Question

A string fixed at both ends has consecutive standing waves for which thedistances between adjacent nodes are 12cm and 10 cm respectively. What is the minimum possible length of the spring?

Answer 1

For the two consecutive waves,
$$\begin{gathered} \mathrm{n}\frac{{\mathrm{\lambda }}_1}{2}=(\mathrm{n}+1)\frac{{\mathrm{\lambda }}_2}{2} \\ \mathrm{where}\mathrm{n}=\mathrm{Number}\mathrm{of}\mathrm{loops}\mathrm{in}\mathrm{first}\mathrm{wave} \\ \mathrm{wave} \\ \mathrm{n}\times (2\times 12)/2=(\mathrm{n}+1)\times (2\times 10)/2 \\ \mathrm{or}\mathrm{n}\times (2\times 12)=(\mathrm{n}+1)\times (2\times 10) \\ 24\mathrm{n}=20\mathrm{n}+20 \\ \mathrm{or}4\mathrm{n}=20 \\ \mathrm{or}\mathrm{n}=5 \\ \mathrm{So}\mathrm{the}\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{string} \\ \mathrm{L}=\mathrm{n}\frac{{\mathrm{\lambda }}_1}{2}=\frac{5\times 24}{2}=60\mathrm{cm} \end{gathered}$$