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.

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 relationRequired 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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