A wave moves with speed $300\ \mathrm{ms}^{-1}$ on a wire, which is under a tension of $500\ \mathrm{N}$.…
A wave moves with speed $300\ \mathrm{ms}^{-1}$ on a wire, which is under a tension of $500\ \mathrm{N}$. Find how much the tension must be changed to increase the speed to $312\ \mathrm{ms}^{-1}$.
Solution
Sol. Speed of a transverse wave on the wire, $v=\sqrt{T/\alpha}\quad...$(i)
Differentiating with respect to tension, we get
$\dfrac{dv}{dT}=\dfrac{1}{2\sqrt{\alpha T}}\quad...$(ii)
On dividing Eq. (ii) by Eq. (i), we get
$\dfrac{dv}{v}=\dfrac{1}{2}\dfrac{dT}{T}\;$ or $\;dT=(2T)\dfrac{dv}{v}$
Substituting the values in above equation, we get
$\displaystyle dT=\dfrac{(2)(500)(312-300)}{300}=40\ \mathrm{N}$
i.e. Tension should be increased by 40 N.
Answer: $40\ \mathrm{N}$