Translated 3.2.1-3.2.17
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| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
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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> | ||
| + | $3.2.16.$ A heavy cart rolls with acceleration $a$ on an inclined plane forming an angle $\alpha$ with the horizon. Find the period of oscillation of a pendulum of length $l$ mounted on the cart. | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/3/3.2.16/3.2.16.png" | ||
| + | loading="lazy" width="180" /> | ||
| + | <figcaption> | ||
| + | For problem $3.2.16$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/3/3.2.16/3.2.16_1.png" | ||
| + | loading="lazy" width="210" /> | ||
| + | <figcaption> | ||
| + | Acceleration vector $\vec{g}^*$ in an inertial frame of reference | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | The period of oscillation of a pendulum on an inclined cart can be found using the well-known formula for the period of oscillation of a mathematical pendulum, by switching to the inertial frame of reference of the cart | ||
| + | $$ T=2\pi\sqrt{\frac{l}{g^*}}\quad(1) $$ | ||
| + | Where $\vec{g}^*$ is the acceleration of the inertial reference frame | ||
| + | $$ \vec{g}^*=\vec{g}-\vec{a} $$ | ||
| + | Using the cosine law, we find the value of the modulus of the vector $\vec{g}^*$ | ||
| + | $$ (g^{*})^2=a^2+g^2-2agcos\alpha $$ | ||
| + | Where | ||
| + | $$ \boxed{g^*=\sqrt{a^2+g^2-2agcos\alpha}} $$ | ||
| + | We substitute $(1)$ into the expression and find the period of oscillation of the pendulum | ||
| + | $$ \boxed{T=2\pi\sqrt{\frac{l}{\sqrt{a^2+g^2-2agcos\alpha}}}} $$ | ||
| + | </p> | ||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
| + | Dzikan Mikita<br> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$T=2\pi\sqrt{l/{\sqrt{a^2+g^2-2agcos\alpha}}}$$ | ||
| + | </p> | ||
| + | |||
| + | |||
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| + | <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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| <html lang="en"> | |||
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| <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 heavy cart rolls with acceleration a on an inclined plane forming an angle \alpha with the horizon. Find the period of oscillation of a pendulum of length l mounted on the cart."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="A heavy cart rolls with acceleration a on an inclined plane forming an angle \alpha with the horizon. Find the period of oscillation of a pendulum of length l mounted on the cart."> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="A heavy cart rolls with acceleration a on an inclined plane forming an angle \alpha with the horizon. Find the period of oscillation of a pendulum of length l mounted on the cart."> | |||
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| <title>A heavy cart rolls with acceleration a on an inclined plane forming an angle \alpha with the horizon. Find the period of oscillation of a pendulum of length l mounted on the cart.</title> | |||
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| showProcessingMessages: false, | |||
| 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> | |||
| $3.2.16.$ A heavy cart rolls with acceleration $a$ on an inclined plane forming an angle $\alpha$ with the horizon. Find the period of oscillation of a pendulum of length $l$ mounted on the cart. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/3/3.2.16/3.2.16.png" | |||
| loading="lazy" width="180" /> | |||
| <figcaption> | |||
| For problem $3.2.16$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/3/3.2.16/3.2.16_1.png" | |||
| loading="lazy" width="210" /> | |||
| <figcaption> | |||
| Acceleration vector $\vec{g}^*$ in an inertial frame of reference | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| The period of oscillation of a pendulum on an inclined cart can be found using the well-known formula for the period of oscillation of a mathematical pendulum, by switching to the inertial frame of reference of the cart | |||
| $$ T=2\pi\sqrt{\frac{l}{g^*}}\quad(1) $$ | |||
| Where $\vec{g}^*$ is the acceleration of the inertial reference frame | |||
| $$ \vec{g}^*=\vec{g}-\vec{a} $$ | |||
| Using the cosine law, we find the value of the modulus of the vector $\vec{g}^*$ | |||
| $$ (g^{*})^2=a^2+g^2-2agcos\alpha $$ | |||
| Where | |||
| $$ \boxed{g^*=\sqrt{a^2+g^2-2agcos\alpha}} $$ | |||
| We substitute $(1)$ into the expression and find the period of oscillation of the pendulum | |||
| $$ \boxed{T=2\pi\sqrt{\frac{l}{\sqrt{a^2+g^2-2agcos\alpha}}}} $$ | |||
| </p> | |||
| <p style="text-align: right; font-style: italic; font-size: 14;"> | |||
| Dzikan Mikita<br> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$T=2\pi\sqrt{l/{\sqrt{a^2+g^2-2agcos\alpha}}}$$ | |||
| </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> | |||