question_answer

A spring is vibrating with frequency under same mass. If it is cut into two equal pieces and same mass is suspended, then the new frequency will be:

A)  \[n\sqrt{2}\]                                    

B)  \[\frac{n}{\sqrt{2}}\]

C)  \[\frac{n}{2}\]                 

D)         n

Correct Answer: A

Solution :

Key Idea: The two parts from which mass is suspended will be in parallel. Frequency of single spring is given by \[n=\frac{1}{2\pi }\sqrt{\frac{k}{m}}\]                    ?(i) When it is cut into two similar parts,     the effective force constant                                            \[k={{k}_{1}}+{{k}_{2}}=k+k=2k\] \[n=\frac{1}{2\pi }\sqrt{\frac{2k}{m}}\] Dividing Eq. (i) by Eq. (ii), we get                                 \[\frac{n}{n}=\sqrt{\frac{k}{2k}}\]             \[\Rightarrow \]   \[n=\sqrt{2}\,n\]