An inclined plane is bent in such a way that the vertical cross-section is given by y=(x²)/4 where y is in vertical and x in horizontal direction. If the upper surface of this curved plane is rough with coefficient of friction μ=0.5, the maximum height in cm at which a stationary block will not sli…

Question

An inclined plane is bent in such a way that the vertical cross-section is given by y=(x²)/4 where y is in vertical and x in horizontal direction. If the upper surface of this curved plane is rough with coefficient of friction μ=0.5, the maximum height in cm at which a stationary block will not slip downward is cm.

Accepted Answer

25. Idea: the surface steepens as you climb, so a block can rest anywhere the local slope is at most μ; the highest such point has tan θ=μ exactly. tan θ=(dy)/(dx)=(2x)/4=x/2. Setting tan θ=μ=0.5, x/2=0.5, so x=1 m. The height there is y=(1²)/4=0.25 m=25 cm. The answer is asked for in centimetres, which is where this question most often goes wrong — 0.25 m is right but 0.25 cm is not.

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