Question

Mass $m_{1}$ moves on a slope making an angle $\theta$ with the horizontal and is attached to mass $m_{2}$ by a string passing over a frictionless pulley as shown in figure. The coefficient of friction between $m_{1}$ and the sloping surface is $\mu$.

Which of the following statements are true?

• A. If $m_{2} > m_{1} \sin \theta$, the body will move up the plane
• B. If $m_{2} > m_{1}(\sin \theta+\mu \cos \theta)$, the body will move up the plane
• C. If $m_{2} < m_{1}(\sin \theta+\mu \cos \theta)$, the body will move up the plane
• D. If $m_{2} < m_{1}(\sin \theta-\mu \cos \theta)$, the body will move down the plane

Answer 1

Answer: (B), (D)

In figure, $f$ is the force of friction. When the body moves up the plane, $f$ acts down the plane.

$f=\mu \,R=\mu\, m_{1}\, g\, \cos \,\theta$
In that event, $m_{2}\, g > m_{1} \,g\, \sin \,\theta+f$
$m_{2} \,g > m_{1}\, g \,\sin\, \theta+\mu\, m_{1}\, g\, \cos \,\theta $
$m_{2} > m_{1}(\sin \,\theta+\mu \,\cos\, \theta)$
Choice $(b)$ is correct.
When the body moves down the plane, $f$ acts up the plane. In that event
$\left(m_{2} \,g+f\right) < m_{1} \,g \,\sin \,\theta $
$m_{2} \,g < m_{1}\, g\, \sin \,\theta-f $
$m_{2}\, g < m_{1}\, g \,\sin\, \theta-\mu \,m_{1} \,g \,\cos\, \theta $
or $m_{2} < m_{1}(\sin \theta-\mu \cos \theta)$