With what tension must a rope with length 2.50 m and mass 0.120 kg be stretched for transverse waves of frequency 40.0 Hz to have a wavelength of 0.750 m?

The speed of a wave is the distance travelled by a given point on a wave ina given interval of time.

Thespeedof a wave on a string in terms of the tension T and the mass per unit length$\mu$ is given by$v=\sqrt{\frac{T}{\mu }}$

And the general speed of a wave in terms of the wavelength $\lambda$and the frequency f is given by:

$v=f\times \lambda$

The length of the rope is l = 2.50m, its mass is m = 0.120kg, the frequency of the wave in the rope is f = 40Hz and its wavelength is $\lambda$= 0.750m.

First, calculate the wave speed by substituting for $\lambda$and f into the wave speed formula:

$$\begin{gathered} v=\left(40s^{-1}\right)\times \left(0.750m\right) \\ =30m/s \end{gathered}$$

$$\begin{gathered} \mu =\frac{m}{l} \\ =\frac{0.12kg}{2.50m} \\ =0.048kg/m \end{gathered}$$

Finally,put in the values for v and f into the wave speed on a string formula and solve for the value of tension on the rope:

$$\begin{gathered} 30m/s=\sqrt{\frac{T}{0.048kg/m}} \\ T=0.048\times {\left(30\right)}^2 \\ =43.2N \end{gathered}$$

Therefore, the required tension on the rope is,$T=43.2N$ .