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 $m_2<m_1(\sin \theta -\mu \cos \theta )$