Question

The coefficient of friction between block of mass $m$ and $2m$ is $\mu =2\tan \theta$. There is no friction between block of mass $2m$ and inclined plane. The maximum amplitude of the two block system for which there is no relative motion between both the block is:

• A. $g\sin \theta \sqrt{\frac{k}{m}}$
• B. $\frac{mg\sin \theta }{k}$
• C. $3\frac{mg\sin \theta }{k}$
• D. $Noneofthese$

Answer 1

The correct option is C $3\frac{mg\sin \theta }{k}$

Here, $\omega =\sqrt{\frac{k}{3m}}$

Maximum static frictional force, $f_{max}=\mu\,

mg\,cos\,\theta=2\,tan\,\theta\,mg\,cos\,\theta=2\,mg\,cos\,\theta$

Applying Newton's second law on the block at lower extreme position, we get

$f-mgcos\theta =m{\omega }^{2}A$

$⟹f=m{\omega }^{2}A+mgsin\theta$

As $f\le f_{max},$

${\omega }^{2}A=gsin\theta$

or

Amplitude, $A=\frac{3mgsin\theta }{k}$