First slide

Standing waves

Question

Two vibrating strings of the same material but lengths L and 2L have radii 2r 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 $n_{1}$ and the other with frequency $n_{2}$ . The ratio $\frac{n_{1}}{n_{2}}$ , is given by

Moderate

A

2
B

4
C

8
D

1

Solution

Fundamental frequency $n = \frac{1}{2 l} \sqrt{\frac{T}{{πr}^{2} ρ}}$
where m = Mass per unit length of wire
$\Rightarrow n \propto = \frac{1}{lr} \Rightarrow \frac{n_{1}}{n_{2}} = \frac{r_{2}}{r_{1}} \times \frac{l_{2}}{l_{1}} = \frac{r}{2 r} \times \frac{2 L}{L} = \frac{1}{1}$

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