A horizontal turn table rotates about its axis at the uniform rate of `2`revolutions per second. Find the maximum distance from the axis at which a small body will remain stationary on the turn table if the coefficient of static friction between the turn table and body is `0.8`. Assume `g=pi^(2)m//s^(2)`.

Explanation

To solve the problem step by step, we will follow the reasoning and calculations presented in the video transcript.

### Step 1: Convert revolutions per second to radians per second
The angular velocity \( \omega \) in radians per second can be calculated from the revolutions per second. Since one revolution is equal to \( 2\pi \) radians, we have:

\[ \omega = 2 \text{ revolutions/second} \times 2\pi \text{ radians/revolution} = 4\pi \text{ radians/second} \]

### Step 2: Write down the forces acting on the body
When a small body of mass \( m \) is placed on the turntable, the forces acting on it are:
- The gravitational force \( mg \) acting downwards.
- The normal force \( N \) acting upwards.
- The frictional force \( f \) acting towards the center of the turntable, providing the necessary centripetal force for circular motion.

### Step 3: Set up the equations for centripetal force and friction
The centripetal force required to keep the body moving in a circle of radius \( r \) is given by:

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

The maximum static friction force that can act on the body is given by:

\[ f_{\text{max}} = \mu_s N \]

Where \( \mu_s \) is the coefficient of static friction and \( N = mg \). Therefore, we can write:

\[ f_{\text{max}} = \mu_s mg \]

### Step 4: Set the centripetal force equal to the maximum frictional force
For the body to remain stationary on the turntable, the centripetal force must equal the maximum static friction force:

\[ m \omega^2 r = \mu_s mg \]

### Step 5: Cancel the mass \( m \) from both sides
Since \( m \) appears on both sides of the equation, we can cancel it out:

\[ \omega^2 r = \mu_s g \]

### Step 6: Solve for \( r \)
Now, we can solve for \( r \):

\[ r = \frac{\mu_s g}{\omega^2} \]

### Step 7: Substitute the values
Now we substitute the values into the equation. We know:
- \( \mu_s = 0.8 \)
- \( g = \pi^2 \, \text{m/s}^2 \)
- \( \omega = 4\pi \, \text{rad/s} \)

Substituting these values gives:

\[ r = \frac{0.8 \cdot \pi^2}{(4\pi)^2} \]

Calculating \( (4\pi)^2 \):

\[ (4\pi)^2 = 16\pi^2 \]

Now substituting back into the equation for \( r \):

\[ r = \frac{0.8 \cdot \pi^2}{16\pi^2} = \frac{0.8}{16} = 0.05 \, \text{m} \]

### Step 8: Convert to centimeters
Finally, converting meters to centimeters:

\[ r = 0.05 \, \text{m} = 5 \, \text{cm} \]

### Final Answer
The maximum distance from the axis at which a small body will remain stationary on the turntable is **5 cm**.

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