Question

As shown in the figure, two blocks, each of mass $3kg$, are connected by a spring, whose spring constant is $200N/m$. 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 $g=10m/s^2$ )

Answer 1

Answer: 3

$m_{1}g\sin \alpha +m_{2}g\sin \alpha -{\mu }_1{m}_{1}g\cos \alpha -{\mu }_2{m}_{2}g\cos \alpha$
$=\left(m_1+m_2\right)\cdot a_{\text{centre }}$
$2mg\sin \alpha -\left({\mu }_1+{\mu }_2\right)mg\cos \alpha =2m{a}_{\text{centre }}$
$a_{\text{centre }}=g\left(\sin \alpha -\frac{{\mu }_1+{\mu }_2}{2}\cos \alpha \right)$
$mg\sin \alpha -{\mu }_{2}mg\cos \alpha -k\Delta l=m{a}_{\text{centre }}$
$mg\sin \alpha +k\Delta l-{\mu }_{1}mg\cos \alpha =m{a}_{\text{centre }}$
$\Delta l=\frac{m}{k}g\frac{{\mu }_1-{\mu }_2}{2}\cdot \cos \alpha$