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Question

A body is moving down a long inclined plane of slope $$37^o$$. The coefficient of friction between the body and plane varies as $$\mu = 0.3 $$x, where x is distance travelled down the plane. The body will have maximum speed $$(sin 37^o=\frac {3}{5}$$ and $$g=10 m/s^2)$$


A
at x=1.16m.
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B
at x=2m.
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C
at bottom of plane.
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D
at x=2.5m.
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Solution

The correct option is D at $$x=2.5 m$$.
Initially the coefficient of friction is 0. It starts increasing as the distance traveled down the plane increases. The friction will act in direction to oppose relative motion i.e in upwards direction. 
As the friction force is less than component of gravity along the slope (downwards) so net acceleration is downwards and hence the block speed will increases. 
It reaches a point where friction force increases than the component of gravity along the slope (downwards). The point where the friction force (upwards) equals the component of gravity along the slope (downwards) at that point velocity is maximum.

$$mgsin\theta =\mu N=\mu mgcos\theta \\ \Rightarrow tan\theta =0.3x\\ \Rightarrow x=\dfrac { 3 }{ 4 } \dfrac { 1 }{ 0.3 } =2.5m$$ 

Physics

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