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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 |
| </p> | | </p> |
| </header> | | </header> |
| | | |
| <h3 id="back-link"><a href="../">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| | | |
| <p> | | <p> |
| $5.5.3$ | | $5.5.3$ |
| 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? | | 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> |
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| 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 |
