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en/2.8.17.md
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| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
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| + | <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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| + | <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="../../#2.8">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $2.8.17^*.$ 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 $\mu$? | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="2.8.17.png" | ||
| + | loading="lazy" width="200" /> | ||
| + | <figcaption> | ||
| + | For problem $2.8.17^*$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="2.8.17_1.png" | ||
| + | loading="lazy" width="300" /> | ||
| + | <figcaption> | ||
| + | For problem $2.8.17^*$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | |||
| + | State of equilibrium for vertical axis | ||
| + | $$(F_0+F_0+dF_0)\cdot\sin\left(\frac{d\theta}{2}\right)=dN$$ | ||
| + | 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 $(dF_0d\theta\to 0)$ | ||
| + | $$F_0\cdot d\theta=dN$$ | ||
| + | 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 | ||
| + | $$-\int_{F_0}^{F}\frac{dF}{F}=\mu \int_0^\theta d\theta$$ | ||
| + | $$\ln \left|\frac{F_0}{F}\right|=\mu\theta$$ | ||
| + | Substituting the minimum force $F$: | ||
| + | $$\frac{F_0}{F}=e^{\mu \theta}\Rightarrow \boxed{F=F_0e^{-\mu \theta}}$$ | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$F=F_0e^{-\mu \theta}$$ | ||
| + | </p> | ||
| + | |||
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| <meta charset="utf-8"> | |||
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |||
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| <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | |||
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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="../../#2.8">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $2.8.17^*.$ 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 $\mu$? | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="2.8.17.png" | |||
| loading="lazy" width="200" /> | |||
| <figcaption> | |||
| For problem $2.8.17^*$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="2.8.17_1.png" | |||
| loading="lazy" width="300" /> | |||
| <figcaption> | |||
| For problem $2.8.17^*$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| State of equilibrium for vertical axis | |||
| $$(F_0+F_0+dF_0)\cdot\sin\left(\frac{d\theta}{2}\right)=dN$$ | |||
| 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 $(dF_0d\theta\to 0)$ | |||
| $$F_0\cdot d\theta=dN$$ | |||
| 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 | |||
| $$-\int_{F_0}^{F}\frac{dF}{F}=\mu \int_0^\theta d\theta$$ | |||
| $$\ln \left|\frac{F_0}{F}\right|=\mu\theta$$ | |||
| Substituting the minimum force $F$: | |||
| $$\frac{F_0}{F}=e^{\mu \theta}\Rightarrow \boxed{F=F_0e^{-\mu \theta}}$$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$F=F_0e^{-\mu \theta}$$ | |||
| </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> | |||
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