The surface on which the blocks are placed is smooth. If the two blocks are displaced by small amount, then determine the time period of oscillation of resulting motion of two blocks.
A) \[2\pi \sqrt{\frac{3m}{7k}}\]
B) \[2\pi \sqrt{\frac{7m}{2k}}\]
C) \[3\pi \sqrt{\frac{m}{7k}}\]
D) \[2\pi \sqrt{\frac{3m}{4k}}\]
Correct Answer: C
Solution :
The given system can be redrawn as follows.
The above simplification has been done by using series and parallel combinations of springs and the reduced mass concept. In series, \[\frac{1}{{{k}_{eq}}}=\frac{1}{{{k}_{1}}}+\frac{1}{{{k}_{2}}}\] In parallel, \[{{k}_{eq}}={{k}_{1}}+{{k}_{2}}\] Reduced mass, \[\mu =\frac{{{m}_{1}}{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}}\] Required time period, \[T=2\pi \sqrt{\frac{\mu }{{{k}_{eqe}}}}=2\pi \sqrt{\frac{3m}{4\times 7k/3}}=3\pi \sqrt{\frac{m}{7k}}\]
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