Translated 2.1.13-2.1.33
en/2.1.25.md
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| + | <meta name="description" content="A thread spanned over a block with a fixed axis is passed through a slot. At the ends of the thread are suspended loads, the mass of which is m_1 and m_2. Determine the acceleration of loads if a constant frictional force F_{tr} acts on the thread from the side of the slot when it moves."> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="A thread spanned over a block with a fixed axis is passed through a slot. At the ends of the thread are suspended loads, the mass of which is m_1 and m_2. Determine the acceleration of loads if a constant frictional force F_{tr} acts on the thread from the side of the slot when it moves."> | ||
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| + | <meta property="og:description" content="A thread spanned over a block with a fixed axis is passed through a slot. At the ends of the thread are suspended loads, the mass of which is m_1 and m_2. Determine the acceleration of loads if a constant frictional force F_{tr} acts on the thread from the side of the slot when it moves."> | ||
| + | <meta name="yandex-verification" content="6cfda41f74038368"> | ||
| + | <title>A thread spanned over a block with a fixed axis is passed through a slot. At the ends of the thread are suspended loads, the mass of which is m_1 and m_2. Determine the acceleration of loads if a constant frictional force F_{tr} acts on the thread from the side of the slot when it moves.</title> | ||
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| + | <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> | ||
| + | $2.1.25.$ A thread spanned over a block with a fixed axis is passed through a slot. At the ends of the thread are suspended loads, the mass of which is $m_1$ and $m_2$. Determine the acceleration of loads if a constant frictional force $F_{tr}$ acts on the thread from the side of the slot when it moves. | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/2/2.1.25/statement.png" | ||
| + | loading="lazy" width="100" /> | ||
| + | <figcaption> | ||
| + | For problem $2.1.25$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/2/2.1.25/sol.jpg" | ||
| + | loading="lazy" width="130" /> | ||
| + | <figcaption> | ||
| + | Forces acting on loads | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | |||
| + | <p> | ||
| + | 1. Due to the weightlessness and non-stretchability of the thread, as well as the ideal properties of the block (no losses and low weight), the problem can be solved in the following approximation) | ||
| + | $$ a_1=a_2=a $$ | ||
| + | |||
| + | $$ T_1=T_2=T $$ | ||
| + | 2. The equations of motion of loads in projection onto the vertical axis in this case are written as follows: | ||
| + | $$ \left\{\begin{matrix} | ||
| + | m_1a = m_2g - T \\ | ||
| + | m_2a =T-m_2g-F_{fr} | ||
| + | \end{matrix}\right. $$ | ||
| + | 3. Solving the equations together, we obtain | ||
| + | $$ a=\frac{(m_1-m_2)g-F_{fr}}{m_1+m_2} $$ | ||
| + | 4. Substituting the acceleration value into the first equation of the system allows us to determine the tension of the thread | ||
| + | $$ \boxed{T=m_1\frac{2m_2g+F_{fr}}{m_1+m_2}} $$ | ||
| + | </p> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$T=m_1\frac{2m_2g+F_{fr}}{m_1+m_2}$$ | ||
| + | </p> | ||
| + | |||
| + | |||
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| + | <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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| <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="A thread spanned over a block with a fixed axis is passed through a slot. At the ends of the thread are suspended loads, the mass of which is m_1 and m_2. Determine the acceleration of loads if a constant frictional force F_{tr} acts on the thread from the side of the slot when it moves."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="A thread spanned over a block with a fixed axis is passed through a slot. At the ends of the thread are suspended loads, the mass of which is m_1 and m_2. Determine the acceleration of loads if a constant frictional force F_{tr} acts on the thread from the side of the slot when it moves."> | |||
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| <meta property="og:description" content="A thread spanned over a block with a fixed axis is passed through a slot. At the ends of the thread are suspended loads, the mass of which is m_1 and m_2. Determine the acceleration of loads if a constant frictional force F_{tr} acts on the thread from the side of the slot when it moves."> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>A thread spanned over a block with a fixed axis is passed through a slot. At the ends of the thread are suspended loads, the mass of which is m_1 and m_2. Determine the acceleration of loads if a constant frictional force F_{tr} acts on the thread from the side of the slot when it moves.</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> | |||
| $2.1.25.$ A thread spanned over a block with a fixed axis is passed through a slot. At the ends of the thread are suspended loads, the mass of which is $m_1$ and $m_2$. Determine the acceleration of loads if a constant frictional force $F_{tr}$ acts on the thread from the side of the slot when it moves. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/2/2.1.25/statement.png" | |||
| loading="lazy" width="100" /> | |||
| <figcaption> | |||
| For problem $2.1.25$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/2/2.1.25/sol.jpg" | |||
| loading="lazy" width="130" /> | |||
| <figcaption> | |||
| Forces acting on loads | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| 1. Due to the weightlessness and non-stretchability of the thread, as well as the ideal properties of the block (no losses and low weight), the problem can be solved in the following approximation) | |||
| $$ a_1=a_2=a $$ | |||
| $$ T_1=T_2=T $$ | |||
| 2. The equations of motion of loads in projection onto the vertical axis in this case are written as follows: | |||
| $$ \left\{\begin{matrix} | |||
| m_1a = m_2g - T \\ | |||
| m_2a =T-m_2g-F_{fr} | |||
| \end{matrix}\right. $$ | |||
| 3. Solving the equations together, we obtain | |||
| $$ a=\frac{(m_1-m_2)g-F_{fr}}{m_1+m_2} $$ | |||
| 4. Substituting the acceleration value into the first equation of the system allows us to determine the tension of the thread | |||
| $$ \boxed{T=m_1\frac{2m_2g+F_{fr}}{m_1+m_2}} $$ | |||
| </p> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$T=m_1\frac{2m_2g+F_{fr}}{m_1+m_2}$$ | |||
| </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> | |||