question_answer

Two blocks of masses m and Mare joined with an ideal spring of spring constant k and kept on a rough surface as shown. The spring is initially unstretched and the coefficient of friction between the blocks and the horizontal surface is \[\mu \]. What should be the maximum speed of the block of mass M such that the smaller block does not move?

A) \[\mu g\sqrt{\frac{Mm}{(M+m)k}}\]       

B) \[\mu g\sqrt{\frac{(M+m)k}{Mm}}\]

C) \[\mu g\sqrt{\frac{(2M+m)m}{kM}}\]     

D) None of these

Correct Answer: C

Solution :

[c] If smaller block does not move, \[\mu mg=kx\] Compression in spring: \[x=\frac{\mu mg}{k}\] Applying conservation of mechanical energy \[{{w}_{s}}+{{w}_{f}}={{k}_{f}}-{{k}_{i}}\] \[-\frac{k}{2}({{x}^{2}}-{{0}^{2}})+(-\mu Mgx)=0-\frac{1}{2}mv_{0}^{2}\] \[-\frac{k}{2}{{\left( \frac{\mu mg}{k} \right)}^{2}}-\mu Mg\frac{\mu mg}{k}=-\frac{1}{2}Mv_{0}^{2}\] \[Mv_{0}^{2}=\frac{{{\mu }^{2}}{{m}^{2}}{{g}^{2}}}{k}+\frac{2{{\mu }^{2}}Mm{{g}^{2}}}{k}\] \[{{v}_{0}}=\mu g\sqrt{\frac{{{m}^{2}}+2Mm}{Mk}}=\mu g\sqrt{\frac{(2M+m)m}{kM}}\]