First slide

Sonometer

Question

Two wires are fixed on a sonometer. Their tensions are in the ratio 8 : 1, their lengths are in the ratio 36 : 35, the diameters are in the ratio 4 : 1 and densities are in the ratio 1 : 2. Find the frequency of the beats produced if the note of the higher pitch has a frequency of 360 Hz.

Moderate

A

10 Hz
B

6 Hz
C

8 Hz
D

4 Hz

Solution

$\frac{T_{1}}{T_{2}} = \frac{8}{1}, \frac{l_{1}}{l_{2}} = \frac{36}{35}, \frac{d_{1}}{d_{2}} = \frac{4}{1}, \frac{ρ_{1}}{ρ_{2}} = \frac{1}{2} ∴ μ_{1} = \frac{{πd}_{1}^{2}}{4} ρ_{1} and μ_{2} = \frac{{πd}_{2}^{2}}{4} ρ_{2} . ∴ \frac{μ_{1}}{μ_{2}} = \frac{8}{1} ∴ \frac{f_{1}}{f_{2}} = \frac{l_{2}}{l_{1}} \times \sqrt{\frac{T_{1}}{T_{2}} . \frac{μ_{2}}{μ_{1}}} = \frac{35}{36} \times \sqrt{\frac{8}{1} \times \frac{1}{8}} = \frac{35}{36} . ∴ f_{2} > f_{1} We have f_{2} = 360 Hz ∴ f_{1} = f_{2} \times \frac{35}{36} = 350 Hz ∴ beat frequency = f_{2} - f_{1} = 10 Hz$

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