A wire of length one metre under a certain initial tension emits a sound of fundamental frequency $256 \,Hz$. When the tension is increased by $1 \,kg$ wt, the frequency of the fundamental node increases to $320 \,Hz$. The initial tension is ........... $kg \,wt$
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(c)
Let tension be $T$
$f_1=\sqrt{\frac{T}{\mu}} \times \frac{1}{2 l}=256$
$f_2=\sqrt{\frac{T+10}{\mu}} \times \frac{1}{2 l}=320$
$\sqrt{\frac{T}{T+10}}=\frac{256}{320}$
$\frac{T}{T+10}=\frac{(16)^2}{(16)^2 \times(20)^2}$
or $\frac{T}{T+10}=\frac{16^2}{(20)^2}$
or $400 T=256 T^2+2560$
or $144 T=2560$
or $T=\frac{2560}{144}$
or $T=\frac{2560}{16 \times 9}$
or $T=\frac{160}{9}$ Newton
$=\frac{16}{9} \,kg -w t$
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