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| ### Statement |
| ### Statement |
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| $3.7.8.$ [Insert the problem statement] |
| $3.7.8.$ 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? |
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| ### Solution |
| ### Solution |
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| 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 |
| 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 |
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| \[P_\perp=(\rho_1\sqrt{(c_1t)^2+(ut)^2}+\rho_2\sqrt{(c_2t)^2+(ut)^2})u.\] |
| \[P_\perp=\left(\rho_1\sqrt{(c_1t)^2+(ut)^2}+\rho_2\sqrt{(c_2t)^2+(ut)^2}\right)u.\] |
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| As a result, |
| As a result, |
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| \[F_\perp=P_\perp/t=\rho_1c_1^2\cdot\frac{u}{c_1}\sqrt{1+\left(\frac{u}{c_1}\right)^2}+\rho_2c_2^2\cdot\frac{u}{c_2}\sqrt{1+\left(\frac{u}{c_2}\right)^2}\] |
| \[F_\perp=P_\perp/t=\rho_1c_1^2\cdot\frac{u}{c_1}\sqrt{1+\left(\frac{u}{c_1}\right)^2}+\rho_2c_2^2\cdot\frac{u}{c_2}\sqrt{1+\left(\frac{u}{c_2}\right)^2}.\] |
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| Under the assumption that $F_\parallel\gg F_\perp=\rho_1c_1^2=\rho_2c_2^2$, it follows that $c_1,c_2\gg u$, and we have the approximation |
| Under the assumption that $F_\perp\ll F_\parallel=\rho_1c_1^2=\rho_2c_2^2$, it follows that $u\ll c_1,c_2$, and we have the approximation |
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| \[F_\perp\approx F_\parallel\left(\frac{u}{c_1}+\frac{u}{c_2}\right).\] |
| \[F_\perp\approx F_\parallel\left(\frac{u}{c_1}+\frac{u}{c_2}\right).\] |
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