question_answer

A block C of mass m is moving with velocity \[{{\operatorname{v}}_{o}}\]and collides elastically with block A of mass m and connected to another block B of mass 2m through spring with spring constant K. What is the value of K, if \[{{\operatorname{x}}_{o}}\] is compression of spring when, velocity of A and B is same?

A) \[\frac{m{{v}^{2}}_{o}}{{{x}^{2}}_{o}}\]               

B) \[\frac{m{{v}^{2}}_{o}}{2{{x}^{2}}_{o}}\]

C) \[\frac{3m{{v}^{2}}_{o}}{2{{x}^{2}}_{o}}\]                       

D) \[\frac{2m{{v}^{2}}_{o}}{3{{x}^{2}}_{o}}\]

Correct Answer: D

Solution :

Conservation of momentum \[{{\operatorname{P}}_{1}}={{P}_{r}}\] \[{{\operatorname{mv}}_{o}}=mv+2mv\] \[\operatorname{v}=\frac{{{v}_{o}}}{3}\] Applying conservation of energy \[\frac{1}{2}m{{v}_{o}}^{2}=\frac{1}{2}k{{x}_{o}}^{2}+\frac{1}{2}(3m){{\left( \frac{{{v}_{o}}}{3} \right)}^{2}}\] \[\frac{1}{2}m{{v}_{o}}^{2}=\frac{K{{x}_{o}}^{2}}{2}+\frac{1}{2}{{\frac{{{v}_{o}}}{3}}^{2}}\] \[m{{v}_{o}}^{2}=K{{x}_{o}}^{2}+{{\frac{{{v}_{o}}}{3}}^{2}}\] \[K=\frac{2m{{v}_{o}}^{2}}{{{x}_{o}}^{2}}\]