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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> | | <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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| <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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? | | $2.1.7.$ 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? |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
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| <img src="https://savchenkosolutions.com/2/2.1.7/astronaut_wheel.jpg" | | <img src="https://savchenkosolutions.com/2/2.1.7/astronaut_wheel.jpg" |
| loading="lazy" width="380" /> | | loading="lazy" width="380" /> |
| <figcaption> | | <figcaption> |
| Tamara Jernigan using the Body Mass Measurement Device (BMMD) | | Tamara Jernigan using the Body Mass Measurement Device (BMMD) |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| Let's write Newton's second law with an astronaut | | Let's write Newton's second law with an astronaut |
| $$ kx_2 = (m_0 + m) a_2 $$ | | $$ kx_2 = (m_0 + m) a_2 $$ |
| And with an empty chair | | And with an empty chair |
| $$ kx_1 = m_0 a_1 $$ | | $$ kx_1 = m_0 a_1 $$ |
| Let's take into account uniformly accelerated motion | | Let's take into account uniformly accelerated motion |
| $$ \left\{\begin{matrix} | | $$ \left\{\begin{matrix} |
| k \frac{a_1t_0^2}{2} = m_0 a_1\\ | | k \frac{a_1t_0^2}{2} = m_0 a_1\\ |
| k \frac{a_2t^2}{2} = (m_0+m) a_2 | | k \frac{a_2t^2}{2} = (m_0+m) a_2 |
| \end{matrix}\right. $$ | | \end{matrix}\right. $$ |
| From the first equation | | From the first equation |
| $$ k = \frac{2m_0}{t_0^2} $$ | | $$ k = \frac{2m_0}{t_0^2} $$ |
| We substitute into the second | | We substitute into the second |
| $$ \fbox{$m = m_0(\frac{t^2}{t^2_0} − 1)$} $$ | | $$ \fbox{$m = m_0(\frac{t^2}{t^2_0} − 1)$} $$ |
| </p> | | </p> |
| </p> | | </p> |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$m = m_0[(t/t_0)^2 − 1]$$ | | $$m = m_0[(t/t_0)^2 − 1]$$ |
| <p style="text-align: right; font-style: italic; font-size: 14;">Kozak Anastasia</p> | | <p style="text-align: right; font-style: italic; font-size: 14;">Kozak Anastasia</p> |
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