Question

Two ends of a string are tied to two rigid walls and the string is oscillating transeversely with maximum amplitude ‘A’ in the first overtone mode. The total energy of oscillation of the string is 0.32 J. If the same string oscillates with maximum amplitude A/2 in the second overtone mode, its total energy of oscillation will be

Moderate

Solution

If $ω_{0}$ be the fundamental frequency of the string, then the first overtone frequency is 2 $ω_{0}$ and the second overtone frequency is 3 $ω_{0}$ .
If ‘A’ be the maximum amplitude and E be the total energy of the string, then total energy $\propto A^{2} ω^{2}$
$\frac{E^{1}}{E} = {(\frac{A^{1}}{A})}^{2} {(\frac{ω^{1}}{ω})}^{2} = {(\frac{1}{2})}^{2} {(\frac{3}{2})}^{2} = \frac{9}{16}$
$\Rightarrow E^{1} = 9 E / 16 = \frac{9 \times 0 .32}{16} J = 0 .18 J$

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