question_answer

Two light springs of force constants \[{{k}_{1}}\] and \[{{k}_{2}}\] a block of mass m are in one line. AB on a smooth horizontal table such that one end of each spring is fixed 10 a required support and other end is attached to block of mass m kg as shown in figure. that frequency of vibration is :

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) \[n=\frac{1}{2\pi }\sqrt{\frac{{{k}_{1}}{{k}_{2}}}{n}}\]    

D)  none of these

Correct Answer: B

Solution :

When the oscillating mass m is at a distnace \[x\] towards right from its equilibrium position, then the spring \[{{k}_{1}}\]is stretched through \[x,\]while \[{{k}_{2}}\] is compressed through x. Hence, the 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 in \[F={{F}_{1}}+{{F}_{2}}=-{{k}_{1}}x-{{k}_{2}}x=-({{k}_{1}}+{{k}_{2}})x=-kx\] Hence, time period is \[T=2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{m}{{{k}_{1}}+{{k}_{2}}}}\] also,       \[T=\frac{1}{n}=\frac{1}{\text{Frequency}}\] \[\therefore \]      \[n=\frac{1}{2\pi }\sqrt{\frac{{{k}_{1}}+{{k}_{2}}}{m}}\]