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| + | <meta name="description" content="At what angle to the horizon will the surface of the liquid in the vessel slide on an inclined plane that makes an angle \alpha to the horizon if the friction coefficient is \mu?"> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="At what angle to the horizon will the surface of the liquid in the vessel slide on an inclined plane that makes an angle \alpha to the horizon if the friction coefficient is \mu?"> | ||
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| + | <title>At what angle to the horizon will the surface of the liquid in the vessel slide on an inclined plane that makes an angle \alpha to the horizon if the friction coefficient is \mu?</title> | ||
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| + | <body style=""> | ||
| + | <header style="text-align:center;"> | ||
| + | <a href="../../" style="text-decoration: none;"> | ||
| + | <div id="logo"> | ||
| + | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | ||
| + | </div> | ||
| + | </a> | ||
| + | <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="../../#4.1">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $4.1.22.$ At what angle to the horizon will the surface of the liquid in the vessel slide on an inclined plane that makes an angle $\alpha$ to the horizon if the friction coefficient is $\mu$? | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="4.1.22_1.png" | ||
| + | loading="lazy" width="300" /> | ||
| + | <figcaption> | ||
| + | Forces acting on the system | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | |||
| + | The vessel with liquid goes downward with acceleration $a$ as shown in the figure</p><p> | ||
| + | Given the reaction force $N=mg\cos\alpha$, the sliding friction force is found as | ||
| + | $$F_\text{fr} = \mu N = \mu mg\cos\alpha$$ | ||
| + | Newton's second law for the vessel | ||
| + | $$ma = mg\sin\alpha-F_\text{fr}$$ | ||
| + | $$\boxed{a=g(\sin\alpha-\mu\cos\alpha)}\quad(1)$$ | ||
| + | Let us introduce the effective acceleration of free fall $\vec{g^*}$ in a non-inertial frame of reference associated with the vessel, then vectorially, it is found as | ||
| + | $$\vec{g^*}=\vec{g}-\vec{a}$$ | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="4.1.22_2.png" | ||
| + | loading="lazy" width="150" /> | ||
| + | <figcaption> | ||
| + | Effective acceleration of free fall $\vec{g^*}$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | |||
| + | From the drawing it could be found that | ||
| + | $$\tan\beta = \frac{a - g\sin\alpha}{g\cos\alpha} = -\mu$$ | ||
| + | The angle between the water level and the horizon will be | ||
| + | $$\boxed{\gamma= \alpha-\operatorname{arctg}\mu}$$ | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$\gamma=\alpha-\operatorname{arctg}\mu $$ | ||
| + | </p> | ||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
| + | Aliaksandr Melnichenka<br> | ||
| + | </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> | ||
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| + | </footer> | ||
| + | </body> | ||
| + | |||
| + | </html> | ||
| @@ -0,0 +1,117 @@ | |||
| <!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="At what angle to the horizon will the surface of the liquid in the vessel slide on an inclined plane that makes an angle \alpha to the horizon if the friction coefficient is \mu?"> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="At what angle to the horizon will the surface of the liquid in the vessel slide on an inclined plane that makes an angle \alpha to the horizon if the friction coefficient is \mu?"> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="At what angle to the horizon will the surface of the liquid in the vessel slide on an inclined plane that makes an angle \alpha to the horizon if the friction coefficient is \mu?"> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>At what angle to the horizon will the surface of the liquid in the vessel slide on an inclined plane that makes an angle \alpha to the horizon if the friction coefficient is \mu?</title> | |||
| <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> | |||
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| MathJax.Hub.Config({ | |||
| extensions: ['tex2jax.js'], | |||
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| }, | |||
| showProcessingMessages: false, | |||
| messageStyle: 'none' | |||
| }); | |||
| </script> | |||
| </head> | |||
| <body style=""> | |||
| <header style="text-align:center;"> | |||
| <a href="../../" style="text-decoration: none;"> | |||
| <div id="logo"> | |||
| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | |||
| </div> | |||
| </a> | |||
| <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="../../#4.1">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $4.1.22.$ At what angle to the horizon will the surface of the liquid in the vessel slide on an inclined plane that makes an angle $\alpha$ to the horizon if the friction coefficient is $\mu$? | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="4.1.22_1.png" | |||
| loading="lazy" width="300" /> | |||
| <figcaption> | |||
| Forces acting on the system | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| The vessel with liquid goes downward with acceleration $a$ as shown in the figure</p><p> | |||
| Given the reaction force $N=mg\cos\alpha$, the sliding friction force is found as | |||
| $$F_\text{fr} = \mu N = \mu mg\cos\alpha$$ | |||
| Newton's second law for the vessel | |||
| $$ma = mg\sin\alpha-F_\text{fr}$$ | |||
| $$\boxed{a=g(\sin\alpha-\mu\cos\alpha)}\quad(1)$$ | |||
| Let us introduce the effective acceleration of free fall $\vec{g^*}$ in a non-inertial frame of reference associated with the vessel, then vectorially, it is found as | |||
| $$\vec{g^*}=\vec{g}-\vec{a}$$ | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="4.1.22_2.png" | |||
| loading="lazy" width="150" /> | |||
| <figcaption> | |||
| Effective acceleration of free fall $\vec{g^*}$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| From the drawing it could be found that | |||
| $$\tan\beta = \frac{a - g\sin\alpha}{g\cos\alpha} = -\mu$$ | |||
| The angle between the water level and the horizon will be | |||
| $$\boxed{\gamma= \alpha-\operatorname{arctg}\mu}$$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$\gamma=\alpha-\operatorname{arctg}\mu $$ | |||
| </p> | |||
| <p style="text-align: right; font-style: italic; font-size: 14;"> | |||
| Aliaksandr Melnichenka<br> | |||
| </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> | |||