A coin, placed on a rotating turntable slips, when it is placed at a distance of `9cm` from the center. If the angular velocity of the turnable is tripled, it will just slip, If its distance from the center is

Options

A. `27cm`
B. `9cm`
C. `3cm`
D. `1cm`

Explanation

To solve the problem step by step, we will analyze the forces acting on the coin placed on the rotating turntable and derive the relationship between the distances and angular velocities.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- A coin is placed on a rotating turntable at a distance \( r_1 = 9 \, \text{cm} \) from the center. At this distance, the coin slips due to insufficient frictional force to provide the necessary centripetal force.

2. **Centripetal Force and Friction:**
- The centripetal force required to keep the coin moving in a circle is given by:
\[ F_c = m r \omega^2 \]
where \( m \) is the mass of the coin, \( r \) is the distance from the center, and \( \omega \) is the angular velocity.
- The maximum static friction force that can act on the coin is:
\[ F_f = \mu m g \]
where \( \mu \) is the coefficient of static friction and \( g \) is the acceleration due to gravity.

3. **Condition for Slipping:**
- The coin will start to slip when the centripetal force equals the maximum static friction force:
\[ \mu m g = m r_1 \omega^2 \]
- Simplifying this, we get:
\[ \mu g = r_1 \omega^2 \quad \text{(1)} \]

4. **Tripling the Angular Velocity:**
- Now, if the angular velocity is tripled, \( \omega' = 3\omega \), we need to find the new distance \( r_2 \) from the center where the coin just begins to slip again.
- The new condition for slipping is:
\[ \mu m g = m r_2 (3\omega)^2 \]
- Simplifying this gives:
\[ \mu g = r_2 (9\omega^2) \quad \text{(2)} \]

5. **Equating the Two Conditions:**
- From equations (1) and (2):
\[ r_1 \omega^2 = r_2 (9\omega^2) \]
- We can cancel \( \omega^2 \) from both sides (assuming \( \omega \neq 0 \)):
\[ r_1 = 9 r_2 \]

6. **Solving for \( r_2 \):**
- Substituting \( r_1 = 9 \, \text{cm} \):
\[ 9 = 9 r_2 \]
- Dividing both sides by 9:
\[ r_2 = 1 \, \text{cm} \]

### Final Answer:
The distance from the center when the angular velocity is tripled, and the coin just slips is \( r_2 = 1 \, \text{cm} \).