3.7.8. A string consisting of two parts with linear densities $\rho_1$ and $\rho_2$ is stretched by longitudinal forces $F_\parallel$. At the point where the parts are connected, the string is pulled by a transverse force $F_\perp$. How does the shape of the string change with time?
Solution
For problem $3.7.8$
The speeds of propagation of the waveform to the left and the right are $c_1=\sqrt{F_\parallel/\rho_1}$ and $c_2=\sqrt{F_\parallel/\rho_2}$, respectively. Consequently, the mass of the portion of the rope that has transverse momentum increases linearly with time, and its transverse speed $u$ must be constant, because the transverse force is constant. After time $t$, transverse momentum of the waveform is
$$P_\perp=(\rho_1c_1t+\rho_2c_2t)u.$$
Note that even though the lengths of the stretched parts of the rope are $\sqrt{(c_1t)^2+(ut)^2}$ and $\sqrt{(c_2t)^2+(ut)^2}$, they used to have lengths $c_1t$ and $c_2t$, so they must have masses $\rho_1c_1t$ and $\rho_2c_2t$, respectively.
