A chain of 100 links is 1 m long and has a mass of 2 kg . With the ends fastened together it is set roation at 3000 rpm.then 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 at 3000 rpm, we can follow these steps:

### Step 1: Determine the mass of each link
The total mass of the chain is given as 2 kg, and there are 100 links in total. Therefore, the mass of each link can be calculated as:

\[ \text{Mass of each link} = \frac{\text{Total mass}}{\text{Number of links}} = \frac{2 \, \text{kg}}{100} = 0.02 \, \text{kg} \]

### Step 2: Convert the rotational speed from rpm to radians per second
The angular velocity \( \omega \) in radians per second can be calculated using the formula:

\[ \omega = \frac{2 \pi n}{60} \]

where \( n \) is the rotational speed in rpm. Substituting \( n = 3000 \):

\[ \omega = \frac{2 \pi \times 3000}{60} = 100 \pi \, \text{rad/s} \]

### Step 3: Determine the radius of the circular path
The length of the chain when straight is 1 meter, which becomes the circumference of the circular path when the chain is formed into a circle. The relationship between circumference \( C \) and radius \( r \) is given by:

\[ C = 2 \pi r \]

Setting \( C = 1 \, \text{m} \):

\[ 1 = 2 \pi r \implies r = \frac{1}{2 \pi} \, \text{m} \]

### Step 4: Calculate the centripetal force on each link
The centripetal force \( F_c \) acting on each link can be calculated using the formula:

\[ F_c = m \omega^2 r \]

Substituting the values we have:

- \( m = 0.02 \, \text{kg} \)
- \( \omega = 100 \pi \, \text{rad/s} \)
- \( r = \frac{1}{2 \pi} \, \text{m} \)

Now substituting these into the centripetal force formula:

\[ F_c = 0.02 \times (100 \pi)^2 \times \frac{1}{2 \pi} \]

Calculating \( (100 \pi)^2 \):

\[ (100 \pi)^2 = 10000 \pi^2 \]

Now substituting this back into the equation for \( F_c \):

\[ F_c = 0.02 \times 10000 \pi^2 \times \frac{1}{2 \pi} \]

Simplifying:

\[ F_c = 0.02 \times 10000 \times \frac{\pi^2}{2 \pi} = 0.02 \times 10000 \times \frac{\pi}{2} = 100 \pi \, \text{N} \]

### Step 5: Calculate the numerical value of the centripetal force
Using \( \pi \approx 3.14 \):

\[ F_c \approx 100 \times 3.14 = 314 \, \text{N} \]

### Final Answer
The centripetal force on each link is approximately **314 N**.

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