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| <h2>Solutions of Savchenko Problems in Physics</h2> | | <h2>Solutions of Savchenko Problems in Physics</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="../../#2.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> |
| <img src="https://savchenkosolutions.com/2/2.1.14/statement.png" | | <img src="https://savchenkosolutions.com/2/2.1.14/statement.png" |
| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| <figcaption> | | <figcaption> |
| For problem $2.1.14$ | | For problem $2.1.14$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.14/draw.png" | | <img src="https://savchenkosolutions.com/2/2.1.14/draw.png" |
| loading="lazy" width="270" /> | | loading="lazy" width="270" /> |
| <figcaption> | | <figcaption> |
| Forces acting on the system | | Forces acting on the system |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| | | |
| <p> | | <p> |
| When the oscillations have already stopped, the balls move with the same acceleration $a$. | | When the oscillations have already stopped, the balls move with the same acceleration $a$. |
| </p> | | </p> |
| <p> | | <p> |
| Let's write Newton's second law | | Let's write Newton's second law |
| $$ | | $$ |
| \left\{\begin{matrix} | | \left\{\begin{matrix} |
| m_1a_1 = -k\Delta x \\ | | m_1a_1 = -k\Delta x \\ |
| m_2a_2 = -F + k\Delta x | | m_2a_2 = -F + k\Delta x |
| \end{matrix}\right. $$ | | \end{matrix}\right. $$ |
| Socratic to $a$ | | Socratic to $a$ |
| $$ \frac{m_1}{m_2} = \frac{-k\Delta x}{-F + k\Delta x} $$ | | $$ \frac{m_1}{m_2} = \frac{-k\Delta x}{-F + k\Delta x} $$ |
| We express $\Delta x$ | | We express $\Delta x$ |
| $$ \boxed{\Delta x = \frac{F}{k} \frac{m_1}{m_1+m_2}} $$ | | $$ \boxed{\Delta x = \frac{F}{k} \frac{m_1}{m_1+m_2}} $$ |
| Immediately after the force ceases to act on the bodies, the elastic force $k \Delta x$ will act, which will cause acceleration | | Immediately after the force ceases to act on the bodies, the elastic force $k \Delta x$ will act, which will cause acceleration |
| $$ | | $$ |
| \left\{\begin{matrix} | | \left\{\begin{matrix} |
| m_1a_1 = -k\Delta x \\ | | m_1a_1 = -k\Delta x \\ |
| m_2a_2 = k\Delta x | | m_2a_2 = k\Delta x |
| \end{matrix}\right. $$ | | \end{matrix}\right. $$ |
| Where do we express $a_1$? | | Where do we express $a_1$? |
| $$ \boxed{a_1 = -\frac{F}{m_{2}+m_{1}}} $$ | | $$ \boxed{a_1 = -\frac{F}{m_{2}+m_{1}}} $$ |
| And similarly $a_2$ | | And similarly $a_2$ |
| $$ \boxed{a_1 = \frac{Fm_{1}}{m_{2}(m_{1}+m_{2})}} $$ | | $$ \boxed{a_1 = \frac{Fm_{1}}{m_{2}(m_{1}+m_{2})}} $$ |
| <p> | | <p> |
| | | |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| <p> | | <p> |
| $$x=-\frac{Fm_{1}}{k(m_{1}+m_{2})}$$ | | $$x=-\frac{Fm_{1}}{k(m_{1}+m_{2})}$$ |
| $$a_{1}=-\frac{F}{m_{2}+m_{1}}$$ | | $$a_{1}=-\frac{F}{m_{2}+m_{1}}$$ |
| $$a_{2}=\frac{Fm_{1}}{m_{2}(m_{1}+m_{2})}$$ | | $$a_{2}=\frac{Fm_{1}}{m_{2}(m_{1}+m_{2})}$$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 16;">Straton</p> | | <p style="text-align: right; font-style: italic; font-size: 16;">Straton</p> |
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