First slide

Beats

Question

Two organ pipes, each closed at one end, give 5 beats per second when emitting their fundamental notes. If their lengths are in the ratio of 50 : 51, their fundamental frequencies (in Hz) are

Moderate

A

250,255
B

255,260
C

260,265
D

265,270

Solution

$\begin{array}{l} v_{1}^{} = \frac{1}{4 L_{1}^{}} \sqrt{\frac{γ P}{ρ}}, v_{2}^{} = \frac{1}{4 L_{2}^{}} \sqrt{\frac{γ P}{ρ}} . T h e r e f o r e \\ \frac{v_{1}^{}}{v_{2}^{}} = \frac{L_{2}^{}}{L_{1}^{}} = \frac{50}{51} . I t i s g i v e n t h a t v_{2}^{} = v_{1}^{} + 5 \end{array}$
$\begin{array}{l} T h e r e f o r e \frac{v_{1}^{}}{v_{1}^{} + 5} = \frac{50}{51} w h i c h g i v e s v_{1}^{} = 250 H z . \\ H e n c e v_{2}^{} = 250 + 5 = 255 H z w h i c h i s c h o i c e (a) \end{array}$

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