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  <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...
    </p>
    <p class="TxtSolutions">
        Initially, considering a massless piston
    </p>
    <center>
        $P_0 = P_a \\;(1)$
    </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,
    </p>
    <center>
      $P'S=(m+\Delta m)g+P_aS \\;(3)$ 
    </center>
    <p class="TxtSolutions">
    From Boyle-Mariotte Law
    </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>
    <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>