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?

Solutions of Savchenko Problems in Physics

Aliaksandr Melnichenka
October 2023

    <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>

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 $$\fbox{$m = m_0(\frac{t^2}{t^2_0} − 1)$}$$

    <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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