question_answer

To make the frequency double of a spring oscillator, we have to:                                              [BHU M-2003]

A)  Reduce the mass to one fourth

B)  Quadruple the mass

C)  Double the mass

D)  Half the mass

Correct Answer: A

Solution :

                 Key Idea: Spring oscillator executes SHM. In motion of a spring oscillator the acceleration is directly proportional to displacement hence motion is SHM. It periodic time is given by                                                 \[T=2\pi \sqrt{\frac{m}{k}}\] Also frequency \[\left( n \right)=\frac{1}{T}\]                 \[\Rightarrow \]                               \[n=\frac{1}{2\pi }\sqrt{\frac{k}{m}}\] For\[{{n}_{2}}=2{{n}_{1}}\], we have                                                 \[\frac{{{n}_{1}}}{{{n}_{2}}}=\sqrt{\frac{{{m}_{2}}}{{{m}_{1}}}}\]                 \[\therefore \]                  \[\frac{{{n}_{1}}}{2{{n}_{2}}}=\sqrt{\frac{{{m}_{2}}}{{{m}_{1}}}}\]                 \[\Rightarrow \]                               \[\frac{1}{4}=\frac{{{m}_{2}}}{{{m}_{1}}}\]                 \[\Rightarrow \]                               \[{{m}_{2}}=\frac{{{m}_{1}}}{4}\] Hence, mass is reduced to one-fourth to make the frequency double.