First slide

Stationary waves

Question

A wire of length $l$ and Mass per uniform length $6.0 \times 10^{- 3} k g / m$ is put under tension of $500 N$ . Two consecutive frequencies it resonates at are $420 H z$ and $490 H z$ . Then $l$ in meters is

Moderate

A

8.21 m
B

5.13 m
C

2.06 m
D

2.25 m

Solution

$S u p p o s e t h e w i r e v i b r a t e s a t 420 H z i n i t s n^{t h} h a r m o n i c a n d a t 490 H z i n i t s {(n + 1)}^{t h} h a r m o n i c . f = \frac{n}{2 l} \sqrt{\frac{T}{μ}} ∴ \frac{490}{420} = \frac{n + 1}{n} \Rightarrow n = 6 N o w, f r o m f i r s t e q u a t i o n ∴ 420 = \frac{6}{2 l} \sqrt{\frac{500}{6 \times 10^{- 3}}} \Rightarrow 70 = \frac{1}{2 \times l} \sqrt{\frac{500 \times 10^{3}}{6}} \Rightarrow 140 l = \sqrt{\frac{5 \times 10^{5}}{6}} \Rightarrow l = \frac{288.67}{140} = 2.0619$

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