A) \[n=\frac{1}{2\pi }\sqrt{\frac{{{k}_{1}}+{{k}_{2}}}{m}}\]
B) \[n=\frac{1}{2\pi }\sqrt{\frac{{{k}_{1}}\,{{k}_{2}}}{m}}\]
C) \[n=\frac{1}{2\pi }\sqrt{\frac{{{k}_{1}}-{{k}_{2}}}{m}}\]
D) none of these
Correct Answer: A
Solution :
When the oscillating mass m is at a distance \[x\]towards right from its equilibrium position, then the spring \[{{k}_{1}}\] is stretched through \[x\], while \[{{k}_{2}}\] is compressed through \[x\]. Hence, restoring force exerted by each spring on the mass m is in the same direction tending to bring it in its equilibrium position. Let \[{{F}_{1}}\] and \[{{F}_{2}}\] be the restoring forces produced in the springs of force constants \[{{k}_{1}}\] and \[{{k}_{2}}\], then \[{{F}_{1}}=-{{k}_{1}}x\] and \[{{F}_{2}}=-{{k}_{2}}x\] Total restoring force acting on the mass is \[F={{F}_{1}}+{{F}_{2}}=-{{k}_{1}}x-{{k}_{2}}x=-({{k}_{1}}+{{k}_{2}})x=-kx\] \[\therefore \] \[k={{k}_{1}}+{{k}_{2}}\] Hence, time period is \[T=2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{m}{{{k}_{1}}+{{k}_{2}}}}\] Also, frequency \[n=\frac{1}{T}=\frac{1}{2\pi }=\sqrt{\frac{{{k}_{1}}+{{k}_{2}}}{m}}\]
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