Added 2.7.44

astrosander edited
View solution History
revision #8792 GitHub e1a1af2 newer →
en/2.7.44.md +112 −0
@@ -0,0 +1,112 @@
+<!DOCTYPE html>
+<html lang="en">
+
+<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="The moment of inertia of a solid body of mass m relative to the O-axis is equal to J. The center of mass of the body is located at a distance R from this axis. Find the force acting on the axis when a force F is briefly applied to a solid body perpendicular to the segment of length x that connects the point of application of the force and the axis. At which x is the smallest force acting on the axis?">
+ <meta name="author" content="Aliaksandr Melnichenka">
+ <meta name="date" content="2023-10" scheme="YYYY-MM">
+ <meta property="og:title" content="The moment of inertia of a solid body of mass m relative to the O-axis is equal to J. The center of mass of the body is located at a distance R from this axis. Find the force acting on the axis when a force F is briefly applied to a solid body perpendicular to the segment of length x that connects the point of application of the force and the axis. At which x is the smallest force acting on the axis?">
+ <meta property="og:image" content="img/logo.png">
+ <meta property="og:description" content="The moment of inertia of a solid body of mass m relative to the O-axis is equal to J. The center of mass of the body is located at a distance R from this axis. Find the force acting on the axis when a force F is briefly applied to a solid body perpendicular to the segment of length x that connects the point of application of the force and the axis. At which x is the smallest force acting on the axis?">
+ <meta name="yandex-verification" content="6cfda41f74038368">
+ <title>The moment of inertia of a solid body of mass m relative to the O-axis is equal to J. The center of mass of the body is located at a distance R from this axis. Find the force acting on the axis when a force F is briefly applied to a solid body perpendicular to the segment of length x that connects the point of application of the force and the axis. At which x is the smallest force acting on the axis?</title>
+ <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">
+ <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 type="text/x-mathjax-config">
+ MathJax.Hub.Config({
+ extensions: ['tex2jax.js'],
+ jax: ['input/TeX', 'output/HTML-CSS'],
+ tex2jax: {
+ inlineMath: [['$', '$'], ['$', '$']],
+ processEscapes: true,
+ processClass: 'tex2jax',
+ ignoreClass: 'html'
+ },
+ 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="../../#2.7">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <p>
+ $2.7.44.$ The moment of inertia of a solid body of mass $m$ relative to the $O$-axis is equal to $J$. The center of mass of the body is located at a distance $R$ from this axis. Find the force acting on the axis when a force $F$ is briefly applied to a solid body perpendicular to the segment of length $x$ that connects the point of application of the force and the axis. At which $x$ is the smallest force acting on the axis?
+</p>
+<center>
+ <figure>
+ <img src="2.7.44.png"
+ loading="lazy" width="130" />
+ <figcaption>
+ For problem $2.7.44$
+ </figcaption>
+ </figure>
+</center>
+<p>
+ </p>
+
+ <h3>Solution</h3>
+ <p>
+
+</p>
+<center>
+ <figure>
+ <img src="2.7.44_1.png"
+ loading="lazy" width="130" />
+ <figcaption>
+ Force $\vec{F}_0$ that will produce the rotational torque
+ </figcaption>
+ </figure>
+</center>
+<p>
+Force $\vec{F}$ creates an angular momentum $\vec{F}\cdot x$ (with $x$ perpendicular to $\vec{F}$):
+$$\tau_0=F\cdot x$$
+$$\tau_f=J\cdot\varepsilon$$
+In steady-state mode, the angular moments will be equal
+$$\tau_0=\tau_f=Fx=J\varepsilon$$
+From where
+$$\varepsilon = \frac{Fx}{J}$$
+Newton's second law for the rotational motion
+$$F+F_0=m\varepsilon R$$
+$$F_0=mR\cdot \frac{Fx}{J}-F\Leftrightarrow \boxed{F_0=F\left(\frac{mRx}{J}-1\right)}$$
+From here, the minimum values of $F_0 = 0$, described by the expression
+$$F_0=0\Leftrightarrow \frac{mRx}{J}-1=0$$
+which will be observed at length that connects the point of application of the force and the axis
+$$\boxed{x=\frac{J}{mR}}$$
+ </p>
+
+ <h4>Answer</h4>
+ <p>
+ $F'=F(mRx/J-1)$. With $x=J/(mR),~F'=0$
+ </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>