A body moves along a circular path of radius `5m`. The coefficient of friction between the surface of the path and the body is `0.5`. The angular velocity in `rad//s` with which the body should move so that it does not leave the path is `(g-10m//s^(-2))`

Options

A. 4
B. 3
C. 2
D. 1

Answer

To solve the problem step by step, we need to analyze the forces acting on the body moving in a circular path and use the concepts of circular motion and friction.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Body:**
- The body is moving in a circular path, which means it experiences a centripetal force directed towards the center of the circle.
- The frictional force is what provides this centripetal force, acting inward.
- The gravitational force acts downward (mg), and the normal force acts upward (N).

2. **Set Up the Equations:**
- The centripetal force required to keep the body moving in a circle is given by:
\[ F_c = m \omega^2 r \]
where \( m \) is the mass of the body, \( \omega \) is the angular velocity, and \( r \) is the radius of the circular path.
- The frictional force, which provides the centripetal force, is given by:
\[ F_f = \mu N \]
where \( \mu \) is the coefficient of friction and \( N \) is the normal force. Since the body is on a horizontal surface, \( N = mg \), so:
\[ F_f = \mu mg \]

3. **Equate the Forces:**
- For the body to not leave the circular path, the frictional force must equal the centripetal force:
\[ \mu mg = m \omega^2 r \]
- We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \mu g = \omega^2 r \]

4. **Substitute Known Values:**
- Given:
- \( r = 5 \, m \)
- \( \mu = 0.5 \)
- \( g = 10 \, m/s^2 \) (as per the problem statement)
- Substitute these values into the equation:
\[ 0.5 \times 10 = \omega^2 \times 5 \]
- Simplifying gives:
\[ 5 = 5 \omega^2 \]

5. **Solve for Angular Velocity (\( \omega \)):**
- Divide both sides by 5:
\[ 1 = \omega^2 \]
- Taking the square root gives:
\[ \omega = 1 \, \text{rad/s} \]

### Final Answer:
The angular velocity with which the body should move so that it does not leave the path is \( \omega = 1 \, \text{rad/s} \).