Translated 3.2.1-3.2.17
en/3.2.13.md
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| + | <title>A straight tunnel is dug in the Earth that does not pass through its center. Determine the time of movement of a train with engines turned off through such a tunnel if the influence of the Earth's rotation on the movement of the train and friction are neglected.</title> | ||
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| + | <div id = "logo"> | ||
| + | <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | ||
| + | </div> | ||
| + | <p class="author"> | ||
| + | Solutions of Savchenko Problems in Physics <br> | ||
| + | <i><b>knowledge must be free</b></i> | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#3.2">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $3.2.13.$ A straight tunnel is dug in the Earth that does not pass through its center. Determine the time of movement of a train with engines turned off through such a tunnel if the influence of the Earth's rotation on the movement of the train and friction are neglected. | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/3/3.2.13/3.2.13.png" | ||
| + | loading="lazy" width="130" /> | ||
| + | <figcaption> | ||
| + | For problem $3.2.13$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/3/3.2.13/3.2.13_1.png" | ||
| + | loading="lazy" width="230" /> | ||
| + | <figcaption> | ||
| + | For problem $3.2.13$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | |||
| + | Similarly to <a href="../3.2.12">3.2.12</a>, we find the external force $F$ as</p><p> | ||
| + | |||
| + | The body, at a distance $x$ from the core, will be affected by the gravitational force of attraction caused by the inner layers of the planet with density $\rho$, forming a sphere of radius $x$. The mass of this part of the earth | ||
| + | $$ M_\oplus = \frac{4}{3} \rho\pi x^3 $$ | ||
| + | Gravitational force acting on a rock at depth $x$ | ||
| + | $$ F_G = \frac{GmM_\oplus}{x^2}=mg\frac{x}{R} $$ | ||
| + | In this case, only the horizontal component of this force will create a moment. Whence, the resulting external force is equal to | ||
| + | $$ F=mg\cos\alpha=mg\frac{x}{R} $$ | ||
| + | From here we find the angular frequency of oscillations | ||
| + | $$ \omega=\sqrt{\frac{g}{R}} $$ | ||
| + | The period of oscillation in a given system | ||
| + | $$ T=2\pi\sqrt{\frac{R}{g}} $$ | ||
| + | Since we are interested in the flight time only in one direction, we take half of this period. | ||
| + | $$ \boxed{t=\frac{T}{2}=\pi\sqrt{\frac{R}{g}}\approx42\text{ min}} $$ | ||
| + | |||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
| + | Dzikan Mikita<br> | ||
| + | Aliaksandr Kanashenka<br> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$t=42\text{ min}$$ | ||
| + | </p> | ||
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| + | <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small> | ||
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| <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | |||
| <meta name="description" content="A straight tunnel is dug in the Earth that does not pass through its center. Determine the time of movement of a train with engines turned off through such a tunnel if the influence of the Earth's rotation on the movement of the train and friction are neglected."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="A straight tunnel is dug in the Earth that does not pass through its center. Determine the time of movement of a train with engines turned off through such a tunnel if the influence of the Earth's rotation on the movement of the train and friction are neglected."> | |||
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| </head> | |||
| <body style=""> | |||
| <header style="text-align:center;"> | |||
| <div id = "logo"> | |||
| <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | |||
| </div> | |||
| <p class="author"> | |||
| Solutions of Savchenko Problems in Physics <br> | |||
| <i><b>knowledge must be free</b></i> | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#3.2">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $3.2.13.$ A straight tunnel is dug in the Earth that does not pass through its center. Determine the time of movement of a train with engines turned off through such a tunnel if the influence of the Earth's rotation on the movement of the train and friction are neglected. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/3/3.2.13/3.2.13.png" | |||
| loading="lazy" width="130" /> | |||
| <figcaption> | |||
| For problem $3.2.13$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/3/3.2.13/3.2.13_1.png" | |||
| loading="lazy" width="230" /> | |||
| <figcaption> | |||
| For problem $3.2.13$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| Similarly to <a href="../3.2.12">3.2.12</a>, we find the external force $F$ as</p><p> | |||
| The body, at a distance $x$ from the core, will be affected by the gravitational force of attraction caused by the inner layers of the planet with density $\rho$, forming a sphere of radius $x$. The mass of this part of the earth | |||
| $$ M_\oplus = \frac{4}{3} \rho\pi x^3 $$ | |||
| Gravitational force acting on a rock at depth $x$ | |||
| $$ F_G = \frac{GmM_\oplus}{x^2}=mg\frac{x}{R} $$ | |||
| In this case, only the horizontal component of this force will create a moment. Whence, the resulting external force is equal to | |||
| $$ F=mg\cos\alpha=mg\frac{x}{R} $$ | |||
| From here we find the angular frequency of oscillations | |||
| $$ \omega=\sqrt{\frac{g}{R}} $$ | |||
| The period of oscillation in a given system | |||
| $$ T=2\pi\sqrt{\frac{R}{g}} $$ | |||
| Since we are interested in the flight time only in one direction, we take half of this period. | |||
| $$ \boxed{t=\frac{T}{2}=\pi\sqrt{\frac{R}{g}}\approx42\text{ min}} $$ | |||
| <p style="text-align: right; font-style: italic; font-size: 14;"> | |||
| Dzikan Mikita<br> | |||
| Aliaksandr Kanashenka<br> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$t=42\text{ min}$$ | |||
| </p> | |||
| <footer class="row container"> | |||
| <br> | |||
| <p> | |||
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | |||
| </p> | |||
| <p> | |||
| <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||