91影视

1. Linear density,$渭=2.0\mathrm{g}/\mathrm{m}\mathrm{or}0.002\mathrm{kg}/\mathrm{m}$
2. From figure, the kinetic energy rate,$\frac{dk}{dt}=10\mathrm{W}$

For a sinusoidal wave, by using the formula of average power, we can calculate the amplitude of the wave.

Formula:

The rate of kinetic energy, $\frac{dk}{dt}=\frac{1}{2}渭谓{蝇}^2{y}_m^2{\cos }^2\left(kx-蝇t\right)$(i)

The angular frequency of a body$蝇=2蟺f$ (ii)

The velocity of a body, $谓=f位$ (iii)

In equation (i), it can be seen that the rate of change of kinetic energy is the function of ${\cos }^2\left(kx-蝇t\right)$. So, using the relationship between the time period of $\cos$and ${\cos }^2$functions and using the time period given in the graph, we can write the frequency as:

$$\begin{gathered} f=\frac{1}{2脳{10}^{-3}\mathrm{s}} \\ =500\mathrm{Hz} \end{gathered}$$

Similarly, the velocity of the wave is given as:

$$\begin{gathered} v=\frac{0.1\mathrm{m}}{1脳{10}^{-3}\mathrm{s}} \\ =100\mathrm{m}/\mathrm{s} \end{gathered}$$

Using equation (iii), the wavelength of a wave is given as:

$$\begin{gathered} 位=v/f \\ =\frac{100}{500} \\ =0.20\mathrm{m} \end{gathered}$$

From equation (i), we can writethat the rate of change of kinetic energy would bemaximum if ${\cos }^2\left(kx-蝇t\right)=1$

localid="1660983930043" $$\begin{gathered} \frac{dk}{dt}=\frac{1}{2}渭谓{蝇}^2{y}_m^2 \\ 10=\frac{1}{2}渭谓{蝇}^2{y}_m^2 \\ {y}_m^2=\frac{2脳10}{2{蟺}^2渭位f^3}\left(鈭�fromgiven\right) \\ {y}_m^2=\frac{2脳10}{渭谓{蝇}^2} \\ {y}_m=\sqrt{\frac{10}{2{蟺}^2渭位f^3}}\left(鈭�fromequation\left(ii\right)\right) \\ =\sqrt{\frac{10}{2脳9.86脳0.002脳0.20脳{500}^3}} \\ =\sqrt{\frac{10}{986000}} \\ =0.0032\mathrm{m} \end{gathered}$$

Hence, the value of the amplitude is 0.0032 m