Question

If the coefficient of friction between $A$ and $B$ is $\mu$, the maximum acceleration of the wedge $A$ for which $B$ remains at rest with respect to the wedge is

• A. $\mu g$
• B. $g\left(\frac{1+\mu }{1-\mu }\right)$
• C. $g\left(\frac{1-\mu }{1+\mu }\right)$
• D. $\frac{g}{\mu }$

Answer 1

The correct option is B $g\left(\frac{1+\mu }{1-\mu }\right)$

The FBD of B is
Here the block $B$ experiences pseudo force towards left in the $A$ frame.
By balancing the equation perpendicular to the incline we get,

$N=ma\sin \theta +mg\cos \theta$

Now balancing the forces along the incline plane,
$ma\cos \theta =mg\sin \theta +f$
In limiting case, $(f=\mu N)$
Hence we get,
$ma\sin \theta =mg\sin \theta +\mu (ma\sin \theta +mg\cos \theta )$

By putting $\theta ={45}^{\text{o}}$ we get,
$a\times \frac{1}{\sqrt{2}}=g\times \frac{1}{\sqrt{2}}+\mu (a\times \frac{1}{\sqrt{2}}+g\times \frac{1}{\sqrt{2}})$

$a=g\left(\frac{1+\mu }{1-\mu }\right)$

Hence option B is the correct answer