Question

The co-efficient of friction between a block of mass $m_2$ and inclined plane is $\mu$. Mass $m_1$ connected to $m_2$ by an inextensible string will starts moving downwards if

• A. $\frac{m_1}{m_2}>\sin \theta +\mu \cos \theta$
• B. $\frac{m_1}{m_2}<\sin \theta +\mu \cos \theta$
• C. $\frac{m_1}{m_2}=\sin \theta +\mu \cos \theta$
• D. $\frac{m_1}{m_2}>\sin \theta -\mu \cos \theta$

Answer 1

The correct option is A $\frac{m_1}{m_2}>\sin \theta +\mu \cos \theta$

If $m_1$ start coming down, then friction on $m_2$ will be downward along inclined plane.


From FBD, the condition for movement of $m_1$ downward will be

$T>m_{2}g\sin \theta +f$

$\Rightarrow T>m_{2}g\sin \theta +\mu m_{2}g\cos \theta .....\left(1\right)$

Also,
$m_{1}g>T.........\left(2\right)$

From eqs. $\left(1\right)$ and $\left(2\right)$

$m_{1}g>m_{2}g\sin \theta +\mu m_{2}g\cos \theta$

$\therefore \frac{m_1}{m_2}>\sin \theta +\mu \cos \theta$

Hence, option $\left(a\right)$ is correct.