Updated spacing between @ latex expressions

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en/2.8.17.md +8 −8
@@ -6,14 +6,14 @@
<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="For one end of the rope covering the pole in an arc with an angle of \theta, pull with a force of F_0. What is the minimum force needed to be applied to the other end of the rope to hold it, if the coefficient of friction of the rope against the post is \mu?">
+ <meta name="description" content="For one end of the rope covering the pole in an arc with an angle of \theta , pull with a force of F_0. What is the minimum force needed to be applied to the other end of the rope to hold it, if the coefficient of friction of the rope against the post is \mu ?">
<meta name="author" content="Aliaksandr Melnichenka">
<meta name="date" content="2023-10" scheme="YYYY-MM">
− <meta property="og:title" content="For one end of the rope covering the pole in an arc with an angle of \theta, pull with a force of F_0. What is the minimum force needed to be applied to the other end of the rope to hold it, if the coefficient of friction of the rope against the post is \mu?">
+ <meta property="og:title" content="For one end of the rope covering the pole in an arc with an angle of \theta , pull with a force of F_0. What is the minimum force needed to be applied to the other end of the rope to hold it, if the coefficient of friction of the rope against the post is \mu ?">
<meta property="og:image" content="img/logo.png">
− <meta property="og:description" content="For one end of the rope covering the pole in an arc with an angle of \theta, pull with a force of F_0. What is the minimum force needed to be applied to the other end of the rope to hold it, if the coefficient of friction of the rope against the post is \mu?">
+ <meta property="og:description" content="For one end of the rope covering the pole in an arc with an angle of \theta , pull with a force of F_0. What is the minimum force needed to be applied to the other end of the rope to hold it, if the coefficient of friction of the rope against the post is \mu ?">
<meta name="yandex-verification" content="6cfda41f74038368">
− <title>For one end of the rope covering the pole in an arc with an angle of \theta, pull with a force of F_0. What is the minimum force needed to be applied to the other end of the rope to hold it, if the coefficient of friction of the rope against the post is \mu?</title>
+ <title>For one end of the rope covering the pole in an arc with an angle of \theta , pull with a force of F_0. What is the minimum force needed to be applied to the other end of the rope to hold it, if the coefficient of friction of the rope against the post is \mu ?</title>
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@@ -84,24 +84,24 @@
Approximation for small angles
$$\sin\left(\frac{d\theta}{2}\right)\approx\frac{d\theta}{2}$$
Let's neglect the second order of magnitude summand $(dF_0d\theta\to 0)$
−$$F_0\cdot d\theta=dN$$
+$$F_0\cdot d\theta =dN$$
Alternatively, the state of equilibrium for the horizontal axis
$$F_0\cos\left(\frac{d\theta}{2}\right)=(F_0+dF_0)\cos\left(\frac{d\theta}{2}\right)+d\vec{F}_\text{fr}$$
Approximation for small angles
$$\cos\left(\frac{d\theta}{2}\right)\approx1$$
$$dF_\text{fr}=-dF_0$$
$$dF_\text{fr}=\mu\, dN$$
$$-dF_0=\mu F_0d\theta\Leftrightarrow \boxed{-\frac{dF_0}{F_0}=\mu d\theta}$$
Integrating of both sides of equation
−$$-\int_{F_0}^{F}\frac{dF}{F}=\mu \int_0^\theta d\theta$$
+$$-\int_{F_0}^{F}\frac{dF}{F}=\mu\int_0^\theta d\theta$$
$$\ln \left|\frac{F_0}{F}\right|=\mu\theta$$
Substituting the minimum force $F$:
−$$\frac{F_0}{F}=e^{\mu \theta}\Rightarrow \boxed{F=F_0e^{-\mu \theta}}$$
+$$\frac{F_0}{F}=e^{\mu\theta}\Rightarrow \boxed{F=F_0e^{-\mu\theta}}$$
</p>
<h4>Answer</h4>
<p>
− $$F=F_0e^{-\mu \theta}$$
+ $$F=F_0e^{-\mu\theta}$$
</p>
<p style="text-align: right; font-style: italic; font-size: 14;">
Almaskhan Arsen<br>
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