Forces on a rough inclined plane: Effort parallel to the plane to move a body upward A body of weight W is to be moved up a rough plane inclined at angle α to the horizontal by a force applied parallel to the plane. The coefficient of friction is μ = tanφ. Choose the correct expression for the required effort P (just impending upward motion).
Mechanical Engineering
Engineering Mechanics
Difficulty: Easy
Choose an option
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AP = W tanα
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BP = W tan(α + φ)
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CP = W (sinα + μ cosα)
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DP = W (cosα + μ sinα)
Answer
Correct Answer: P = W (sinα + μ cosα)
Explanation
GivenIncline angle α, weight W, coefficient of friction μ, effort P applied parallel to plane to move body up the plane.
Free-body equations (limiting equilibrium upward) Along plane: P − W sinα − F = 0 Normal: R = W cosα At impending upward motion, friction acts down the plane with magnitude F = μ R = μ W cosα.
Compute P P = W sinα + μ W cosα = W (sinα + μ cosα)
Alternative angle of friction formUsing μ = tanφ, the same result can be expressed via trigonometric identities if needed, but the simplest correct form is above.
Final AnswerP = W (sinα + μ cosα)