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Question
Show that when a string fixed at its two ends vibrates in 1 loop, 2 loops, 3 loops and 4 loops, the frequencies are in the ratio 1:2:3:4.
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Solution
Let, there is n number of loops in the string.
The length corresponding to each loop is `λ/2`.
Now, we can write
L = `(nλ)/2`
⇒ λ = `(2L)/n` .....[For n loops]
⇒ `v/v = (2L)/n` ......[∵ v = vλ]
⇒ v = `n/(2L)`v = `n/(2L) sqrt(T/mu)` .....[∵ Velocity of transverse waves = `sqrt(T/mu)`]
⇒ v ∝ n ......[∵ Length and speed are constants]
So, v1 : v2 : v3: v4 = n1 : n2 : n3 : n4
= 1 : 2 : 3 : 4
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