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| ho_0. The piston, which acts on a constant force F, moving in the pipe, compacts the substance to a density | | ho_0. The piston, which acts on a constant force F, moving in the pipe, compacts the substance to a density |
| ho. At what speed does the piston move if the compaction of the substance occurs in a jump, i.e. the interface moves at a certain speed in the pipe, to the right of which the density of the substance is | | ho. At what speed does the piston move if the compaction of the substance occurs in a jump, i.e. the interface moves at a certain speed in the pipe, to the right of which the density of the substance is |
| ho, and to the left- | | ho, and to the left- |
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| ho_0. The piston, which acts on a constant force F, moving in the pipe, compacts the substance to a density | | ho_0. The piston, which acts on a constant force F, moving in the pipe, compacts the substance to a density |
| ho. At what speed does the piston move if the compaction of the substance occurs in a jump, i.e. the interface moves at a certain speed in the pipe, to the right of which the density of the substance is | | ho. At what speed does the piston move if the compaction of the substance occurs in a jump, i.e. the interface moves at a certain speed in the pipe, to the right of which the density of the substance is |
| ho, and to the left- | | ho, and to the left- |
| ho_0? At the initial moment, this boundary coincides with the surface of the piston."> | | ho_0? At the initial moment, this boundary coincides with the surface of the piston."> |
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| <meta property="og:description" content="The tube of radius r is filled with a porous substance of density | | <meta property="og:description" content="The tube of radius r is filled with a porous substance of density |
| ho_0. The piston, which acts on a constant force F, moving in the pipe, compacts the substance to a density | | ho_0. The piston, which acts on a constant force F, moving in the pipe, compacts the substance to a density |
| ho. At what speed does the piston move if the compaction of the substance occurs in a jump, i.e. the interface moves at a certain speed in the pipe, to the right of which the density of the substance is | | ho. At what speed does the piston move if the compaction of the substance occurs in a jump, i.e. the interface moves at a certain speed in the pipe, to the right of which the density of the substance is |
| ho, and to the left- | | ho, and to the left- |
| ho_0? At the initial moment, this boundary coincides with the surface of the piston."> | | ho_0? At the initial moment, this boundary coincides with the surface of the piston."> |
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| ho_0. The piston, which acts on a constant force F, moving in the pipe, compacts the substance to a density | | ho_0. The piston, which acts on a constant force F, moving in the pipe, compacts the substance to a density |
| ho. At what speed does the piston move if the compaction of the substance occurs in a jump, i.e. the interface moves at a certain speed in the pipe, to the right of which the density of the substance is | | ho. At what speed does the piston move if the compaction of the substance occurs in a jump, i.e. the interface moves at a certain speed in the pipe, to the right of which the density of the substance is |
| ho, and to the left- | | ho, and to the left- |
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| <h2>Solutions of Savchenko Physics Textbook</h2> | | <h2>Solutions of Savchenko Physics Textbook</h2> |
| <p class="author"> | | <p class="author"> |
| Aliaksandr Melnichenka <br/> | | Aliaksandr Melnichenka <br/> |
| October 2023 | | October 2023 |
| $2.2.39.$ The tube of radius $r$ is filled with a porous substance of density $\rho_0$. The piston, which acts on a constant force $F$, moving in the pipe, compacts the substance to a density $\rho$. At what speed does the piston move if the compaction of the substance occurs in a jump, i.e. the interface moves at a certain speed in the pipe, to the right of which the density of the substance is $\rho$, and to the left-$\rho_0$? At the initial moment, this boundary coincides with the surface of the piston. | | $2.2.39.$ The tube of radius $r$ is filled with a porous substance of density $\rho_0$. The piston, which acts on a constant force $F$, moving in the pipe, compacts the substance to a density $\rho$. At what speed does the piston move if the compaction of the substance occurs in a jump, i.e. the interface moves at a certain speed in the pipe, to the right of which the density of the substance is $\rho$, and to the left-$\rho_0$? At the initial moment, this boundary coincides with the surface of the piston. |
| </p> | | </p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
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| <center> | | <center> |
| <figure> | | <figure> |
| <img src="draw.png" | | <img src="draw.png" |
| loading="lazy" width="280" /> | | loading="lazy" width="280" /> |
| <figcaption> | | <figcaption> |
| Geometry of the problem | | Geometry of the problem |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| | | |
| From the drawing | | From the drawing |
| $$\Delta L=L_0-L$$ | | $$\Delta L=L_0-L$$ |
| During the time $\Delta t$ the piston with velocity $v$ has moved by the amount $\Delta l$ | | During the time $\Delta t$ the piston with velocity $v$ has moved by the amount $\Delta l$ |
| $$\Delta L=v\Delta t$$ | | $$\Delta L=v\Delta t$$ |
| Given a constant cross-sectional area $S=\pi r^2$, the expression from above can be rewritten as | | Given a constant cross-sectional area $S=\pi r^2$, the expression from above can be rewritten as |
| $$\frac{m}{\rho_0S}-\frac{m}{\rho S}=v\Delta t$$ | | $$\frac{m}{\rho_0S}-\frac{m}{\rho S}=v\Delta t$$ |
| $$\frac{m}{\Delta t}=Sv\frac{\rho\rho_0}{\rho-\rho_0}$$ | | $$\frac{m}{\Delta t}=Sv\frac{\rho\rho_0}{\rho-\rho_0}$$ |
| Because the mass remains constant ($m=\text{const}$): | | Because the mass remains constant ($m=\text{const}$): |
| $$m=\rho_0SL_0=\rho SL$$ | | $$m=\rho_0SL_0=\rho SL$$ |
| From where | | From where |
| $$\frac{\rho}{\rho_0}=\frac{L}{L_0}$$ | | $$\frac{\rho}{\rho_0}=\frac{L}{L_0}$$ |
| Representing the force by a small change in momentum | | Representing the force by a small change in momentum |
| $$F_\text{average}=\frac{mv_\text{system}}{\Delta t}$$ | | $$F_\text{average}=\frac{mv_\text{system}}{\Delta t}$$ |
| Substituting the value of $\frac{m}{\Delta t}$: | | Substituting the value of $\frac{m}{\Delta t}$: |
| $$F_\text{average}=Sv\frac{\rho\rho_0}{\rho-\rho_0}v_\text{system}$$ | | $$F_\text{average}=Sv\frac{\rho\rho_0}{\rho-\rho_0}v_\text{system}$$ |
| $$v_\text{system}=\frac{0+v}{2}=\frac{v}{2}$$ | | $$v_\text{system}=\frac{0+v}{2}=\frac{v}{2}$$ |
| $$F_\text{average}=\frac{0+F}{2}=\frac{F}{2}$$ | | $$F_\text{average}=\frac{0+F}{2}=\frac{F}{2}$$ |
| Substitute into the formula for $F_\text{average}$: | | Substitute into the formula for $F_\text{average}$: |
| $$\frac{F}{2}=Sv\frac{\rho\rho_0}{\rho-\rho_0}\frac{v}{2}$$ | | $$\frac{F}{2}=Sv\frac{\rho\rho_0}{\rho-\rho_0}\frac{v}{2}$$ |
| From where | | From where |
| $$\boxed{v=\sqrt{\frac{F(\rho-\rho_0)}{\pi r^2\rho\rho_0}}}$$ | | $$\boxed{v=\sqrt{\frac{F(\rho-\rho_0)}{\pi r^2\rho\rho_0}}}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$v=\sqrt{F(\rho-\rho_0)/(\pi r^2\rho\rho_0)}$$ | | $$v=\sqrt{F(\rho-\rho_0)/(\pi r^2\rho\rho_0)}$$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 16;">Almaskhan Arsen</p> | | <p style="text-align: right; font-style: italic; font-size: 16;">Almaskhan Arsen</p> |
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