A chain of 100 links is 1m long and has a mass of 2kg. With the ends fastened together it is set rotating at 3000 rpm, in a horizontal plane. The centripetal force on each link is

Options

A. `3.14 N`
B. `31.4 N`
C. `314 N`
D. `3140 N`

Answer

To solve the problem of finding the centripetal force on each link of a chain rotating in a horizontal plane, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Number of links, \( n = 100 \)
- Total length of the chain, \( L = 1 \, \text{m} \)
- Total mass of the chain, \( m = 2 \, \text{kg} \)
- Rotational speed, \( \text{RPM} = 3000 \)

2. **Convert RPM to Revolutions per Second:**
\[ F = \frac{3000 \, \text{RPM}}{60} = 50 \, \text{revolutions per second} \]

3. **Calculate the Radius of the Circle:**
- The chain forms a circle when the ends are fastened together. The circumference of the circle is equal to the length of the chain.
\[ L = 2 \pi r \implies r = \frac{L}{2 \pi} = \frac{1}{2 \pi} \, \text{m} \]

4. **Calculate Angular Velocity (\( \omega \)):**
\[ \omega = 2 \pi F = 2 \pi \times 50 = 100 \pi \, \text{rad/s} \]

5. **Calculate the Total Centripetal Force (\( F \)):**
- The formula for centripetal force is given by:
\[ F = m r \omega^2 \]
Substituting the values:
\[ F = 2 \times \left(\frac{1}{2 \pi}\right) \times (100 \pi)^2 \]
Simplifying this:
\[ F = 2 \times \frac{1}{2 \pi} \times 10000 \pi^2 = 10000 \, \text{N} \]

6. **Calculate the Centripetal Force on Each Link:**
\[ \text{Centripetal force on each link} = \frac{F}{n} = \frac{10000}{100} = 100 \, \text{N} \]

### Final Result:
The centripetal force on each link is **100 N**.