A string of length 1 m is fixed at one end and carries a mass of 100 gm at the other end. The string makes (2/`pi`) revolutions per second around vertical axis through the fixed end. Calculate the tension in the string-

Options

A. 1.0 N
B. 1.6 N
C. 2N
D. 4N

Answer

To solve the problem of finding the tension in the string, we can follow these steps:

### Step 1: Understand the Given Information
- Length of the string (L) = 1 m
- Mass (m) = 100 gm = 0.1 kg (convert grams to kilograms)
- Revolutions per second (f) = \( \frac{2}{\pi} \) rev/s

### Step 2: Convert Revolutions per Second to Radians per Second
To find the angular velocity (ω) in radians per second, we use the conversion:
\[ \omega = 2\pi f \]
Substituting the value of f:
\[ \omega = 2\pi \left(\frac{2}{\pi}\right) = 4 \text{ rad/s} \]

### Step 3: Identify the Radius of Circular Motion
Since the string is fixed at one end and the mass is at the other end, the radius (r) of the circular motion is equal to the length of the string:
\[ r = L = 1 \text{ m} \]

### Step 4: Calculate the Centripetal Force
The tension in the string provides the necessary centripetal force for the circular motion. The formula for centripetal force (F_c) is:
\[ F_c = \frac{mv^2}{r} \]
where \( v \) is the linear velocity. We can express \( v \) in terms of \( \omega \) and \( r \):
\[ v = r\omega \]
Substituting \( r = 1 \text{ m} \) and \( \omega = 4 \text{ rad/s} \):
\[ v = 1 \times 4 = 4 \text{ m/s} \]

### Step 5: Substitute Values into the Centripetal Force Formula
Now we can substitute \( v \) back into the centripetal force formula:
\[ F_c = \frac{m(v^2)}{r} = \frac{0.1 \times (4^2)}{1} \]
Calculating \( v^2 \):
\[ v^2 = 16 \text{ m}^2/\text{s}^2 \]
Now substituting:
\[ F_c = \frac{0.1 \times 16}{1} = 1.6 \text{ N} \]

### Step 6: Conclusion
The tension in the string, which is equal to the centripetal force required for the circular motion, is:
\[ T = 1.6 \text{ N} \]

### Final Answer
The tension in the string is **1.6 N**.

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