Question

A disc is kept on a smooth horizontal plane with its plane parallel to horizontal plane. A groove is mass in the is as shown in the figure. The coefficient of friction between mass $m$ and surface of the ground is $2/5$ and $\sin \theta =3/5$. Find the acceleration of mass with respect to the frame of reference of the disc

Answer 1

Here in this question there will be two frictional force acting on the body as object have two surfaces in contact with the groove , one lower surface($f_1$) and another is side surface opposite to the direction of motion($f_2$).

Also, w.r.t frame of disc there will be a pseudo force on the block as shown in the figure.

According to Newton's second law , from FBD we can write

for block moving downwards along groove :

$ma\cos \theta -f_1-f_2=m{a}^{\prime }$

Here $f_1=\mu mg$ and$f_2=\mu ma\sin \theta$

using these we get : $ma\cos \theta -\mu m(g+a\sin \theta =m{a}^{\prime }$ (a' = acceleration w.r.t frame of disc)

solving this we get : $a^{\prime }=a\cos \theta -\mu (g+a\sin \theta )$

from question we know that $a=25\frac{m}{s^2},\mu =\frac{2}{5}$ and $\sin \theta =\frac{3}{5}$, this gives $\cos \theta =\frac{4}{5}$ and using $g=10\frac{m}{s^2}$

we get , $a^{\prime }=25\times \frac{4}{5}-\frac{2}{5}(10+25\times \frac{3}{5})=10\frac{m}{s^2}$