Translated 3.2.1-3.2.17
en/3.2.6.md
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| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
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| + | <meta property="og:description" content="A pendulum is a light and rigid rod of length l with a weight at its end. To make the period of oscillations of the pendulum large without excessive increase in the size of the pendulum itself, its axis is set at an angle \alpha to the vertical. Determine the period of oscillation."> | ||
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| + | <title>A pendulum is a light and rigid rod of length l with a weight at its end. To make the period of oscillations of the pendulum large without excessive increase in the size of the pendulum itself, its axis is set at an angle \alpha to the vertical. Determine the period of oscillation.</title> | ||
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| + | <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> | ||
| + | $3.2.6.$ A pendulum is a light and rigid rod of length $l$ with a weight at its end. To make the period of oscillations of the pendulum large without excessive increase in the size of the pendulum itself, its axis is set at an angle $\alpha$ to the vertical. Determine the period of oscillation. | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/3/3.2.6/3.2.6.png" | ||
| + | loading="lazy" width="200" /> | ||
| + | <figcaption> | ||
| + | For problem $3.2.6$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/3/3.2.6/3.2.6_1.png" | ||
| + | loading="lazy" width="260" /> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | |||
| + | We will split $\vec{g}$ into two components $\vec{g_x}$ and $\vec{g_y}$</p><p> | ||
| + | In this case, only $\vec{g_x}$ creates a moment of force, since $\vec{g_y}$ tries to break it out of the mount | ||
| + | $$ g_x=g\cos\alpha $$ | ||
| + | We substitute the acceleration into the formula for the period of oscillation of a mathematical pendulum | ||
| + | $$ \boxed{T=2\pi\sqrt{\frac{l}{g_x}}=2\pi\sqrt{\frac{l}{g\cos\alpha}}} $$ | ||
| + | </p> | ||
| + | |||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
| + | Aliaksandr Kanashenka<br> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$T=2\pi\sqrt{\frac{l}{g\cos\alpha}}$$ | ||
| + | </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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| <head> | |||
| <meta charset="utf-8"> | |||
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |||
| <meta http-equiv="content-language" content="en"> | |||
| <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | |||
| <meta name="description" content="A pendulum is a light and rigid rod of length l with a weight at its end. To make the period of oscillations of the pendulum large without excessive increase in the size of the pendulum itself, its axis is set at an angle \alpha to the vertical. Determine the period of oscillation."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="A pendulum is a light and rigid rod of length l with a weight at its end. To make the period of oscillations of the pendulum large without excessive increase in the size of the pendulum itself, its axis is set at an angle \alpha to the vertical. Determine the period of oscillation."> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="A pendulum is a light and rigid rod of length l with a weight at its end. To make the period of oscillations of the pendulum large without excessive increase in the size of the pendulum itself, its axis is set at an angle \alpha to the vertical. Determine the period of oscillation."> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>A pendulum is a light and rigid rod of length l with a weight at its end. To make the period of oscillations of the pendulum large without excessive increase in the size of the pendulum itself, its axis is set at an angle \alpha to the vertical. Determine the period of oscillation.</title> | |||
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| messageStyle: 'none' | |||
| }); | |||
| </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> | |||
| $3.2.6.$ A pendulum is a light and rigid rod of length $l$ with a weight at its end. To make the period of oscillations of the pendulum large without excessive increase in the size of the pendulum itself, its axis is set at an angle $\alpha$ to the vertical. Determine the period of oscillation. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/3/3.2.6/3.2.6.png" | |||
| loading="lazy" width="200" /> | |||
| <figcaption> | |||
| For problem $3.2.6$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/3/3.2.6/3.2.6_1.png" | |||
| loading="lazy" width="260" /> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| We will split $\vec{g}$ into two components $\vec{g_x}$ and $\vec{g_y}$</p><p> | |||
| In this case, only $\vec{g_x}$ creates a moment of force, since $\vec{g_y}$ tries to break it out of the mount | |||
| $$ g_x=g\cos\alpha $$ | |||
| We substitute the acceleration into the formula for the period of oscillation of a mathematical pendulum | |||
| $$ \boxed{T=2\pi\sqrt{\frac{l}{g_x}}=2\pi\sqrt{\frac{l}{g\cos\alpha}}} $$ | |||
| </p> | |||
| <p style="text-align: right; font-style: italic; font-size: 14;"> | |||
| Aliaksandr Kanashenka<br> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$T=2\pi\sqrt{\frac{l}{g\cos\alpha}}$$ | |||
| </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> | |||