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                 \[k={{k}_{1}}+{{k}_{2}}=k+k=2\,k\]                 \[n=\frac{1}{2\pi }\sqrt{\frac{2\,k}{m}}\] ?. (ii) Dividing Eq. (i) by Eq. (ii), we get                 \[\frac{n}{n}=\sqrt{\frac{k}{2\,k}}\] \[\Rightarrow \] \[n=\sqrt{2}\,n\]