Added Luis's English solution of 2.6.36
en/2.6.36.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="In the case of a central force acting on a body, the radius vector drawn to it from the center describes equal areas at regular intervals. (This, in fact, is Kepler's second law with respect to the motion of the planets.) What area will the radius vector drawn from the Sun to the planet describe in time $t$, if at the initial moment the distance from it to the Sun is $r$, the speed is $v$, and the angle between the speed of the planet and the radius vector is $\alpha$?"> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="In the case of a central force acting on a body, the radius vector drawn to it from the center describes equal areas at regular intervals. (This, in fact, is Kepler's second law with respect to the motion of the planets.) What area will the radius vector drawn from the Sun to the planet describe in time $t$, if at the initial moment the distance from it to the Sun is $r$, the speed is $v$, and the angle between the speed of the planet and the radius vector is $\alpha$?"> | ||
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| + | <title>In the case of a central force acting on a body, the radius vector drawn to it from the center describes equal areas at regular intervals. (This, in fact, is Kepler's second law with respect to the motion of the planets.) What area will the radius vector drawn from the Sun to the planet describe in time $t$, if at the initial moment the distance from it to the Sun is $r$, the speed is $v$, and the angle between the speed of the planet and the radius vector is $\alpha$?</title> | ||
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| + | <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="../../#2.6">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $2.6.36*$ | ||
| + | In the case of a central force acting on a body, the radius vector drawn to it from the center describes equal areas at regular intervals. (This, in fact, is Kepler's second law with respect to the motion of the planets.) What area will the radius vector drawn from the Sun to the planet describe in time $t$, if at the initial moment the distance from it to the Sun is $r$, the speed is $v$, and the angle between the speed of the planet and the radius vector is $\alpha$? | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="analysis.png" | ||
| + | loading="lazy" width="300" /> | ||
| + | <figcaption> | ||
| + | Analysis | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | Graphically, it can be seen that | ||
| + | $$\Delta S = \frac{r^2\Delta \theta}{2}\;(1)$$ | ||
| + | considering a small variation of angle swept by radiovector. But, | ||
| + | $$\Delta \theta = \omega~\Delta t\;(2)$$ | ||
| + | As angular momentum $L$ is related to angular velocity, $L = I\omega$, where $I$ is inertia's momentum, in this case $I = mr^2$, so | ||
| + | $$\omega = \frac{L}{mr^2}\;(3)$$ | ||
| + | By definition, $\vec{L} = m \vec{r}\times\vec{v}$, then | ||
| + | $$L = mrv\sin{\alpha}\;(4)$$ | ||
| + | Putting (4) into (3) | ||
| + | $$\omega = \frac{v}{r}\sin{\alpha}\;(5)$$ | ||
| + | Substituting (5) into (2) | ||
| + | $$\Delta \theta = \frac{v\Delta t}{r}\sin{\alpha}\;(6)$$ | ||
| + | From (6) and (1) | ||
| + | $$\Delta S = \frac{rv\Delta t}{2}\sin{\alpha}$$ | ||
| + | where $\Delta S = S - S_0 = S$ ($S_0=0$) and $\Delta t = t - t_0 = t$ ($t_0=0$). Finally, | ||
| + | </p> | ||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$S = \frac{rvt}{2}\sin{\alpha}$$ | ||
| + | </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="description" content="In the case of a central force acting on a body, the radius vector drawn to it from the center describes equal areas at regular intervals. (This, in fact, is Kepler's second law with respect to the motion of the planets.) What area will the radius vector drawn from the Sun to the planet describe in time $t$, if at the initial moment the distance from it to the Sun is $r$, the speed is $v$, and the angle between the speed of the planet and the radius vector is $\alpha$?"> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="In the case of a central force acting on a body, the radius vector drawn to it from the center describes equal areas at regular intervals. (This, in fact, is Kepler's second law with respect to the motion of the planets.) What area will the radius vector drawn from the Sun to the planet describe in time $t$, if at the initial moment the distance from it to the Sun is $r$, the speed is $v$, and the angle between the speed of the planet and the radius vector is $\alpha$?"> | |||
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| <meta property="og:description" content="In the case of a central force acting on a body, the radius vector drawn to it from the center describes equal areas at regular intervals. (This, in fact, is Kepler's second law with respect to the motion of the planets.) What area will the radius vector drawn from the Sun to the planet describe in time $t$, if at the initial moment the distance from it to the Sun is $r$, the speed is $v$, and the angle between the speed of the planet and the radius vector is $\alpha$?"> | |||
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| <title>In the case of a central force acting on a body, the radius vector drawn to it from the center describes equal areas at regular intervals. (This, in fact, is Kepler's second law with respect to the motion of the planets.) What area will the radius vector drawn from the Sun to the planet describe in time $t$, if at the initial moment the distance from it to the Sun is $r$, the speed is $v$, and the angle between the speed of the planet and the radius vector is $\alpha$?</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="../../#2.6">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $2.6.36*$ | |||
| In the case of a central force acting on a body, the radius vector drawn to it from the center describes equal areas at regular intervals. (This, in fact, is Kepler's second law with respect to the motion of the planets.) What area will the radius vector drawn from the Sun to the planet describe in time $t$, if at the initial moment the distance from it to the Sun is $r$, the speed is $v$, and the angle between the speed of the planet and the radius vector is $\alpha$? | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <center> | |||
| <figure> | |||
| <img src="analysis.png" | |||
| loading="lazy" width="300" /> | |||
| <figcaption> | |||
| Analysis | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| Graphically, it can be seen that | |||
| $$\Delta S = \frac{r^2\Delta \theta}{2}\;(1)$$ | |||
| considering a small variation of angle swept by radiovector. But, | |||
| $$\Delta \theta = \omega~\Delta t\;(2)$$ | |||
| As angular momentum $L$ is related to angular velocity, $L = I\omega$, where $I$ is inertia's momentum, in this case $I = mr^2$, so | |||
| $$\omega = \frac{L}{mr^2}\;(3)$$ | |||
| By definition, $\vec{L} = m \vec{r}\times\vec{v}$, then | |||
| $$L = mrv\sin{\alpha}\;(4)$$ | |||
| Putting (4) into (3) | |||
| $$\omega = \frac{v}{r}\sin{\alpha}\;(5)$$ | |||
| Substituting (5) into (2) | |||
| $$\Delta \theta = \frac{v\Delta t}{r}\sin{\alpha}\;(6)$$ | |||
| From (6) and (1) | |||
| $$\Delta S = \frac{rv\Delta t}{2}\sin{\alpha}$$ | |||
| where $\Delta S = S - S_0 = S$ ($S_0=0$) and $\Delta t = t - t_0 = t$ ($t_0=0$). Finally, | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$S = \frac{rvt}{2}\sin{\alpha}$$ | |||
| </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> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
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