Friction

If the same wedge is made rough then time taken by it to come down becomes times more. The coefficient of friction

To solve this problem, we need to consider the forces acting on the body when it is on the inclined plane.
Step 1: Forces Acting on the Body When a body is on an inclined plane with friction, the forces that act on the body include: 1. Normal Force ( ): The perpendicular force from the surface of the plane. 2. Frictional Force ( ): This opposes the motion and is given by: 3. Gravitational Force ( ): The force due to gravity acting vertically downwards.
Step 2: Minimum Force to Start Moving ( ) The minimum force required to start pushing the body up the inclined plane is the force that overcomes both the frictional force and the component of the gravitational force parallel to the plane. - The component of the gravitational force parallel to the incline is . - The frictional force opposing the motion is . Thus, the minimum force to move the body up the plane is:
Step 3: Minimum Force to Prevent Sliding ( ) The minimum force required to prevent the body from sliding down the incline is the force that exactly balances the component of the gravitational force pulling the body down the plane. - The component of the gravitational force parallel to the incline is . - The frictional force opposes sliding, and it has a maximum value of . Thus, the minimum force required to prevent sliding is:
Step 4: Calculating the Ratio Now, we can calculate the ratio : We can simplify the expression:
Step 5: Substituting Given that , we can use the identity to express and in terms of . From , we get: Substitute this into the equation for : Simplifying:
Step 6: Conclusion The ratio is 3. Therefore, the correct answer is: