m \omega^2 r = 0.5 \cdot (4^2) \cdot 1 = 0.5 \cdot 16 = 8 \, \text{N} \]

### Step 6: Substitute back into the equation
Now substitute these values back into the equation:
\[ T - 5 = 8 \]

### Step 7: Solve for tension \( T \)
Now, solve for \( T \):
\[ T = 8 + 5 = 13 \, \text{N} \]

### Conclusion
The tension in the string when the body is at the lower most point of its motion is **13 N**.

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