Question

In the arrangements shown in the figure below, block $A$ is connected to a spring and another block $B$ rests on $A$ and the system performs simple harmonic motion with a time period of $2\text{sec}$. The coefficient of static friction between $A$ and $B$ is $0.6$. The maximum amplitude of oscillation which the system can have, so that there is no relative motion between $A$ and $B$ is:
[Take $g=10{\text{m/s}}^2\&{\pi }^2\approx 10$]

• A. $0.3\text{m}$
• B. $0.6\text{m}$
• C. $0.4\text{m}$
• D. $0.52\text{m}$

Answer 1

The correct option is B $0.6\text{m}$

FBD of block B:


Let $A$ be the amplitude of oscillation,
At extreme position, the acceleration is maximum. So, there is the highest chance of slipping between the blocks.
From the FBD of block B, we can say that,
$f=m{\omega }^{2}A$ at extreme position
For no relative motion between the blocks,
$f\le f_L$
$\Rightarrow m{\omega }^{2}A\le {\mu }_{s}mg[f_L={\mu }_{s}mg]$
From the data given in the question and by using $\omega =\frac{2\pi }{T}$, we get
$A\le \frac{0.6\times g}{\frac{4{\pi }^2}{4}}\Rightarrow A_{\text{max}}=0.6\text{m}$
Thus, option (b) is the correct answer.