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+ <meta name="description" content="What force should be applied to the upper weight of mass m_1 so that the lower weight of mass m_2, connected to the upper one by a spring, breaks off from the floor after this force ceases?">
+ <meta name="author" content="Aliaksandr Melnichenka">
+ <meta name="date" content="2023-10" scheme="YYYY-MM">
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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="../../#2.3">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <p>
+ $2.3.46.$ What force should be applied to the upper weight of mass $m_1$ so that the lower weight of mass $m_2$, connected to the upper one by a spring, breaks off from the floor after this force ceases?
+</p>
+<center>
+ <figure>
+ <img src="2.3.46.png"
+ loading="lazy" width="170" />
+ <figcaption>
+ For problem $2.3.46$
+ </figcaption>
+ </figure>
+</center>
+<p>
+
+ </p>
+
+ <h3>Solution</h3>
+<center>
+ <figure>
+ <img src="2.3.46_1.png"
+ loading="lazy" width="300" />
+ <figcaption>
+ Forces acting on the system
+ </figcaption>
+ </figure>
+</center>
+<p>
+
+Forces acting on the upper body in the force-less situation ($F=0$)
+In the case where no external force $F$ is applied, we have the equation:
+$$N+k\Delta x'=m_2g $$
+As $N=0$:
+$$\Delta x' = \frac{m_2g}{k}$$
+For the case where there is a force $F$
+$$\Delta x = \frac{F+m_1g}{k}$$
+Law of energy preservation
+$$m_1g\left(l-\Delta x\right)+\frac{k\Delta x^2}{2} = m_1g\left(l+\Delta x'\right)+\frac{k{\Delta x'}^2}{2}$$
+$$-m_1g\left(\Delta x + \Delta x'\right)=\frac{k}{2}\left({\Delta x'}^2-{\Delta x}^2\right)$$
+The expression $ {\Delta x'}^2 - {\Delta x}^2 $ can be rewritten using the difference of squares:
+$$2m_1g=k\Delta x - k\Delta x'$$
+Substituting the previously obtained values
+$$2m_1g=F+m_1g-m_2g$$
+From where the final result could be obtained
+$$\boxed{F=m_1g+m_2g}$$
+ </p>
+
+ <h4>Answer</h4>
+ <p>
+ $$F=(m_1+m_2)g$$
+ </p>
+ <p style="text-align: right; font-style: italic; font-size: 14;">
+ Almaskhan Arsen<br>
+ </p>
+
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