Added Luis's English solution of 2.4.34
en/2.4.34.md
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| + | <title>Two loads of mass $m_1$ and $m_2$ ($m_1 > m_2$) are connected by a thread thrown over a fixed block. At the initial moment, the load of mass $m_1$ is held at a height $h$ above the floor. Then it is released without a push. How much heat will be released when the load hits the floor? The impact is absolutely inelastic.</title> | ||
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| + | <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="../../#2.4">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $2.4.34$ | ||
| + | Two loads of mass $m_1$ and $m_2$ ($m_1 \geq m_2$) are connected by a thread thrown over a fixed block. At the initial moment, the load of mass $m_1$ is held at a height $h$ above the floor. Then it is released without a push. How much heat will be released when the load hits the floor? The impact is absolutely inelastic. | ||
| + | </p> | ||
| + | |||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="statement.png" | ||
| + | loading="lazy" width="250" /> | ||
| + | <figcaption> | ||
| + | For problem $2.4.34$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | Applying Energy Conservation Law: | ||
| + | $$m_1 g h = Q + m_2 g H\;(1)$$ | ||
| + | where $Q$ is lost energy by heat and $H$ is the height achieved by body of mass $m_2$ after the impact of body of mass $m_1$, and is given by, | ||
| + | $$H = h + \frac{v^2}{2g}\;(2)$$ | ||
| + | Why? Potential energy of body 1 is converted in kinetic energy of body 1 and kinetic energy of body 2 with same velocity due to the constraint ($v_1=v_2=v$), after totally inelastic collision with ground, part of this energy is transformed in heat and the other one in potential energy for body 2, which behaves as a projectile launched vertically upwards from height $h$. For determining this velocity $v$, it's necessary to know what acceleration did the system have before impact. Applying Newton Second Law, for body 1: | ||
| + | $$m_1 g - T = m_1 a \;(3)$$ | ||
| + | and for body 2, | ||
| + | $$T - m_2 g = m_2 a\;(4)$$ | ||
| + | Suming up equations (3) and (4) side by side, | ||
| + | $$a = \frac{m_1-m_2}{m_1+m_2}g\;(5)$$ | ||
| + | As $h = \frac{v^2}{2a}$, so, | ||
| + | $$v = \sqrt{2ah}\;(6)$$ | ||
| + | Substituting (5) into (6), | ||
| + | $$v = \sqrt{2gh\left(\frac{m_1-m_2}{m_1+m_2}\right)}\;(7)$$ | ||
| + | Putting (7) into (2) | ||
| + | $$H = \frac{2m_1}{m_1+m_2}h\;(8)$$ | ||
| + | Finally, substituting (8) into (1) and separating $Q$, | ||
| + | </p> | ||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$Q = m_1gh \frac{m_1-m_2}{m_1+m_2}$$ | ||
| + | </p> | ||
| + | |||
| + | |||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
| + | BSc. Luis Daniel Fernández Quintana<br> | ||
| + | Physics Department (FCNE)<br> | ||
| + | Universidad de Oriente, Cuba<br> | ||
| + | </p> | ||
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| <meta name="author" content="Aliaksandr Melnichenka"> | |||
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| <title>Two loads of mass $m_1$ and $m_2$ ($m_1 > m_2$) are connected by a thread thrown over a fixed block. At the initial moment, the load of mass $m_1$ is held at a height $h$ above the floor. Then it is released without a push. How much heat will be released when the load hits the floor? The impact is absolutely inelastic.</title> | |||
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| <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="../../#2.4">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $2.4.34$ | |||
| Two loads of mass $m_1$ and $m_2$ ($m_1 \geq m_2$) are connected by a thread thrown over a fixed block. At the initial moment, the load of mass $m_1$ is held at a height $h$ above the floor. Then it is released without a push. How much heat will be released when the load hits the floor? The impact is absolutely inelastic. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="statement.png" | |||
| loading="lazy" width="250" /> | |||
| <figcaption> | |||
| For problem $2.4.34$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| Applying Energy Conservation Law: | |||
| $$m_1 g h = Q + m_2 g H\;(1)$$ | |||
| where $Q$ is lost energy by heat and $H$ is the height achieved by body of mass $m_2$ after the impact of body of mass $m_1$, and is given by, | |||
| $$H = h + \frac{v^2}{2g}\;(2)$$ | |||
| Why? Potential energy of body 1 is converted in kinetic energy of body 1 and kinetic energy of body 2 with same velocity due to the constraint ($v_1=v_2=v$), after totally inelastic collision with ground, part of this energy is transformed in heat and the other one in potential energy for body 2, which behaves as a projectile launched vertically upwards from height $h$. For determining this velocity $v$, it's necessary to know what acceleration did the system have before impact. Applying Newton Second Law, for body 1: | |||
| $$m_1 g - T = m_1 a \;(3)$$ | |||
| and for body 2, | |||
| $$T - m_2 g = m_2 a\;(4)$$ | |||
| Suming up equations (3) and (4) side by side, | |||
| $$a = \frac{m_1-m_2}{m_1+m_2}g\;(5)$$ | |||
| As $h = \frac{v^2}{2a}$, so, | |||
| $$v = \sqrt{2ah}\;(6)$$ | |||
| Substituting (5) into (6), | |||
| $$v = \sqrt{2gh\left(\frac{m_1-m_2}{m_1+m_2}\right)}\;(7)$$ | |||
| Putting (7) into (2) | |||
| $$H = \frac{2m_1}{m_1+m_2}h\;(8)$$ | |||
| Finally, substituting (8) into (1) and separating $Q$, | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$Q = m_1gh \frac{m_1-m_2}{m_1+m_2}$$ | |||
| </p> | |||
| <p style="text-align: right; font-style: italic; font-size: 14;"> | |||
| BSc. Luis Daniel Fernández Quintana<br> | |||
| Physics Department (FCNE)<br> | |||
| Universidad de Oriente, Cuba<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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