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Question

The upper half of an inclined plane of inclination $$\theta$$ is perfectly smooth while lower half is rough. A block starting from rest at the top of the plane will again come to rest at the bottom, if the coefficient of friction between the block and lower half of the plane is given by 


A
2θ
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B
2tanθ
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C
2cosθ
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D
2sinθ
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Solution

The correct option is B $$2\tan{\theta}$$
For the desires outcome, the force applied on the block should be same in magnitude but opposite in direction, on the upper half and the lower half of the incline plane.
Let force in upper half be $${ f }_{ u }$$ and the force be lower half be $${ f }_{ l }$$
$${ f }_{ u }=mgsin\theta $$
$${ f }_{ l }=mgsin\theta -\mu mgcos\theta $$
$${ f }_{ u }$$+$${ f }_{ l }$$ should be 0.
Hence, $$\mu mgcos\theta =2mgsin\theta $$
$$\mu =2tan\theta $$

Physics

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