<meta name="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. 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_0? At the initial moment, this boundary coincides with the surface of the piston.">
<meta property="og:title" 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. 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_0? At the initial moment, this boundary coincides with the surface of the piston.">
<meta property="og:image" content="img/logo.png">
<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. 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_0? At the initial moment, this boundary coincides with the surface of the piston.">
<title>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
@@ -28,9 +28,9 @@
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_0? At the initial moment, this boundary coincides with the surface of the piston.</title>
$2.2.39.$ 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>
<h3>Solution</h3>
<p>
</p>
<center>
<figure>
<img src="draw.png"
loading="lazy" width="280" />
<figcaption>
Geometry of the problem
</figcaption>
</figure>
</center>
<p>
From the drawing
$$\Delta L=L_0-L$$
During the time $\Delta t$ the piston with velocity $v$ has moved by the amount $\Delta l$
$$\Delta L=v\Delta t$$
Given a constant cross-sectional area $S=\pi r^2$, the expression from above can be rewritten as
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
<meta name="description" content="The tube of radius r is filled with a porous substance of density
<meta name="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.">
<meta property="og:title" content="The tube of radius r is filled with a porous substance of density
<meta property="og:title" 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.">
<meta property="og:image" content="img/logo.png">
<meta property="og:image" content="img/logo.png">
<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.">
<title>The tube of radius r is filled with a porous substance of density
<title>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
@@ -28,9 +28,9 @@
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.</title>
ho_0? At the initial moment, this boundary coincides with the surface of the piston.</title>
$2.2.39.$ 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>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
</p>
</p>
<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
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
