A rope, under a tension of 200 Nand fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the rope is given by $\mathit{y}\mathbf{=}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{10}\mathbf{m}\mathbf{)}\mathbf{(}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\pi x}\mathbf{/}\mathbf{2}\mathbf{)}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathbf{12}\mathbf{}\mathbf{\pi t}$ , where x = 0at one end of the rope, x is in meters, andis in seconds. What are (a) the length of the rope, (b) the speed of the waves on the rope, and (d) the mass of the rope? (d) If the rope oscillates in a third-harmonic standing wave pattern, what will be the period of oscillation?

Tension in the rope, T = 200 N

Displacement of the rope,$y=\left(0.10m\right)\sin \left(\frac{\mathrm{\pi x}}{2}\right)\sin 12\mathrm{\pi t}$.

Rope is oscillating in second-harmonic.

We can get the value of the wavenumber and angular velocity by comparing the given equation of wave with the general wave equation. From it, we can find the wavelength and then the length of the rope using the corresponding formulae for standing waves. From the angular velocity, we can find the frequency and then the speed and period from it using the corresponding relations.

Formulae:

The wavelength of an oscillation,$\lambda =\frac{2\mathrm{\pi }}{k}.............\left(1\right)$

The wavelength of a standing wave,$\lambda =\frac{2L}{n}.............\left(2\right)$

The angular frequency of the wave, $\omega =2\mathrm{\pi f}.............\left(3\right)$

The velocity of the wave, $v=\lambda f................\left(4\right)$ The velocity of the wave, $v=\sqrt{\frac{T}{\mu }}.................\left(5\right)$

The time period of the standing wave, $t=\frac{2L}{nv}.................\left(6\right)$

Displacement of the rope is,$y=\left(0.10m\right)\sin \left(\frac{\mathrm{\pi x}}{2}\right)\sin 12\mathrm{\pi t}$comparing it with the general equation of wave we get the wavenumber as:

$k=\frac{\mathrm{\pi }}{2}{m}^{-1}$

Using equation (1), we get the wavelength as given:

$$\begin{gathered} \lambda =\frac{2\mathrm{\pi }}{\left({\displaystyle \frac{\mathrm{\pi }}{2}}\right)} \\ =4 \end{gathered}$$

Wavelength of the standing wave using equation (2) is given as:

$$\begin{gathered} L=\frac{n\lambda }{2} \\ =\frac{2\left(4\right)}{2} \\ =4.0m \end{gathered}$$

Therefore, the length of the rope is 4.0 m .

Comparing the given wave equation with the general equation of wave we get the angular frequency as given:

$\omega =12{\mathrm{\pi s}}^{-1}$

Using equation (3), we get the frequency of the wave as given;

$$\begin{gathered} f=\frac{12\mathrm{\pi }}{2\mathrm{\pi }} \\ =6Hz \end{gathered}$$

Again, the speed of the waves using equation (4) can be written as:

$$\begin{gathered} v=6\left(4\right) \\ =24m/s \end{gathered}$$

Therefore, speed of the waves on the rope is 24 m/s .

Using equation (5), we get the mass of the rope as:

$$\begin{gathered} m=\frac{\tau L}{v^2}\left(\because \mu =\frac{m}{L}\right) \\ =\frac{\left(200\right)\left(4\right)}{{24}^2} \\ =1.388 \\ ~1.4kg \end{gathered}$$

Therefore, mass of the rope is 1.4 kg .

For the third harmonic standing pattern , n = 3

Using equation (vi) and the given values, we get the period of oscillation is given as;

$$\begin{gathered} T=\frac{2\left(2\right)}{3\left(24\right)} \\ =0.11s \end{gathered}$$

Therefore, the period of oscillation if the rope oscillates in a third harmonic standing wave pattern is 0.11 s.