Translated 2.1.1-2.1.7

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+ <meta name="description" content="To measure the mass of an astronaut on an orbital station, a movable seat of known mass m_0 attached to a spring is used. With the same initial spring deformation (compression), the empty seat returns to its original position after a time t_0, but if the astronaut is on the seat, it returns after a time t > t_0. What is the mass of an astronaut?">
+ <meta name="author" content="Aliaksandr Melnichenka">
+ <meta name="date" content="2023-10" scheme="YYYY-MM">
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+ <title>To measure the mass of an astronaut on an orbital station, a movable seat of known mass m_0 attached to a spring is used. With the same initial spring deformation (compression), the empty seat returns to its original position after a time t_0, but if the astronaut is on the seat, it returns after a time t > t_0. What is the mass of an astronaut?</title>
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+ <h2>Solutions of Savchenko Problems in Physics</h2>
+ <p class="author">
+ Aliaksandr Melnichenka <br/>
+ October 2023
+ </p>
+ </header>
+
+ <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <p>
+ $2.1.7.$ To measure the mass of an astronaut on an orbital station, a movable seat of known mass $m_0$ attached to a spring is used. With the same initial spring deformation (compression), the empty seat returns to its original position after a time $t_0$, but if the astronaut is on the seat, it returns after a time $t > t_0$. What is the mass of an astronaut?
+ </p>
+
+ <h3>Solution</h3>
+ <p>
+ <center>
+ <figure>
+ <img src="https://savchenkosolutions.com/2/2.1.7/astronaut_wheel.jpg"
+ loading="lazy" width="380" />
+ <figcaption>
+ Tamara Jernigan using the Body Mass Measurement Device (BMMD)
+ </figcaption>
+ </figure>
+ </center>
+<p>
+Let's write Newton's second law with an astronaut
+$$ kx_2 = (m_0 + m) a_2 $$
+And with an empty chair
+$$ kx_1 = m_0 a_1 $$
+Let's take into account uniformly accelerated motion
+$$ \left\{\begin{matrix}
+ k \frac{a_1t_0^2}{2} = m_0 a_1\\
+ k \frac{a_2t^2}{2} = (m_0+m) a_2
+ \end{matrix}\right. $$
+From the first equation
+$$ k = \frac{2m_0}{t_0^2} $$
+We substitute into the second
+$$ \fbox{$m = m_0(\frac{t^2}{t^2_0} − 1)$} $$
+</p>
+ </p>
+
+ <h4>Answer</h4>
+ <p>
+ $$m = m_0[(t/t_0)^2 − 1]$$
+ <p style="text-align: right; font-style: italic; font-size: 14;">Kozak Anastasia</p>
+ </p>
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