Translated 3.2.1-3.2.17

astrosander edited Sep 28, 2024, 1:02 AM

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		+	<h3> Statement </h3>
		+	<p>
		+	$3.2.10.$ Measurements of the circular frequency of oscillations of a body of mass $m$, fixed in the middle of a tensioned string with length $2 l$, gave the value $\omega$. Find the force of tension of the string.
		+	</p>
		+
		+	<h3>Solution</h3>
		+	<p>
		+	The total external force acting on the system
		+	$$ F=2T\sin\varphi $$
		+	Newton's second law
		+	$$ m\ddot{x}(t)+2T\sin\varphi=0 $$
		+	Let's use the approximation for small angles $(\varphi \ll 1)$:
		+	$$ \sin\varphi\approx\varphi=\frac{x}{l} $$
		+	Harmonic oscillation equation
		+	$$ \ddot{x}(t)+\frac{2T}{ml}x(t)=0 $$
		+	Where does the period of oscillation come from?
		+	$$ \omega^{2}=\frac{2T}{ml}\Rightarrow \boxed{T=\frac{ml\omega^{2}}{2}} $$
		+	</p>
		+	<p style="text-align: right; font-style: italic; font-size: 14;">
		+	Dzikan Mikita<br>
		+	Aliaksandr Kanashenka
		+	</p>
		+
		+	<h4>Answer</h4>
		+	<p>
		+	$$T=\frac{ml\omega^{*2}}{2}$$
		+	</p>
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			<meta name="author" content="Aliaksandr Melnichenka">
			<meta name="date" content="2023-10" scheme="YYYY-MM">
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			<div id = "logo">
			<span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span>
			</div>
			<p class="author">
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			<i><b>knowledge must be free</b></i>
			</p>
			</header>

			<h3 id="back-link"><a href="../../#3.2">$\leftarrow$Back</a></h3>

			<h3> Statement </h3>
			<p>
			$3.2.10.$ Measurements of the circular frequency of oscillations of a body of mass $m$, fixed in the middle of a tensioned string with length $2 l$, gave the value $\omega$. Find the force of tension of the string.
			</p>

			<h3>Solution</h3>
			<p>
			The total external force acting on the system
			$$ F=2T\sin\varphi $$
			Newton's second law
			$$ m\ddot{x}(t)+2T\sin\varphi=0 $$
			Let's use the approximation for small angles $(\varphi \ll 1)$:
			$$ \sin\varphi\approx\varphi=\frac{x}{l} $$
			Harmonic oscillation equation
			$$ \ddot{x}(t)+\frac{2T}{ml}x(t)=0 $$
			Where does the period of oscillation come from?
			$$ \omega^{2}=\frac{2T}{ml}\Rightarrow \boxed{T=\frac{ml\omega^{2}}{2}} $$
			</p>
			<p style="text-align: right; font-style: italic; font-size: 14;">
			Dzikan Mikita<br>
			Aliaksandr Kanashenka
			</p>

			<h4>Answer</h4>
			<p>
			$$T=\frac{ml\omega^{*2}}{2}$$
			</p>


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