Question

Two masses $m$ and $2m$ are connected by a string passing over a pulley fixed at the top of an inclined plane. The first mass moves vertically downwards drawing the second up the incline. If the angle of inclination is $\theta$ and the coefficient of friction is $\mu$, the acceleration of the system is less than it would be if the plane went rough by an amount:

• A. $2\mu g$
• B. $\frac{2}{3}\mu \sin \theta g$
• C. $\frac{(1-2\mu \sin \theta )g}{3}$
• D. $\frac{2}{3}\mu \cos \theta g$

Answer 1

The correct option is D $\frac{2}{3}\mu \cos \theta g$

Free body diagram for the problem is shown in the attached figure.

For mass $m$,

$mg-T=ma..........\left(i\right)$

For mass $2m$ with friction,

$N=2mg\cos \theta$

$F_r=\mu N=2\mu mg\cos \theta$

$T-2mg\sin \theta -F_r=2ma$

$T-2mg\sin \theta -2\mu mg\cos \theta =2ma...............\left(ii\right)$

Adding $\left(i\right)\&\left(ii\right),$

$mg-2mg\sin \theta -2\mu mg\cos \theta =3ma$

$a_1=g/3-2g\sin \theta /3-2\mu g\cos \theta /3.............\left(iii\right)$

For mass $2m$ without friction,

$T-2mg\sin \theta =2ma.....................\left(iv\right)$

Adding $\left(i\right)\&\left(iv\right),$

$a_2=g/3-2g\sin \theta /3...............\left(v\right)$

Reduction in acceleration due to friction is:

$a_2-a_1=2\mu g\cos \theta /3$