A gramophone record is revolving with an angular velocity `omega`. A coin is placed at a distance `R` from the centre of the record. The static coefficient of friction is `mu`. The coin will revolve with the record if

Options

A. `r= mu g omega^(2)`
B. `r lt omega^(2)/(mu g)`
C. `r le (mu g)/omega^(2)`
D. `r ge (mu g)/omega^(2)`

Answer

To solve the problem, we need to analyze the forces acting on the coin placed on a gramophone record that is rotating with an angular velocity \( \omega \). Here’s a step-by-step solution:

### Step 1: Identify the Forces Acting on the Coin
The coin experiences two main forces:
1. **Centrifugal Force**: This is the outward force experienced due to the circular motion of the coin. It can be expressed as:
\[ F_{\text{centrifugal}} = m \omega^2 R \]
where \( m \) is the mass of the coin, \( \omega \) is the angular velocity, and \( R \) is the distance from the center of the record to the coin.

2. **Frictional Force**: This is the force that provides the necessary centripetal force to keep the coin moving in a circle. The maximum static frictional force can be expressed as:
\[ F_{\text{friction}} = \mu m g \]
where \( \mu \) is the coefficient of static friction and \( g \) is the acceleration due to gravity.

### Step 2: Set Up the Condition for the Coin to Revolve
For the coin to revolve with the record without slipping, the frictional force must be equal to or greater than the centrifugal force. Thus, we have the inequality:
\[ \mu m g \geq m \omega^2 R \]

### Step 3: Simplify the Inequality
We can cancel the mass \( m \) from both sides of the inequality (assuming \( m \neq 0 \)):
\[ \mu g \geq \omega^2 R \]

### Step 4: Rearrange to Find the Maximum Radius
Rearranging the inequality gives us:
\[ R \leq \frac{\mu g}{\omega^2} \]
This means that the coin will revolve with the record if the distance \( R \) from the center of the record is less than or equal to \( \frac{\mu g}{\omega^2} \).

### Conclusion
Thus, the condition for the coin to revolve with the gramophone record is:
\[ R \leq \frac{\mu g}{\omega^2} \]