Translated 3.2.1-3.2.17
en/3.2.17.md
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| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="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."> | ||
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| + | <meta property="og:description" content="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."> | ||
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| + | <title>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.</title> | ||
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| + | <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> | ||
| + | $3.2.17.$ 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> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/3/3.2.17/3.2.17_1.png" | ||
| + | loading="lazy" width="60" /> | ||
| + | <figcaption> | ||
| + | The pendulum deflected a small distance $x$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | When rotating along an arc of a circle of radius $l$, the ball is subject to centripetal acceleration | ||
| + | $$ a = \omega^2(R+l) $$ | ||
| + | Newton's second law for a ball: | ||
| + | $$ m\ddot{x}(t)=-m\omega^2(R+l)\sin\varphi $$ | ||
| + | Neglecting gravitational interaction with the Earth, we write Newton's second law | ||
| + | $$ m\ddot{x}(t)+m\omega^2(R+l)\sin\varphi=0\quad(1) $$ | ||
| + | Let's use the approximation for small angles $(\varphi \ll 1)$: | ||
| + | $$ \sin\varphi\approx\varphi=\frac{x}{l} $$ | ||
| + | Using the approximation for $(1)$, we obtain the equation of harmonic oscillations | ||
| + | $$ \ddot{x}(t)+\frac{\omega^2(R+l)}{l}x(t)=0 $$ | ||
| + | We solve the equation of harmonic oscillations of the form $\ddot{x}+\omega ^2x(t)=0$ using the standard method and obtain the desired oscillation period | ||
| + | $$ \boxed{T=\frac{2\pi}{\omega}\sqrt{\frac{l}{R+l}}} $$ | ||
| + | </p> | ||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
| + | Dzikan Mikita<br> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$T=\frac{2\pi}{\omega}\sqrt{\frac{l}{R+l}}$$ | ||
| + | </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> | ||
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| @@ -0,0 +1,102 @@ | |||
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| <meta charset="utf-8"> | |||
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |||
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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 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."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="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."> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="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."> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>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.</title> | |||
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| }); | |||
| </script> | |||
| </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> | |||
| $3.2.17.$ 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> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/3/3.2.17/3.2.17_1.png" | |||
| loading="lazy" width="60" /> | |||
| <figcaption> | |||
| The pendulum deflected a small distance $x$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| When rotating along an arc of a circle of radius $l$, the ball is subject to centripetal acceleration | |||
| $$ a = \omega^2(R+l) $$ | |||
| Newton's second law for a ball: | |||
| $$ m\ddot{x}(t)=-m\omega^2(R+l)\sin\varphi $$ | |||
| Neglecting gravitational interaction with the Earth, we write Newton's second law | |||
| $$ m\ddot{x}(t)+m\omega^2(R+l)\sin\varphi=0\quad(1) $$ | |||
| Let's use the approximation for small angles $(\varphi \ll 1)$: | |||
| $$ \sin\varphi\approx\varphi=\frac{x}{l} $$ | |||
| Using the approximation for $(1)$, we obtain the equation of harmonic oscillations | |||
| $$ \ddot{x}(t)+\frac{\omega^2(R+l)}{l}x(t)=0 $$ | |||
| We solve the equation of harmonic oscillations of the form $\ddot{x}+\omega ^2x(t)=0$ using the standard method and obtain the desired oscillation period | |||
| $$ \boxed{T=\frac{2\pi}{\omega}\sqrt{\frac{l}{R+l}}} $$ | |||
| </p> | |||
| <p style="text-align: right; font-style: italic; font-size: 14;"> | |||
| Dzikan Mikita<br> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$T=\frac{2\pi}{\omega}\sqrt{\frac{l}{R+l}}$$ | |||
| </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> | |||