Question

A body starts sliding down an inclined plane, the top half of which is perfectly smooth and lower half is rough. Find the ratio of force of friction and weight of the body, if the body comes to rest just when it reaches the bottom. Given that inclination of plane with the horizontal is 30.

Answer 1

$h_2=lsin{30}^0$

$=\frac{l}{n}$

$h_2=2l\sin {90}^0=l$

Now friction present in the lower

Let us assume the body starts from rest at the top so the change in energy at point $P$ is

Initially $Pϵ-mg\left(h_1\right)=mgl$

final $Pϵ=mg(h_1-h_2)=\frac{mgl}{2}$

$\Delta pe=\frac{mgl}{2}=KE$ at point $p$

Now from point $P$; initially $KE=\frac{mgl}{2}$ and

$pe=\frac{mgl}{2}$

so total energy $=mgl$

at the bottom total energy $=(KE+pe)final$

$=0+0$

dissipation =$\Delta$ Total energy

$mgl=$ work done by friction

So, $\frac{l}{2}\left(umg\cos {30}^0\right)=-mgl$

$\left(f_s\right)\frac{f}{2}=wt$

$\frac{f_s}{w}=\frac{2}{1}$