Question

A body is moving down an inclined plane of slope ${37}^o$. The coefficient of friction between the body and the plane varies as $\mu =0.3x$, where $x$ is the distance traveled down the plane by the body. The body will have maximum speed at $(\sin {37}^o=\frac{3}{5})$.

• A. at $x=1.16m$
• B. at $x=2m$
• C. at bottom most point of the plane
• D. at $x=2.5m$

Answer 1

The correct option is D at $x=2.5m$

$F_{net}=mg\sin \theta -\mu mg\cos \theta$

Thus, $F_{net}=mg\sin \theta -0.3xmg\cos \theta$ ....(i)

At maximum speed $F_{net}=0$

Because after this net force will become negative and speed will decrease.

From Eq. (i), $F_{net}=0$

we get $mgsin\theta -0.3xmgcos\theta =0$

where $\theta ={37}^o$

Or $x=\frac{\tan \theta }{0.3}$

We know $cos\theta =\sqrt{1-{\sin }^2\theta }=\sqrt{1-\frac{9}{25}}=\frac{4}{5}$

$⟹tan\theta =\frac{sin\theta }{cos\theta }=\frac{3}{4}$

So, $x=\frac{3/4}{0.3}=2.5m$