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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> |
| $4.2.12$ | | $4.2.12$ |
| Determine the force of the tension of the thread connecting two balls of volume 10 cm$^3$ each, if the upper ball floats half-immersed in water. The lower ball is three times heavier than the upper one. | | Determine the force of the tension of the thread connecting two balls of volume 10 cm$^3$ each, if the upper ball floats half-immersed in water. The lower ball is three times heavier than the upper one. |
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| loading="lazy" alt="4.2.12" width="200" /> | | loading="lazy" alt="4.2.12" width="200" /> |
| <figcaption> | | <figcaption> |
| For problem 4.2.12 | | For problem 4.2.12 |
| </figcaption> | | </figcaption> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| Applying Newton's Second Law... | | Applying Newton's Second Law... |
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| loading="lazy" alt="4.2.12" width="200" /> | | loading="lazy" alt="4.2.12" width="200" /> |
| <figcaption> | | <figcaption> |
| Force analysis | | Force analysis |
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| <p> | | <p> |
| For upper ball | | For upper ball |
| $$m_{\uparrow}~g+T = F_{\uparrow}$$ | | $$m_{\uparrow}~g+T = F_{\uparrow}$$ |
| $$m_{\uparrow}~g+T = \rho_0~g~\frac{V}{2} \;(1)$$ | | $$m_{\uparrow}~g+T = \rho_0~g~\frac{V}{2} \;(1)$$ |
| For lower ball | | For lower ball |
| $$m_{\downarrow}~g = T+F_{\downarrow}$$ | | $$m_{\downarrow}~g = T+F_{\downarrow}$$ |
| $$m_{\downarrow}~g = T+\rho_0~g~V \;(2)$$ | | $$m_{\downarrow}~g = T+\rho_0~g~V \;(2)$$ |
| Separating the gravity's force term from $(1)$ | | Separating the gravity's force term from $(1)$ |
| $$m_{\uparrow}~g = \rho_0~g~\frac{V}{2}-T \;(3)$$ | | $$m_{\uparrow}~g = \rho_0~g~\frac{V}{2}-T \;(3)$$ |
| Dividing $(2)$ by $(3)$, and taking account that $m_{\downarrow}=3~m_{\uparrow}$ | | Dividing $(2)$ by $(3)$, and taking account that $m_{\downarrow}=3~m_{\uparrow}$ |
| $$\frac{\rho_0~g~V+T}{\rho_0~g~\frac{V}{2}-T}=3$$ | | $$\frac{\rho_0~g~V+T}{\rho_0~g~\frac{V}{2}-T}=3$$ |
| and separating $T$ | | and separating $T$ |
| $$T = \frac{\rho_0~g~V}{8}$$ | | $$T = \frac{\rho_0~g~V}{8}$$ |
| Taking $\rho_0 = 1~\rm{g/cm^3} = 10^{-3}~\rm{kg/cm^3}$ (water's density) and $g=9.8~\rm{m/s^2}$ (acceleration due to gravity at Earth's surface), | | Taking $\rho_0 = 1~\rm{g/cm^3} = 10^{-3}~\rm{kg/cm^3}$ (water's density) and $g=9.8~\rm{m/s^2}$ (acceleration due to gravity at Earth's surface), |
| $$\boxed{T \simeq 1.2 \cdot 10^{-2}~\rm{N}}$$ | | $$\boxed{T \simeq 1.2 \cdot 10^{-2}~\rm{N}}$$ |
| </p> | | </p> |
| <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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