Translated 3.1.1-3.1.17

astrosander edited Sep 24, 2024, 2:16 AM

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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="../../#3.1">$\leftarrow$Back</a></h3>
		+
		+	<h3> Statement </h3>
		+	<p>
		+	$3.1.4.$ Why does a rapidly oscillating bulb on a spring seem to flash at the extreme points of its trajectory?
		+	</p>
		+
		+	<h3>Solution</h3>
		+	<p>
		+	</p>
		+	<center>
		+	<figure>
		+	<img src="https://savchenkosolutions.com/3/3.1.4/animation.gif"
		+	loading="lazy" width="350" />
		+	<figcaption>
		+	Harmonic motion of a light bulb using a mathematical pendulum as an example
		+	</figcaption>
		+	</figure>
		+	</center>
		+	<p>
		+
		+	The power of the light bulb, and as a consequence the intensity of the light bulb, remains constant, so the longer the light bulb is at a certain moment, the more energy comes to the eye. Because the human eye perceives and accumulates light energy at a particularly high frequency of change (more details: <a href="https://doi.org/10.1016/S0042-6989(99)00084-X">doi.org/10.1016/S0042-6989(99)00084-X</a>), then the longer the body is at a certain point, the brighter it seems to the retina of our eye</p><p>
		+	The perceived energy is inversely proportional to the time spent in a given place:
		+	$$ W\sim\Delta t $$
		+	In harmonic motion, the position of the light bulb is described by the expression
		+	$$ x(t) = x_0 \sin\omega t $$
		+	From here, the speed value changes as
		+	$$ v(t) = \dot{x}(t) =\omega \cdot x_0\cos \omega t $$
		+	The lower the speed, the more time it takes for the light bulb to cover equal intervals of time.
		+	$$ v\sim\frac{1}{\Delta t} $$
		+	Thus, when the speed of the light bulb during harmonic motion is minimal, the energy perceived by the eye becomes greater.
		+	$$ W\sim \frac{1}{v} $$
		+	Thus, the bulb appears brighter in places where its speed is minimal, i.e. at its extreme points.
		+	</p>
		+
		+	<p style="text-align: right; font-style: italic; font-size: 14;">
		+	Aliaksandr Melnichenka<br>
		+	</p>
		+
		+
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	@@ -0,0 +1,98 @@
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			<head>
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			<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="Why does a rapidly oscillating bulb on a spring seem to flash at the extreme points of its trajectory?">
			<meta name="author" content="Aliaksandr Melnichenka">
			<meta name="date" content="2023-10" scheme="YYYY-MM">
			<meta property="og:title" content="Why does a rapidly oscillating bulb on a spring seem to flash at the extreme points of its trajectory?">
			<meta property="og:image" content="img/logo.png">
			<meta property="og:description" content="Why does a rapidly oscillating bulb on a spring seem to flash at the extreme points of its trajectory?">
			<meta name="yandex-verification" content="6cfda41f74038368">
			<title>Why does a rapidly oscillating bulb on a spring seem to flash at the extreme points of its trajectory?</title>
			<link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css">
			<link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png">
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			extensions: ['tex2jax.js'],
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			ignoreClass: 'html'
			},
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			messageStyle: 'none'
			});
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			</head>
			<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="../../#3.1">$\leftarrow$Back</a></h3>

			<h3> Statement </h3>
			<p>
			$3.1.4.$ Why does a rapidly oscillating bulb on a spring seem to flash at the extreme points of its trajectory?
			</p>

			<h3>Solution</h3>
			<p>
			</p>
			<center>
			<figure>
			<img src="https://savchenkosolutions.com/3/3.1.4/animation.gif"
			loading="lazy" width="350" />
			<figcaption>
			Harmonic motion of a light bulb using a mathematical pendulum as an example
			</figcaption>
			</figure>
			</center>
			<p>

			The power of the light bulb, and as a consequence the intensity of the light bulb, remains constant, so the longer the light bulb is at a certain moment, the more energy comes to the eye. Because the human eye perceives and accumulates light energy at a particularly high frequency of change (more details: <a href="https://doi.org/10.1016/S0042-6989(99)00084-X">doi.org/10.1016/S0042-6989(99)00084-X</a>), then the longer the body is at a certain point, the brighter it seems to the retina of our eye</p><p>
			The perceived energy is inversely proportional to the time spent in a given place:
			$$ W\sim\Delta t $$
			In harmonic motion, the position of the light bulb is described by the expression
			$$ x(t) = x_0 \sin\omega t $$
			From here, the speed value changes as
			$$ v(t) = \dot{x}(t) =\omega \cdot x_0\cos \omega t $$
			The lower the speed, the more time it takes for the light bulb to cover equal intervals of time.
			$$ v\sim\frac{1}{\Delta t} $$
			Thus, when the speed of the light bulb during harmonic motion is minimal, the energy perceived by the eye becomes greater.
			$$ W\sim \frac{1}{v} $$
			Thus, the bulb appears brighter in places where its speed is minimal, i.e. at its extreme points.
			</p>

			<p style="text-align: right; font-style: italic; font-size: 14;">
			Aliaksandr Melnichenka<br>
			</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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