| <!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="Find the period of small vibrations of a body of mass m, whose charge is q, inside a smooth sphere of radius R, if the charge Q is fixed at the top point of the sphere."> | | <meta name="description" content="Find the period of small vibrations of a body of mass m, whose charge is q, inside a smooth sphere of radius R, if the charge Q is fixed at the top point of the sphere."> |
| <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="Find the period of small vibrations of a body of mass m, whose charge is q, inside a smooth sphere of radius R, if the charge Q is fixed at the top point of the sphere."> | | <meta property="og:title" content="Find the period of small vibrations of a body of mass m, whose charge is q, inside a smooth sphere of radius R, if the charge Q is fixed at the top point of the sphere."> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="Find the period of small vibrations of a body of mass m, whose charge is q, inside a smooth sphere of radius R, if the charge Q is fixed at the top point of the sphere."> | | <meta property="og:description" content="Find the period of small vibrations of a body of mass m, whose charge is q, inside a smooth sphere of radius R, if the charge Q is fixed at the top point of the sphere."> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>Find the period of small vibrations of a body of mass m, whose charge is q, inside a smooth sphere of radius R, if the charge Q is fixed at the top point of the sphere.</title> | | <title>Find the period of small vibrations of a body of mass m, whose charge is q, inside a smooth sphere of radius R, if the charge Q is fixed at the top point of the sphere.</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> |
| </head> | | </head> |
| <body style=""> | | <body style=""> |
| <header style="text-align:center;"> | | <header style="text-align:center;"> |
| <a href="../../" style="text-decoration: none;"> | | <a href="../../" style="text-decoration: none;"> |
| <div id="logo"> | | <div id="logo"> |
| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> |
| </div> | | </div> |
| </a> | | </a> |
| <p class="author"> | | <p class="author"> |
| 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> |
| </p> | | </p> |
| </header> | | </header> |
| | | |
| <h3 id="back-link"><a href="../../#7.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#7.1">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $7.1.23^*.$ Find the period of small vibrations of a body of mass $m$, whose charge is $q$, inside a smooth sphere of radius $R$, if the charge $Q$ is fixed at the top point of the sphere. | | $7.1.23^*.$ Find the period of small vibrations of a body of mass $m$, whose charge is $q$, inside a smooth sphere of radius $R$, if the charge $Q$ is fixed at the top point of the sphere. |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="7.1.23.png" | | <img src="7.1.23.png" |
| loading="lazy" width="200" /> | | loading="lazy" width="200" /> |
| <figcaption> | | <figcaption> |
| For problem $7.1.23^*$ | | For problem $7.1.23^*$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| | | |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="7.1.23_1.png" | | <img src="7.1.23_1.png" |
| loading="lazy" width="200" /> | | loading="lazy" width="200" /> |
| <figcaption> | | <figcaption> |
| Forces acting on the system of two charges | | Forces acting on the system of two charges |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| | | |
| Considering the fact that oscillations are small, $\varphi \ll 1$, we could use the approximations for small angle $\varphi$ | | Considering the fact that oscillations are small, $\varphi \ll 1$, we could use the approximations for small angle $\varphi$ |
| $$\sin\varphi \approx \varphi; \quad\cos\varphi \approx 1\quad(1)$$ | | $$\sin\varphi \approx \varphi; \quad\cos\varphi \approx 1\quad(1)$$ |
| From where, we could approximate that the distance between charges remains approximately the same | | From where, we could approximate that the distance between charges remains approximately the same |
| $$r\approx2R$$ | | $$r\approx2R$$ |
| Thus, the сoulomb's law for two charges | | Thus, the сoulomb's law for two charges |
| $$F_c = \frac{1}{4\pi\varepsilon_0}\frac{qQ}{4R^2}$$ | | $$F_c = \frac{1}{4\pi\varepsilon_0}\frac{qQ}{4R^2}$$ |
| Let's write Newton's second law on a tangential trajectory, in the direction of motion of a charge of mass $m$ | | Let's write Newton's second law on a tangential trajectory, in the direction of motion of a charge of mass $m$ |
| $$ma = mg \sin\varphi + F_c \sin\frac{\varphi}{2}$$ | | $$ma = mg \sin\varphi + F_c \sin\frac{\varphi}{2}$$ |
| Considering the approximation $(1)$ and the expression | | Considering the approximation $(1)$ and the expression |
| $$ma = -mg\frac{x}{R} - \frac{x}{2R} \frac{1}{4\pi\varepsilon_0}\frac{qQ}{4R^2}$$ | | $$ma = -mg\frac{x}{R} - \frac{x}{2R} \frac{1}{4\pi\varepsilon_0}\frac{qQ}{4R^2}$$ |
| $$a = -x\left(\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}\right)$$ | | $$a = -x\left(\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}\right)$$ |
| Solving the equation of harmonic oscillations $(\ddot{x}+\omega^2x(t)=0)$, we could obtain the value for angular velocity | | Solving the equation of harmonic oscillations $(\ddot{x}+\omega^2x(t)=0)$, we could obtain the value for angular velocity |
| $$\omega=\sqrt{\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}}$$ | | $$\omega=\sqrt{\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}}$$ |
| From here the period of oscillation | | From here the period of oscillation |
| $$\boxed{T=\frac{2\pi}{\omega} = \frac{2\pi}{\sqrt{\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}}}}$$ | | $$\boxed{T=\frac{2\pi}{\omega} = \frac{2\pi}{\sqrt{\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3}}}}$$ |
| Since the expression under the square root cannot be less than zero, the ratio at which there will be no oscillations | | Since the expression under the square root cannot be less than zero, the ratio at which there will be no oscillations |
| $$\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3} > 0 \Leftrightarrow \boxed{\frac{qQ}{32\pi\varepsilon_0R^2}>-mg}$$ | | $$\frac{g}{R}+\frac{qQ}{32\pi\varepsilon_0mR^3} > 0 \Leftrightarrow \boxed{\frac{qQ}{32\pi\varepsilon_0R^2}>-mg}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <div class="scroll-wrapper"> | | <div class="scroll-wrapper"> |
| <p> | | <p> |
