Question

A wedge M and a block N are subjected to forces P and Q as shown in the figure. If force P is sufficiently large, then the block N can be raised. The weights of the wedge and the block are negligible compared to the forces P and Q. The coefficient of friction \( \mu \) along the inclined surface between the wedge and the block is 0.2. All other surfaces are frictionless. The wedge angle is 30°.

The limiting force \( P \), in terms of \( Q \), required for impending motion of block N to just move it in the upward direction is given as \( P = \alpha Q \). The value of the coefficient \( \alpha \) (round off to one decimal place) is:

• A. 0.6
• B. 0.5
• C. 2.0
• D. 0.9

Hint:

For a block on an inclined plane with friction, use equilibrium equations considering forces along and perpendicular to the plane to find the required limiting force for motion.

Answer 1

The Correct Option is D

We are given a wedge and a block, with forces \( P \) and \( Q \) applied, and we are asked to find the value of \( \alpha \) in the equation \( P = \alpha Q \) required for the impending motion of block N. The coefficient of friction between the wedge and the block is \( \mu = 0.2 \) and the wedge angle is \( 30^\circ \). The motion is considered to be impending, so we use the condition for limiting friction.

Step 1: Analyze the forces on the block N.
The frictional force acting on the block due to the surface of the wedge is given by: \[ F_f = \mu N, \] where \( N \) is the normal force, which is the component of the weight of the block perpendicular to the surface of the wedge.

Step 2: Resolve the forces along the direction of the incline.
The forces acting along the inclined plane include the applied force \( P \) and the frictional force \( F_f \), while the weight of the block \( Q \) can be resolved into components along and perpendicular to the plane. Using equilibrium conditions and solving the force balance equations, we get the relationship between \( P \) and \( Q \) in terms of the coefficient of friction and the wedge angle.

Step 3: Solve for \( \alpha \).
After solving, we find that the value of \( \alpha \) is approximately 0.9.

Final Answer: \[ \boxed{0.9}. \]