A rotor of radius (r) is rotating about its own vertical axis and a person in contact with innerwall of rotor remains in equilibrium without slipping down. If 'w' is angular velocity of rotor and `mu` is minimum coefficient of friction between persona and the wall of rotor. Then following is correct
(A) `mu alpha w^(2)` (B) `mu alpha (1)/(r)` (C) `mu alpha(1)/(w^(2))` (D) `mu alpha r`

Options

A. A and B are true
B. A and D are true
C. C and B are true
D. C and D are true

Answer

To solve the problem, we need to analyze the forces acting on a person standing inside a rotating rotor. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Forces Acting on the Person
When the rotor is rotating, the person experiences a centrifugal force acting outward due to the rotation. This force must be balanced by the frictional force between the person and the wall of the rotor, as well as the gravitational force acting downward.

### Step 2: Identify the Forces
1. **Centrifugal Force (F_c)**: This is given by the formula:
\[ F_c = m \omega^2 r \]
where \( m \) is the mass of the person, \( \omega \) is the angular velocity, and \( r \) is the radius of the rotor.

2. **Gravitational Force (F_g)**: This is simply the weight of the person:
\[ F_g = mg \]
where \( g \) is the acceleration due to gravity.

3. **Frictional Force (F_r)**: This is the force that prevents the person from slipping down and is given by:
\[ F_r = \mu F_n \]
where \( \mu \) is the coefficient of friction and \( F_n \) is the normal force which, in this case, is equal to the centrifugal force (since the person is in equilibrium).

### Step 3: Set Up the Equations
Since the person is in equilibrium, the centrifugal force must equal the frictional force:
\[ F_c = F_r \]
Substituting the expressions for these forces, we have:
\[ m \omega^2 r = \mu m g \]

### Step 4: Simplify the Equation
We can cancel the mass \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \omega^2 r = \mu g \]

### Step 5: Solve for the Coefficient of Friction \( \mu \)
Rearranging the equation gives:
\[ \mu = \frac{\omega^2 r}{g} \]

### Step 6: Analyze the Proportionality
From the equation \( \mu = \frac{\omega^2 r}{g} \), we can see that:
- \( \mu \) is directly proportional to \( \omega^2 \) (as \( r \) and \( g \) are constants).
- \( \mu \) is directly proportional to \( r \) (as \( \omega \) is constant).

### Conclusion
Thus, we can conclude:
- \( \mu \propto \omega^2 \)
- \( \mu \propto r \)

The correct options from the given choices are:
- (A) \( \mu \propto \omega^2 \)
- (D) \( \mu \propto r \)