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| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> |
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| Solutions of Savchenko Problems in Physics <br> | | Solutions of Savchenko Problems in Physics <br> |
| <i><b>knowledge must be free</b></i> | | <i><b>knowledge must be free</b></i> |
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| <h3 id="back-link"><a href="../../#3.2">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#3.2">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $3.2.17.$ A spacecraft rotates around its axis with angular velocity $\omega$. How does the period of oscillation of a pendulum of length $l$ depend on the distance $R$ of the suspension point to the axis of rotation? The plane of oscillation passes through the axis of rotation. | | $3.2.17.$ A spacecraft rotates around its axis with angular velocity $\omega$. How does the period of oscillation of a pendulum of length $l$ depend on the distance $R$ of the suspension point to the axis of rotation? The plane of oscillation passes through the axis of rotation. |
| </p> | | </p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
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| The pendulum deflected a small distance $x$ | | The pendulum deflected a small distance $x$ |
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| When rotating along an arc of a circle of radius $l$, the ball is subject to centripetal acceleration | | When rotating along an arc of a circle of radius $l$, the ball is subject to centripetal acceleration |
| $$ a = \omega^2(R+l) $$ | | $$ a = \omega^2(R+l) $$ |
| Newton's second law for a ball: | | Newton's second law for a ball: |
| $$ m\ddot{x}(t)=-m\omega^2(R+l)\sin\varphi $$ | | $$ m\ddot{x}(t)=-m\omega^2(R+l)\sin\varphi $$ |
| Neglecting gravitational interaction with the Earth, we write Newton's second law | | Neglecting gravitational interaction with the Earth, we write Newton's second law |
| $$ m\ddot{x}(t)+m\omega^2(R+l)\sin\varphi=0\quad(1) $$ | | $$ m\ddot{x}(t)+m\omega^2(R+l)\sin\varphi=0\quad(1) $$ |
| Let's use the approximation for small angles $(\varphi \ll 1)$: | | Let's use the approximation for small angles $(\varphi \ll 1)$: |
