question_answer

Fig. shows two vibrating modes of an air column. Find the ratio of frequencies of the two modes.

Answer:

                                Let L be the length of the air column                 For mode (a),                 \[L=3\lambda /4\] or \[\lambda =4L/3\]                 \[\therefore \] Frequency                 \[{{v}_{1}}=\frac{\upsilon }{\lambda }=\frac{3\upsilon }{4L}\]                 For mode (b),                 \[L=5\lambda /4\] or \[\lambda =4L/5\]                 \[\therefore \] Frequency                 \[{{v}_{2}}=\frac{\upsilon }{\lambda }=\frac{5\upsilon }{4L}\]                 Hence \[{{v}_{1}}.{{v}_{2}}=\mathbf{3:5}\]