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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="../../#2.2">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.2">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.2.23.$ To create artificial gravity, two compartments of the orbital station (mass ratio $1 : 2$) were separated by a distance $R$ from each other and spun around their common center of mass. Determine the time of complete rotation of the compartments if, in a more massive compartment, the artificial gravity is half the force of gravity on the Ground. | | $2.2.23.$ To create artificial gravity, two compartments of the orbital station (mass ratio $1 : 2$) were separated by a distance $R$ from each other and spun around their common center of mass. Determine the time of complete rotation of the compartments if, in a more massive compartment, the artificial gravity is half the force of gravity on the Ground. |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| | | |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.2.23_1.png" | | <img src="2.2.23_1.png" |
| loading="lazy" width="230" /> | | loading="lazy" width="230" /> |
| <figcaption> | | <figcaption> |
| Velocities of the system | | Velocities of the system |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| | | |
| $$\sum \vec{R}_\text{external} = \vec{0}$$ | | $$\sum \vec{R}_\text{external} = \vec{0}$$ |
| $F_3$ - The force of gravity on the ground acting on an object of mass $m$<br> | | $F_3$ - The force of gravity on the ground acting on an object of mass $m$<br> |
| $F_2$ - The force of gravity on a massive compartment<br> | | $F_2$ - The force of gravity on a massive compartment<br> |
| $F_1$ - The force of gravity on the less massive compartment <br> | | $F_1$ - The force of gravity on the less massive compartment <br> |
| From the statement | | From the statement |
| $$\frac{F_3}{F_2} = 2; \frac{m_xg}{m_xg_2}=2 \Rightarrow g_2 = \frac{g}{2}$$ | | $$\frac{F_3}{F_2} = 2; \frac{m_xg}{m_xg_2}=2 \Rightarrow g_2 = \frac{g}{2}$$ |
| $$\frac{m_1}{m_2} = \frac{1}{2}$$ | | $$\frac{m_1}{m_2} = \frac{1}{2}$$ |
| $$\frac{F_1}{F_2} = \frac{m_1g_1}{m_2g_2} = \frac{g_1}{2g_2} = \frac{g_1}{g}$$ | | $$\frac{F_1}{F_2} = \frac{m_1g_1}{m_2g_2} = \frac{g_1}{2g_2} = \frac{g_1}{g}$$ |
| Since centrifugal forces $F_1$ and $F_2$ are internal forces, they are equal to | | Since centrifugal forces $F_1$ and $F_2$ are internal forces, they are equal to |
| $$F_1 = F_2 \Rightarrow g_1 = g$$ | | $$F_1 = F_2 \Rightarrow g_1 = g$$ |
| Centripetal acceleration could be found as | | Centripetal acceleration could be found as |
| $$g_1 = \frac{v_1^2}{x_1}; \quad g_2 = \frac{v_2^2}{x_2}$$ | | $$g_1 = \frac{v_1^2}{x_1}; \quad g_2 = \frac{v_2^2}{x_2}$$ |
| From where | | From where |
| $$x_1 = \frac{2mR}{3m}=\frac{2}{3}R; \quad x_2 = R-x_1 = \frac{1}{3}R$$ | | $$x_1 = \frac{2mR}{3m}=\frac{2}{3}R; \quad x_2 = R-x_1 = \frac{1}{3}R$$ |
| Alternatively | | Alternatively |
| $$v_1 = \sqrt{\frac{2gR}{3}}; \quad v_2 = \sqrt{\frac{gR}{6}}$$ | | $$v_1 = \sqrt{\frac{2gR}{3}}; \quad v_2 = \sqrt{\frac{gR}{6}}$$ |
| Then we find the rotation period as the ratio of the trajectory circle length to the velocity of the first body | | Then we find the rotation period as the ratio of the trajectory circle length to the velocity of the first body |
| $$ T = \frac{2\pi x_1}{v_1} $$ | | $$ T = \frac{2\pi x_1}{v_1} $$ |
| $$ T = 2\pi \cdot \frac{2}{3}R \cdot \sqrt{\frac{3}{2gR}}$$ | | $$ T = 2\pi \cdot \frac{2}{3}R \cdot \sqrt{\frac{3}{2gR}}$$ |
| $$\boxed{T=2\pi\sqrt{\frac{2R}{3g}}}$$ | | $$\boxed{T=2\pi\sqrt{\frac{2R}{3g}}}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
