A body moves along a circular path of radius 10m and the coefficient of friction is 0.5.What should be its angular speed `(in rad s^(-1))`, if is not to slip from the surface ? `(Given,g =9.8ms^(-2))`

Options

A. 5
B. 10
C. 0.1
D. 0.7

Answer

To solve the problem of determining the angular speed of a body moving along a circular path without slipping, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces**:
- The body is moving in a circular path, which means it experiences a centripetal force directed towards the center of the circle. This force is provided by the friction between the body and the surface.
- The frictional force \( f \) can be expressed as:
\[ f = \mu \cdot N \]
where \( \mu \) is the coefficient of friction and \( N \) is the normal force. For a horizontal circular motion, \( N = mg \), where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity.

2. **Set Up the Equation**:
- The centripetal force required for circular motion is given by:
\[ F_c = m \cdot \frac{v^2}{r} \]
where \( v \) is the linear speed and \( r \) is the radius of the circular path.
- The frictional force must equal the centripetal force for the body to not slip:
\[ \mu \cdot mg = m \cdot \frac{v^2}{r} \]

3. **Cancel Mass**:
- Since mass \( m \) appears on both sides of the equation, we can cancel it out:
\[ \mu g = \frac{v^2}{r} \]

4. **Rearrange for Linear Speed**:
- Rearranging gives:
\[ v^2 = \mu g r \]

5. **Substitute Values**:
- Given:
- \( \mu = 0.5 \)
- \( g = 9.8 \, \text{m/s}^2 \)
- \( r = 10 \, \text{m} \)
- Substitute these values into the equation:
\[ v^2 = 0.5 \cdot 9.8 \cdot 10 \]
\[ v^2 = 49 \]
\[ v = \sqrt{49} = 7 \, \text{m/s} \]

6. **Convert Linear Speed to Angular Speed**:
- The relationship between linear speed \( v \) and angular speed \( \omega \) is given by:
\[ v = r \cdot \omega \]
- Rearranging gives:
\[ \omega = \frac{v}{r} \]
- Substitute \( v = 7 \, \text{m/s} \) and \( r = 10 \, \text{m} \):
\[ \omega = \frac{7}{10} = 0.7 \, \text{rad/s} \]

### Final Answer:
The angular speed required for the body to not slip from the surface is \( \omega = 0.7 \, \text{rad/s} \).