Question

The speed of a transverse wave on a string is170m/swhen the string tension is 120N. To what value must the tension be changed to raise the wave speed to180m/s?

Short answer

The value of change in tension to raise the velocity to 180 m/s in wave speed is 135 N

Step-by-step solution

The given data

• Speed of the wave,$v_1=170\mathrm{m}/\mathrm{s}$
• Tension in the string, $T_1=120\mathrm{N}$
• Speed of the wave,$v_2=180\mathrm{m}/\mathrm{s}$

Understanding the concept of the wave equation

The wave speed in a string will be equal to the square root of tension in the string having unit linear density.

Formula:

The velocity of a wave in terms of tension and linear density,$v=\sqrt{\frac{T}{\mu }}$ (i)

Calculations the change in tension

First, we find the linear density of the string from the tension 120 N when the wave speed is 170 m/s.

Again, as the value of linear density is same for the given string. Hence, considering equation (i), we can compare the two velocities, hence

To find the tension in the string when wave speed is 180 m/s we can use the formula

$$\begin{gathered} T_2=v_2^2\times \mu \\ =\left(\frac{v_2^2}{v_1^2}\right)T_1\left(\because \mu =\frac{T_1}{v_1^2}\right) \\ =\left(\frac{180}{170}\right)\times 120 \\ =134.5N \\ \approx 135N \end{gathered}$$

Hence, the value of tension should be changed to 135 N