Q. A block is kept on an inclined plane of inclination $\theta$ and length $l.$ The velocity of particle at the bottom of incline is (the coefficient of friction is $\mu$)

A) $\sqrt{2gl(\mu\cos{\theta - \sin{\theta)}}}$
B) $\sqrt{2gl(\sin{\theta - {\mu\ cos}{\theta)}}}$
C) $\sqrt{2gl(\sin{\theta + {\mu\ cos}{\theta)}}}$
D) $\sqrt{2gl(\cos{\theta - \mu\ \sin{\theta)}}}$
Explanation

(b) The various forces acting on the block are as shown
From Newton’s law
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$mg\sin{\theta - f = ma\ \ \ldots.(i)}$
Where $f$ is frictional force and $a$ the acceleration downwards.
Since, there is no motion perpendicular to surface, we have
$R - mg\cos{\theta = 0}$
$\Longrightarrow \ \ R = mg\cos{\theta\ \ \ \ldots.(ii)}$
Also, $f = \mu R = \mu\ mg\cos\theta$
Putting the value in Eq. (i) we get
$mg\sin{\theta - \mu\ mg\cos{\theta = ma}}$
$\Longrightarrow \ \ a = g\sin{\theta - \mu\ g\cos\theta}$
Now, velocity at bottom
$v^{2} = u^{2} - 2as$
Since, $v = 0$
$\therefore u = \sqrt{2as}$
$Given,\ \ s = l,\ \ a = g\sin{\theta - g\mu\cos\theta}$
$\therefore\ \ u = \sqrt{2l(g\sin{\theta - g\mu\cos\theta})}$
$u = \sqrt{2gl(\sin{\theta - \mu\cos\theta}})$

Dr. Abubacker
Reviewed by Dr. Abubacker

Physics Educator & NEET/JEE Specialist

Ph.D. in Physics with 19+ years of teaching experience.

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