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<h2>Solutions of Savchenko Physics Textbook</h2>
<p class="author">
Aliaksandr Melnichenka <br/>
October 2023
</p>
</header>
− <main>
− <article>
− <h3 id="back-link"><a href="../">$\leftarrow$Back</a></h3>
−
− <h4>Statement</h4>
− <p>
− $5.5.3$
− 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>
− <h4>Solution</h4>
− <p class="TxtSolutions">
− Applying Newton's Second Law...
+ <h3 id="back-link"><a href="../">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+
+ <p>
+ $5.5.3$
+ 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 class="TxtSolutions">
− Initially, considering a massless piston
− </p>
<center>
− $P_0 = P_a \;(1)$
+ <figure>
+ <img src="statement.png"
+ loading="lazy" alt="5.5.3" width="200" />
+ <figcaption>
+ For problem 5.5.3
+ </figcaption>
+ </figure>
</center>
− When load of mass $m$ is putted over
− <center>
− $PS=mg+P_aS \;(2)$
− </center>
− <p class="TxtSolutions">
− After the addition of the another load,
+
+ <h3>Solution</h3>
+ <p>
+ Applying Newton's Second Law...
</p>
− <center>
− $P'S=(m+\Delta m)g+P_aS \;(3)$
− </center>
− <p class="TxtSolutions">
− From Boyle-Mariotte Law
+ <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
+ $$\left(\frac{mg}{S}+P_a\right)(k-1)=\frac{\Delta mg}{S} \;(7)$$
+ Putting $(5)$ into $(7)$ and separating $\Delta m$
+ $$\boxed{\Delta m=m\frac{(k-1)n}{(n-1)}}$$
</p>
− <center>
− $P_0V_0 = P\frac{V_0}{n}$
− </center>
− <center>
− $P_0 = \frac{P}{n} \;(4)$
− </center>
− <p class="TxtSolutions">
− Substituting $(2)$ into $(3)$, according to $(1)$ and separating $P_a$
+ <p style="text-align: right; font-style: italic; font-size: 14;">
+ BSc. Luis Daniel Fernández Quintana<br>
+ Physics Department (FCNE)<br>
+ Universidad de Oriente, Cuba<br>
</p>
− <center>
− $P_a = \frac{mg}{S(n-1)} \;(5)$
− </center>
− <p class="TxtSolutions">
− Applying Boyle-Mariotte Law again
− </p>
− <center>
− $P\frac{V_0}{n}=P'\frac{V_0}{nk}$
− </center>
− <center>
− $P=\frac{P'}{k} \;(6)$
− </center>
− <p class="TxtSolutions">
− Substituting $(2)$ and $(3)$ into $(6)$ and developing algebraically
− </p>
− <center>
− $\left(\frac{mg}{S}+P_a\right)(k-1)=\frac{\Delta mg}{S} \;(7)$
− </center>
− <p class="TxtSolutions">
− Putting $(5)$ into $(7)$ and separating $\Delta m$
− </p>
− <center>
− $\boxed{\Delta m=m\frac{(k-1)n}{(n-1)}}$
− </center>
− <p class="TxtSolutions" style="text-align: right; font-style: italic; font-size: 14;">
− BSc. Luis Daniel Fernández Quintana<br>
− Physics Department (FCNE)<br>
− Universidad de Oriente, Cuba<br>
− </p>
− </article>
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