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<meta name="description" content="For diminishing isothermally n times the volume of gas in a cylinder with piston, over this piston is putted a load of mass m. What load should be added such that the volume of gas decreases isothermally k times more?">
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<meta property="og:description" content="For diminishing isothermally n times the volume of gas in a cylinder with piston, over this piston is putted a load of mass m. What load should be added such that the volume of gas decreases isothermally k times more?">
<title>For diminishing isothermally n times the volume of gas in a cylinder with piston, over this piston is putted a load of mass m. What load should be added such that the volume of gas decreases isothermally k times more?</title>
For diminishing isothermally $n$ times the volume of gas in a cylinder with piston, over this piston is putted a load of mass $m$. What load should be added such that the volume of gas decreases isothermally $k$ times more?
</p>
<center>
<figure>
<img src="statement.png"
loading="lazy" alt="5.5.3" width="200" />
<figcaption>
For problem 5.5.3
</figcaption>
</figure>
</center>
<h3>Solution</h3>
<p>
Applying Newton's Second Law...
</p>
<p>
Initially, considering a massless piston
$$P_0 = P_a \;(1)$$
When load of mass $m$ is putted over
$$PS=mg+P_aS \;(2)$$
After the addition of the another load,
$$P'S=(m+\Delta m)g+P_aS \;(3)$$
From Boyle-Mariotte Law
$$P_0V_0 = P\frac{V_0}{n}$$
$$P_0 = \frac{P}{n} \;(4)$$
Substituting $(2)$ into $(3)$, according to $(1)$ and separating $P_a$
$$P_a = \frac{mg}{S(n-1)} \;(5)$$
Applying Boyle-Mariotte Law again
$$P\frac{V_0}{n}=P'\frac{V_0}{nk}$$
$$P=\frac{P'}{k} \;(6)$$
Substituting $(2)$ and $(3)$ into $(6)$ and developing algebraically
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
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<meta name="description" content="For diminishing isothermally n times the volume of gas in a cylinder with piston, over this piston is putted a load of mass m. What load should be added such that the volume of gas decreases isothermally k times more?">
<meta name="description" content="For diminishing isothermally n times the volume of gas in a cylinder with piston, over this piston is putted a load of mass m. What load should be added such that the volume of gas decreases isothermally k times more?">
<meta property="og:title" content="For diminishing isothermally n times the volume of gas in a cylinder with piston, over this piston is putted a load of mass m. What load should be added such that the volume of gas decreases isothermally k times more?">
<meta property="og:title" content="For diminishing isothermally n times the volume of gas in a cylinder with piston, over this piston is putted a load of mass m. What load should be added such that the volume of gas decreases isothermally k times more?">
<meta property="og:image" content="img/logo.png">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<meta property="og:description" content="For diminishing isothermally n times the volume of gas in a cylinder with piston, over this piston is putted a load of mass m. What load should be added such that the volume of gas decreases isothermally k times more?">
<meta property="og:description" content="For diminishing isothermally n times the volume of gas in a cylinder with piston, over this piston is putted a load of mass m. What load should be added such that the volume of gas decreases isothermally k times more?">
<title>For diminishing isothermally n times the volume of gas in a cylinder with piston, over this piston is putted a load of mass m. What load should be added such that the volume of gas decreases isothermally k times more?</title>
<title>For diminishing isothermally n times the volume of gas in a cylinder with piston, over this piston is putted a load of mass m. What load should be added such that the volume of gas decreases isothermally k times more?</title>
For diminishing isothermally $n$ times the volume of gas in a cylinder with piston, over this piston is putted a load of mass $m$. What load should be added such that the volume of gas decreases isothermally $k$ times more?
For diminishing isothermally $n$ times the volume of gas in a cylinder with piston, over this piston is putted a load of mass $m$. What load should be added such that the volume of gas decreases isothermally $k$ times more?
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="statement.png"
<img src="statement.png"
loading="lazy" alt="5.5.3" width="200" />
loading="lazy" alt="5.5.3" width="200" />
<figcaption>
<figcaption>
For problem 5.5.3
For problem 5.5.3
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
Applying Newton's Second Law...
Applying Newton's Second Law...
</p>
</p>
<p>
<p>
Initially, considering a massless piston
Initially, considering a massless piston
$$P_0 = P_a \;(1)$$
$$P_0 = P_a \;(1)$$
When load of mass $m$ is putted over
When load of mass $m$ is putted over
$$PS=mg+P_aS \;(2)$$
$$PS=mg+P_aS \;(2)$$
After the addition of the another load,
After the addition of the another load,
$$P'S=(m+\Delta m)g+P_aS \;(3)$$
$$P'S=(m+\Delta m)g+P_aS \;(3)$$
From Boyle-Mariotte Law
From Boyle-Mariotte Law
$$P_0V_0 = P\frac{V_0}{n}$$
$$P_0V_0 = P\frac{V_0}{n}$$
$$P_0 = \frac{P}{n} \;(4)$$
$$P_0 = \frac{P}{n} \;(4)$$
Substituting $(2)$ into $(3)$, according to $(1)$ and separating $P_a$
Substituting $(2)$ into $(3)$, according to $(1)$ and separating $P_a$
$$P_a = \frac{mg}{S(n-1)} \;(5)$$
$$P_a = \frac{mg}{S(n-1)} \;(5)$$
Applying Boyle-Mariotte Law again
Applying Boyle-Mariotte Law again
$$P\frac{V_0}{n}=P'\frac{V_0}{nk}$$
$$P\frac{V_0}{n}=P'\frac{V_0}{nk}$$
$$P=\frac{P'}{k} \;(6)$$
$$P=\frac{P'}{k} \;(6)$$
Substituting $(2)$ and $(3)$ into $(6)$ and developing algebraically
Substituting $(2)$ and $(3)$ into $(6)$ and developing algebraically
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
