A small objective placed on a rotating horizontal turn table just slips when it is placed at a distance 4cm from the axis of rotation. If the angular velocity of the trun-table doubled, the objective slip when its distance from the axis of ratation is.

Options

A. 1cm
B. 2cm
C. 4cm
D. 8cm

Explanation

To solve the problem step by step, we will analyze the forces acting on the object placed on the rotating turntable and how they change when the angular velocity is doubled.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Initial distance from the axis of rotation, \( R = 4 \) cm.
- Initial angular velocity, \( \omega \).
- When the object just slips, the centripetal force equals the limiting frictional force.

2. **Write the Equation for Just Slipping Condition:**
- The centripetal force acting on the object is given by:
\[ F_c = m R \omega^2 \]
- The limiting frictional force is given by:
\[ F_L = \mu N = \mu mg \]
- Setting these two forces equal gives:
\[ \mu mg = m R \omega^2 \]
- Canceling \( m \) from both sides:
\[ \mu g = R \omega^2 \quad \text{(1)} \]

3. **Consider the New Angular Velocity:**
- When the angular velocity is doubled, the new angular velocity is:
\[ \omega' = 2\omega \]
- The new centripetal force when the object slips at a new radius \( R' \) is:
\[ F_c' = m R' \omega'^2 = m R' (2\omega)^2 = m R' \cdot 4\omega^2 \]
- The limiting frictional force remains the same:
\[ F_L = \mu mg \]
- Setting these two forces equal gives:
\[ \mu mg = m R' \cdot 4\omega^2 \]
- Canceling \( m \) from both sides:
\[ \mu g = R' \cdot 4\omega^2 \quad \text{(2)} \]

4. **Divide Equation (1) by Equation (2):**
- From equations (1) and (2):
\[ \frac{\mu g}{\mu g} = \frac{R \omega^2}{R' \cdot 4\omega^2} \]
- This simplifies to:
\[ 1 = \frac{R}{R' \cdot 4} \]
- Rearranging gives:
\[ R' = \frac{R}{4} \]

5. **Substituting the Value of R:**
- Since \( R = 4 \) cm, we have:
\[ R' = \frac{4 \text{ cm}}{4} = 1 \text{ cm} \]

### Final Answer:
The object will slip when its distance from the axis of rotation is **1 cm**.

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