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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> |
| <p> | | <p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.13/draw.png" | | <img src="https://savchenkosolutions.com/2/2.1.13/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> |
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| <p> | | <p> |
| Let's write the equilibrium condition for the two lower balls on the vertical and horizontal axes | | Let's write the equilibrium condition for the two lower balls on the vertical and horizontal axes |
| $$ | | $$ |
| \left\{\begin{matrix} | | \left\{\begin{matrix} |
| mg = F_1 \sin \alpha \\ | | mg = F_1 \sin \alpha \\ |
| F_x = F_1 \cos \alpha & | | F_x = F_1 \cos \alpha & |
| \end{matrix}\right. $$ | | \end{matrix}\right. $$ |
| where $P$ is the painter's pressure force on the chair. </p> | | where $P$ is the painter's pressure force on the chair. </p> |
| And for the upper ball | | And for the upper ball |
| $$ T = mg + 2F_1 \sin \alpha $$ | | $$ T = mg + 2F_1 \sin \alpha $$ |
| | | |
| $$ T = 3mg $$ | | $$ T = 3mg $$ |
| Accordingly, when the thread burns out, the upper ball will be acted upon downwards by a force of $T=3mg$. From Newton's second law, we find its initial acceleration as | | Accordingly, when the thread burns out, the upper ball will be acted upon downwards by a force of $T=3mg$. From Newton's second law, we find its initial acceleration as |
| $$ a = \frac{T}{m} = 3g $$ | | $$ a = \frac{T}{m} = 3g $$ |
| The lower balls will be acted upon in the horizontal direction by the force $F_x$, which will be compensated by the force $F_1 \cos \alpha$, and the force of gravity $mg$ — $F_1 \sin \alpha$ | | The lower balls will be acted upon in the horizontal direction by the force $F_x$, which will be compensated by the force $F_1 \cos \alpha$, and the force of gravity $mg$ — $F_1 \sin \alpha$ |
| <p> | | <p> |
| <p> | | <p> |
| Thus, the lower balls will be in zero gravity | | Thus, the lower balls will be in zero gravity |
| $$ a=0 $$ | | $$ a=0 $$ |
| NO: Something similar happens when a Slinky falls | | NO: Something similar happens when a Slinky falls |
| </p> | | </p> |
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| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.13/slinki.gif" | | <img src="https://savchenkosolutions.com/2/2.1.13/slinki.gif" |
| loading="lazy" width="270" /> | | loading="lazy" width="270" /> |
| <figcaption> | | <figcaption> |
| Falling Slinky | | Falling Slinky |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| I recommend an interesting problem about Slinky <a href="https://s3.eu-central-1.amazonaws.com/physprob.com/files/ipho/2019_Israel_p1.pdf" target="_blank">IPhO 2019 "Springs and Slinky"</a> | | I recommend an interesting problem about Slinky <a href="https://s3.eu-central-1.amazonaws.com/physprob.com/files/ipho/2019_Israel_p1.pdf" target="_blank">IPhO 2019 "Springs and Slinky"</a> |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| The acceleration of the upper ball is $3g$, and the acceleration of the lower balls is zero. | | The acceleration of the upper ball is $3g$, and the acceleration of the lower balls is zero. |
| </p> | | </p> |
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