Answer to Question #199104 in Mechanics | Relativity for jessie

Question

A wave can travel at a speed 11.1 m/s on a wire that is under a tension of 5.3 N. What tension is required for a wave speed of 35.7 m/s in the same wire?

Answer 1

Explabations & Calculations

• Wave speed on a tensed thread is given by

$\qquad\qquad \begin{aligned} \small v&=\small \sqrt{\frac{T}{\mu}} \end{aligned}$

• Since it is the same thread used in both the cases, linear density — $\small \mu$ — remains unchanged.
• Then the speed appears to depend only on the tension in the thread.

$\qquad\qquad \begin{aligned} \small v\propto \sqrt T \end{aligned}$

• Therefore, if the required tension for the second case is $\small T_1$,

$\qquad\qquad \begin{aligned} \small \frac{v_1}{v_0}&=\small \frac{\sqrt {T_1}}{\sqrt{T_0}}\\ \small T_1&=\small T_0\cdot\big(\frac{v_1}{v_0}\big)^2\\ &=\small 5.3\,N\times \big(\frac{35.7\,ms^{-1}}{11.1\,ms^{-1}}\big)^2\\ &=\small \bold{54.8\,N} \end{aligned}$

• Notice the increased tension as higher the wave speed expected. The thread should be able to withstand this tension for a successful behavior.