Two vibrating strings of the same material but lengths and have radii and respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental modes, the one of length with frequency and the other with frequency . T

Question

Two vibrating strings of the same material but lengths $L$ and $2 L$ have radii $2 r$ and $r$ respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental modes, the one of length $L$ with frequency $ν_{1}$ and the other with frequency $ν_{2}$ . The ratio $ν_{1} / ν_{2}$ is given by [2000]

Smart Achievers · Accepted Answer

(4)

$n_{1} = \frac{1}{2 ℓ} \sqrt{(\frac{T}{4 π r^{2} ρ})}$ $and$ $n_{2} = \frac{1}{4 ℓ} \sqrt{(\frac{T}{π r^{2} ρ})}$

$n = \frac{v}{λ} = \frac{1}{λ} \sqrt{\frac{T}{m}}$ $[where \frac{λ}{2} = length of string]$

$∴$ $\frac{n_{1}}{n_{2}} = 2 \times \frac{1}{2} = 1$ $[∵ m = \frac{mass}{length} = \frac{ρ \times A \times length}{length} = ρ A]$

Solution

1 2

2 4

3 8

4 1

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Solution

1 2

2 4

3 8

4 1

Ans. (4) n1=12ℓ(T4πr2ρ) and n2=14ℓ(Tπr2ρ) n=vλ=1λTm [where λ2=length of string] ∴ n1n2=2×12=1 [∵m=masslength=ρ×A×lengthlength=ρA]

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