Question

A body takes time $t$ to reach the bottom of an inclined plane of angle θ will be horizontal. if the plane is rough, the time it takes now is $2t$. The coefficient of friction of the rough surface is ?

Answer 1

Hint: In the question, a condition is already given and we can easily calculate the length of the incline and eventually after having two equations, we can find out the coefficient of friction of the rough surface.

Complete step by step answer:
As per the question, we can see the body takes time \[t\] to reach the bottom of an inclined plate of angle $\theta $ with the horizontal. Therefore,
Initial velocity, $u=0$
And acceleration down the incline, $a=g\sin \theta $
As the plane is initially smooth, the length of incline will be
$s=\dfrac{1}{2}g\sin \theta {{t}^{2}}$ ---$(1)$
Now when there is a presence of friction in the second case, let the coefficient of friction be $\mu $
Therefore, the frictional force, $f=\mu N=\mu mg\cos \theta $
Here, $m$ is the mass of the body

Now, by equating the force equation , we get acceleration as
$a=(\sin \theta -\mu \cos \theta )g$
Also, it is given that the time taken in the 2nd case is $2t$
Therefore, the distance equation becomes
$s=\dfrac{1}{2}\times (\sin \theta -\mu \cos \theta )g\times {{(2t)}^{2}}$ ---$(2)$
After solving equation one and two, we get
$\sin \theta =(\sin \theta -\mu \cos \theta )4 \\
\therefore \mu =\dfrac{3}{4}\tan \theta \\ $
Therefore the final answer to the solution is $\dfrac{3}{4}\tan \theta $ , so, the coefficient of friction in the second case is $\dfrac{3}{4}\tan \theta $.

Note: If the initial situation was changed and there was no smooth surface then such problems are just put to trick students as the coefficient of friction of a body is constant and will be the same in all rough surfaces if we keep the rule of exception aside.