Minimum effort to just move a block up a rough inclined plane (force applied parallel to plane) A block of weight W rests on a plane inclined at angle θ to the horizontal. A force P is applied parallel to the plane and just sufficient to move the block up the plane. If the angle of friction is φ (with μ = tan φ), what is the minimum P?
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AP = W(sinθ + μ cosθ)
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BP = W(sinθ − μ cosθ)
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CP = W(cosθ + μ sinθ)
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DP = W(cosθ − μ sinθ)
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EP = W tan(θ − φ)
Answer
Correct Answer: P = W(sinθ + μ cosθ)
Explanation
Introduction / Context:For impending upward motion on a rough incline, friction acts down the plane and adds to the component of weight along the plane. This classic statics question tests correct identification of directions and use of friction-limited equilibrium at the verge of motion.
Given Data / Assumptions:
- Weight W on a plane with inclination θ.
- Coefficient of friction μ, or friction angle φ with μ = tanφ.
- Force P applied parallel to the plane to just start upward motion (impending slip up the plane).
- Rigid block, Coulomb friction model.
Concept / Approach:
Resolve forces along and normal to the plane. At impending upward motion, friction direction is down the plane with magnitude F = μN. Use equilibrium of forces along the plane to find the minimum P that balances the downhill components (weight component plus friction).
Step-by-Step Solution:
Normal reaction N = W cosθ.Friction F = μ N = μ W cosθ (down the plane).Component of weight along plane = W sinθ (down the plane).For impending upward motion: P − (W sinθ + μ W cosθ) = 0.Therefore, P_min = W(sinθ + μ cosθ). Equivalently, using μ = tanφ, P = W sin(θ + φ)/cosφ.Verification / Alternative check:
Check limiting cases: μ = 0 ⇒ P = W sinθ (smooth plane). For steep θ approaching 90°, expression approaches W, consistent with vertical lift.
Why Other Options Are Wrong:
(b) uses the wrong sign for friction; (c) and (d) mix sine/cosine placements; (e) does not match the correct parallel-to-plane loading form for impending upward motion.
Common Pitfalls:
Reversing friction direction; using N = W instead of N = W cosθ; forgetting friction acts at its limiting value only at impending motion.
Final Answer:
P = W(sinθ + μ cosθ)