Question

As shown in the figure, two blocks, each of mass $3\, kg$, are connected by a spring, whose spring constant is $200 \, N / m$. They are placed onto an inclined plane of angle $37^{\circ}$. The coefficient of friction between the upper block and the inclined plane is $0.6$, while between the lower block and the inclined plane is $0.1$. After a while, the two blocks move together with the same acceleration. Find the extension (in $cm$ ) of the spring. (Use $g=10\, m / s ^{2}$ )

Answer 1

$m_{1} g \sin \alpha+m_{2} g \sin \alpha-\mu_{1} m_{1} g \cos \alpha-\mu_{2} m_{2} g \cos \alpha$
$=\left(m_{1}+m_{2}\right) \cdot a_{\text {centre }}$
$2 m g \sin \alpha-\left(\mu_{1}+\mu_{2}\right) m g \cos \alpha=2 m a_{\text {centre }}$
$a_{\text {centre }}=g\left(\sin \alpha-\frac{\mu_{1}+\mu_{2}}{2} \cos \alpha\right)$
$m g \sin \alpha-\mu_{2} m g \cos \alpha-k \Delta l=m a_{\text {centre }}$
$m g \sin \alpha+k \Delta l-\mu_{1} m g \cos \alpha=m a_{\text {centre }}$
$\Delta l=\frac{m}{k} g \frac{\mu_{1}-\mu_{2}}{2} \cdot \cos \alpha$