Pulling a body up an inclined plane: angle of applied force for minimum effort
A force is applied at an angle to the plane to move a body upward.
Let the angle of friction be φ (tan φ = μ).
Complete the statement: the required force is minimum when the angle with the plane is ______ φ.
Difficulty: Medium
Correct Answer: equal to
Explanation:
Given data
Rough plane with coefficient of friction μ; angle of friction φ where tan φ = μ.
Concept/Approach
Resolve the pull P into components along and normal to the plane: P cos θ (up the plane) and P sin θ (reduces the normal reaction).
Required P is minimized when dP/dθ = 0, which gives θ = φ.
Step-by-step relation Required P = (W sin α + μ W cos α)/(cos θ + μ sin θ)Denominator is maximum when tan θ = μ → θ = φ.
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