| <!DOCTYPE html> | | <!DOCTYPE html> |
| <html lang="en"> | | <html lang="en"> |
| | | |
| <head> | | <head> |
| <meta charset="utf-8"> | | <meta charset="utf-8"> |
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | | <meta name="viewport" content="width=device-width, initial-scale=1.0"> |
| <meta http-equiv="content-language" content="en"> | | <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="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> |
| <meta name="description" content="The length of the string of a mathematical pendulum is l, the mass of the ball is m. Determine the force acting on the ball as it deviates from the equilibrium position by x in the case where x \ll l. How does the potential energy of the ball depend on x?"> | | <meta name="description" content="The length of the string of a mathematical pendulum is l, the mass of the ball is m. Determine the force acting on the ball as it deviates from the equilibrium position by x in the case where x \ll l. How does the potential energy of the ball depend on x?"> |
| <meta name="author" content="Aliaksandr Melnichenka"> | | <meta name="author" content="Aliaksandr Melnichenka"> |
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | | <meta name="date" content="2023-10" scheme="YYYY-MM"> |
| <meta property="og:title" content="The length of the string of a mathematical pendulum is l, the mass of the ball is m. Determine the force acting on the ball as it deviates from the equilibrium position by x in the case where x \ll l. How does the potential energy of the ball depend on x?"> | | <meta property="og:title" content="The length of the string of a mathematical pendulum is l, the mass of the ball is m. Determine the force acting on the ball as it deviates from the equilibrium position by x in the case where x \ll l. How does the potential energy of the ball depend on x?"> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="The length of the string of a mathematical pendulum is l, the mass of the ball is m. Determine the force acting on the ball as it deviates from the equilibrium position by x in the case where x \ll l. How does the potential energy of the ball depend on x?"> | | <meta property="og:description" content="The length of the string of a mathematical pendulum is l, the mass of the ball is m. Determine the force acting on the ball as it deviates from the equilibrium position by x in the case where x \ll l. How does the potential energy of the ball depend on x?"> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>The length of the string of a mathematical pendulum is l, the mass of the ball is m. Determine the force acting on the ball as it deviates from the equilibrium position by x in the case where x \ll l. How does the potential energy of the ball depend on x?</title> | | <title>The length of the string of a mathematical pendulum is l, the mass of the ball is m. Determine the force acting on the ball as it deviates from the equilibrium position by x in the case where x \ll l. How does the potential energy of the ball depend on x?</title> |
| <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> | | <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> |
| <link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png"> | | <link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png"> |
| <script src="https://savchenkosolutions.com/js/jquery-1.10.1.min.js"></script> | | <script src="https://savchenkosolutions.com/js/jquery-1.10.1.min.js"></script> |
| <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script> | | <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script> |
| <script type="text/x-mathjax-config"> | | <script type="text/x-mathjax-config"> |
| MathJax.Hub.Config({ | | MathJax.Hub.Config({ |
| extensions: ['tex2jax.js'], | | extensions: ['tex2jax.js'], |
| jax: ['input/TeX', 'output/HTML-CSS'], | | jax: ['input/TeX', 'output/HTML-CSS'], |
| tex2jax: { | | tex2jax: { |
| inlineMath: [['$', '$'], ['$', '$']], | | inlineMath: [['$', '$'], ['$', '$']], |
| processEscapes: true, | | processEscapes: true, |
| processClass: 'tex2jax', | | processClass: 'tex2jax', |
| ignoreClass: 'html' | | ignoreClass: 'html' |
| }, | | }, |
| showProcessingMessages: false, | | showProcessingMessages: false, |
| messageStyle: 'none' | | messageStyle: 'none' |
| }); | | }); |
| </script> | | </script> |
| </p> | | </p> |
| </header> | | </header> |
| | | |
| <h3 id="back-link"><a href="../../#3.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#3.1">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $3.1.5.$ The length of the string of a mathematical pendulum is $l$, the mass of the ball is $m$. Determine the force acting on the ball as it deviates from the equilibrium position by $x$ in the case where $x \ll l$. How does the potential energy of the ball depend on $x$? | | $3.1.5.$ The length of the string of a mathematical pendulum is $l$, the mass of the ball is $m$. Determine the force acting on the ball as it deviates from the equilibrium position by $x$ in the case where $x \ll l$. How does the potential energy of the ball depend on $x$? |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/3/3.1.5/statement.png" | | <img src="https://savchenkosolutions.com/3/3.1.5/statement.png" |
| loading="lazy" width="130" /> | | loading="lazy" width="130" /> |
| <figcaption> | | <figcaption> |
| For problem $3.1.5$ | | For problem $3.1.5$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| When the pendulum swings away, it is acted upon by a restoring force $F$ acting in the direction opposite to the direction of the velocity, which is restoring and equal to | | When the pendulum swings away, it is acted upon by a restoring force $F$ acting in the direction opposite to the direction of the velocity, which is restoring and equal to |
| $$ OX:F=-mg\sin\varphi=\fbox{$-mg\frac{x}{l}$} $$ | | $$ OX:F=-mg\sin\varphi=\fbox{$-mg\frac{x}{l}$} $$ |
| Considering that the oscillations are harmonic, their amplitude is not large, and the magnitude of the deviation is much less than the length of the thread: $(x\ll l)$. Let us use the expression for small angles $\cos\alpha\approx1-\frac{\alpha^2}{2};\quad\sin\alpha\approx\alpha$ | | Considering that the oscillations are harmonic, their amplitude is not large, and the magnitude of the deviation is much less than the length of the thread: $(x\ll l)$. Let us use the expression for small angles $\cos\alpha\approx1-\frac{\alpha^2}{2};\quad\sin\alpha\approx\alpha$ |
| $$ 1-\cos\alpha\approx\frac{x^2}{2l^2} $$ | | $$ 1-\cos\alpha\approx\frac{x^2}{2l^2} $$ |
| Changing the position of the ball vertically | | Changing the position of the ball vertically |
| $$ \Delta h = l \cdot (1-\cos\alpha)=\frac{x^2}{2l} $$ | | $$ \Delta h = l \cdot (1-\cos\alpha)=\frac{x^2}{2l} $$ |
| Where does the change in potential energy come from? | | Where does the change in potential energy come from? |
| $$ \Delta U = mg\cdot\Delta h = \boxed{mg\frac{x^2}{2l}} $$ | | $$ \Delta U = mg\cdot\Delta h = \boxed{mg\frac{x^2}{2l}} $$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Aliaksandr Melnichenka | | Aliaksandr Melnichenka |
| </p> | | </p> |
| <h3>Alternative solution</h3> | | <h3>Alternative solution</h3> |
| <p> | | <p> |
| Alternatively, one could write the change in energy as an integral of the work done on the interval $dx$: | | Alternatively, one could write the change in energy as an integral of the work done on the interval $dx$: |
| $$ U=\int_{0}^{U}dU=\int_{0}^{x}mg\frac{x}{l}dx=\boxed{\frac{mgx^2}{2l}} $$ | | $$ U=\int_{0}^{U}dU=\int_{0}^{x}mg\frac{x}{l}dx=\boxed{\frac{mgx^2}{2l}} $$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Dzikan Mikita<br> | | Dzikan Mikita<br> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$F=-mg\frac{x}{l};\quad U=\frac{mgx^2}{2l}$$ | | $$F=-mg\frac{x}{l};\quad U=\frac{mgx^2}{2l}$$ |
| </p> | | </p> |
| | | |
| | | |
| <footer class="row container"> | | <footer class="row container"> |
| <br> | | <br> |
| <p> | | <p> |
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> |
| </p> | | </p> |
| <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> | | <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> | | </p> |
| </footer> | | </footer> |
| </body> | | </body> |
| | | |
| </html> | | </html> |
