A block of mass m is placed on a smooth wedge of inclination θ. The whole system is accelerated horizontally so that the block does not slip. The acceleration is:
For the block to remain stationary relative to the wedge, the pseudo force ma must balance the component of gravity along the incline. ma cosθ = mg sinθ => a = g tanθ.
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Step-by-Step Solution
Draw a free body diagram:
Draw the free body diagram of the block in the accelerating frame of reference. The forces acting on the block are:
Weight (mg) acting vertically downwards.
Normal reaction (N) from the wedge, acting perpendicular to the inclined surface.
Pseudo force (ma) acting horizontally opposite to the direction of acceleration of the wedge.
Resolve the forces:
Resolve the normal reaction N into its horizontal and vertical components:
Vertical component: N cos θ
Horizontal component: N sin θ
Apply Newton's Second Law:
Since the block does not slip, the net force on the block in both horizontal and vertical directions must be zero.
Vertical equilibrium: N cos θ = mg
Horizontal equilibrium: N sin θ = ma
Solve for acceleration (a):
From the vertical equilibrium equation, we have: N = mg / cos θ
Substitute this value of N into the horizontal equilibrium equation: (mg / cos θ) * sin θ = ma
Simplify to find the acceleration: a = g tan θ
Correct Answer: g tan θ
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Pedagogical Audit
Bloom's Analysis:
This is an APPLY question because the student needs to apply the concepts of Newton's laws of motion and pseudo forces to solve the problem. They must use the given information to derive the required acceleration.
Knowledge Dimension:PROCEDURAL
Justification:The question requires a step-by-step solution involving force diagrams, application of Newton's second law, and algebraic manipulation to arrive at the final answer. This involves knowing the procedures to solve dynamics problems.
Syllabus Audit:
In the context of JEE, this is classified as KNOWLEDGE. The question directly tests the understanding and application of concepts related to Newton's laws of motion, which are fundamental knowledge areas in the JEE syllabus.