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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> |
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| <h3 id="back-link"><a href="../">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
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| <p> | | <p> |
| $2.8.4$ | | $2.8.4$ |
| Through three holes made in a desktop pass threads united by a common knot. At the free ending of each thread is hanged equal loads. Determine the angles between threads. There is no friction. | | Through three holes made in a desktop pass threads united by a common knot. At the free ending of each thread is hanged equal loads. Determine the angles between threads. There is no friction. |
| </p> | | </p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| Tension in each thread is the same ($T=mg$). Moreover, note that the absolute value of the sum of two any tensions must be equal to the third one which is also $T$ and be directed in opposed sense for maintain the equilibrium. So, consider for example the angle $\gamma$ | | Tension in each thread is the same ($T=mg$). Moreover, note that the absolute value of the sum of two any tensions must be equal to the third one which is also $T$ and be directed in opposed sense for maintain the equilibrium. So, consider for example the angle $\gamma$ |
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| loading="lazy" alt="2.8.4" width="300" /> | | loading="lazy" alt="2.8.4" width="300" /> |
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| Top view | | Top view |
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| <p> | | <p> |
| $$T_R = T = 2T\cos{\frac{\gamma}{2}}$$ | | $$T_R = T = 2T\cos{\frac{\gamma}{2}}$$ |
| Hence | | Hence |
| $$\cos{\frac{\gamma}{2}} = 0.5$$ | | $$\cos{\frac{\gamma}{2}} = 0.5$$ |
| $$\gamma = \frac{2\pi}{3} = 120^\circ$$ | | $$\gamma = \frac{2\pi}{3} = 120^\circ$$ |
| But, since the above condition of the sum of two any tensions always is true, the three angles must be equal. | | But, since the above condition of the sum of two any tensions always is true, the three angles must be equal. |
| $$\boxed{\alpha=\beta=\gamma=120^\circ}$$ | | $$\boxed{\alpha=\beta=\gamma=120^\circ}$$ |
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| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| BSc. Luis Daniel Fernández Quintana<br> | | BSc. Luis Daniel Fernández Quintana<br> |
| Physics Department (FCNE)<br> | | Physics Department (FCNE)<br> |
| Universidad de Oriente, Cuba<br> | | Universidad de Oriente, Cuba<br> |
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