A body is kept on a horizontal disc of radius 2 m at a distance of 1 m from the centre . The coefficient of friction between the body and the surface of disc is 0.4. The speed of rotation of the disc at which the body starts slipping is (g=10`m//s^2`)

Options

A. 2`rad//s`
B. 4`rad//s`
C. 0.2`rad//s`
D. 0.4`rad//s`

Answer

To solve the problem step by step, we will analyze the forces acting on the body placed on the rotating disc and determine the speed of rotation at which the body starts to slip.

### Step 1: Identify the forces acting on the body
The body experiences two main forces:
1. The gravitational force acting downwards, \( F_g = mg \).
2. The frictional force acting upwards, which provides the centripetal force necessary for circular motion.

### Step 2: Determine the maximum static friction force
The maximum static friction force, \( F_s \), can be calculated using the formula:
\[ F_s = \mu \cdot N \]
where \( \mu \) is the coefficient of friction and \( N \) is the normal force. Since the body is on a horizontal surface, the normal force \( N \) is equal to the gravitational force \( mg \):
\[ F_s = \mu \cdot mg \]

### Step 3: Set up the equation for centripetal force
The centripetal force required to keep the body moving in a circle is provided by the frictional force. The centripetal force \( F_c \) can be expressed as:
\[ F_c = m \cdot r \cdot \omega^2 \]
where \( r \) is the distance from the center of the disc to the body (1 m in this case) and \( \omega \) is the angular velocity in radians per second.

### Step 4: Equate the centripetal force to the maximum static friction force
At the point of slipping, the maximum static friction force equals the centripetal force:
\[ \mu \cdot mg = m \cdot r \cdot \omega^2 \]

### Step 5: Cancel out the mass \( m \)
Since \( m \) appears on both sides of the equation, we can cancel it out:
\[ \mu \cdot g = r \cdot \omega^2 \]

### Step 6: Solve for \( \omega^2 \)
Rearranging the equation gives:
\[ \omega^2 = \frac{\mu \cdot g}{r} \]

### Step 7: Substitute the known values
We know:
- \( \mu = 0.4 \)
- \( g = 10 \, \text{m/s}^2 \)
- \( r = 1 \, \text{m} \)

Substituting these values into the equation:
\[ \omega^2 = \frac{0.4 \cdot 10}{1} = 4 \]

### Step 8: Calculate \( \omega \)
Taking the square root gives:
\[ \omega = \sqrt{4} = 2 \, \text{rad/s} \]

### Final Answer
The speed of rotation of the disc at which the body starts slipping is \( \omega = 2 \, \text{rad/s} \).

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