Translated 2.1.9-2.1.20
en/2.1.16.md
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| + | <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | ||
| + | <meta name="description" content="The body of mass m is connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls."> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="The body of mass m is connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls."> | ||
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| + | <meta property="og:description" content="The body of mass m is connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls."> | ||
| + | <meta name="yandex-verification" content="6cfda41f74038368"> | ||
| + | <title>The body of mass m is connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls.</title> | ||
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| + | </head> | ||
| + | <body style=""> | ||
| + | <header style="text-align:center;"> | ||
| + | <h2>Solutions of Savchenko Problems in Physics</h2> | ||
| + | <p class="author"> | ||
| + | Aliaksandr Melnichenka <br/> | ||
| + | October 2023 | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $2.1.16.$ The body of mass m is connected by two springs of stiffness $k_1$ and $k_2$ with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is $a$. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls. | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/2/2.1.16/statement.png" | ||
| + | loading="lazy" width="250" /> | ||
| + | <figcaption> | ||
| + | For problem $2.1.16$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/2/2.1.16/sol.png" | ||
| + | loading="lazy" width="350" /> | ||
| + | <figcaption> | ||
| + | Series spring system | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | |||
| + | <p> | ||
| + | 1. When springs are connected in series, their deformation will be different with the same acting force, this circumstance allows us to determine the overall stiffness of the springs as follows | ||
| + | $$ \Delta x_0 = \Delta x_1 + \Delta x_2 = \frac{F}{k_1}+\frac{F}{k_2} = \frac{F}{k_0} $$ | ||
| + | |||
| + | $$ \boxed{k_0 = \frac{k_1 k_2}{k_1 + k_2}} $$ | ||
| + | 2. The combined effect on the mass of the springs when the mass is at rest will be equal to the applied force $F$ | ||
| + | $$ k_0 \Delta x_0 = F $$ | ||
| + | |||
| + | $$ \boxed{ \Delta x_0 = \frac{F(k_1 + k_2)}{k_1 k_2}} $$ | ||
| + | </p> | ||
| + | <h3> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$x = F(k_1 + k_2)/(k_1k_2)$$ | ||
| + | </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> | ||
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| + | </html> | ||
| @@ -0,0 +1,105 @@ | |||
| <!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 body of mass m is connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="The body of mass m is connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls."> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="The body of mass m is connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls."> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>The body of mass m is connected by two springs of stiffness k_1 and k_2 with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is a. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls.</title> | |||
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| }, | |||
| showProcessingMessages: false, | |||
| messageStyle: 'none' | |||
| }); | |||
| </script> | |||
| </head> | |||
| <body style=""> | |||
| <header style="text-align:center;"> | |||
| <h2>Solutions of Savchenko Problems in Physics</h2> | |||
| <p class="author"> | |||
| Aliaksandr Melnichenka <br/> | |||
| October 2023 | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $2.1.16.$ The body of mass m is connected by two springs of stiffness $k_1$ and $k_2$ with fixed walls, the springs are not initially deformed. When vibrations occur, the greatest acceleration of the body is $a$. Find the maximum deviation of the body from the equilibrium position and the maximum forces with which the springs act on the walls. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/2/2.1.16/statement.png" | |||
| loading="lazy" width="250" /> | |||
| <figcaption> | |||
| For problem $2.1.16$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/2/2.1.16/sol.png" | |||
| loading="lazy" width="350" /> | |||
| <figcaption> | |||
| Series spring system | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| 1. When springs are connected in series, their deformation will be different with the same acting force, this circumstance allows us to determine the overall stiffness of the springs as follows | |||
| $$ \Delta x_0 = \Delta x_1 + \Delta x_2 = \frac{F}{k_1}+\frac{F}{k_2} = \frac{F}{k_0} $$ | |||
| $$ \boxed{k_0 = \frac{k_1 k_2}{k_1 + k_2}} $$ | |||
| 2. The combined effect on the mass of the springs when the mass is at rest will be equal to the applied force $F$ | |||
| $$ k_0 \Delta x_0 = F $$ | |||
| $$ \boxed{ \Delta x_0 = \frac{F(k_1 + k_2)}{k_1 k_2}} $$ | |||
| </p> | |||
| <h3> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$x = F(k_1 + k_2)/(k_1k_2)$$ | |||
| </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> | |||