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.

Savchenko Solutions

Solutions of Savchenko Problems in Physics
knowledge must be free

    <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.

    <h3>Solution</h3>
    <p>

Similarly to 3.2.12, we find the external force $F$ as

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>