Question

As we've seen, astronauts measure their mass by measuring the period of oscillation when sitting in a chair connected to a spring. The Body Mass Measurement Device on Skylab, a 1970s space station, had a spring constant of 606 N/m. The empty chair oscillated with a period of 0.848 s.

What is the mass of an astronaut who oscillates with a period of 2.26 s when sitting in the chair?

Answer 1

Concepts and reason

The concept required to solve this problem is period of oscillation of an object in simple harmonic motion.

Initially, apply the formula for the period of oscillation and solve for the mass of the object. After that repeat, the same with new mass. Then for the mass of astronaut, subtract the masses obtained in both the steps.

Fundamentals

The period of oscillation $T$ of an object oscillating is given as follows:

$T = 2\pi \sqrt {\frac{m}{k}}$

Here, $m$ is the mass of the object and $k$ is the spring constant.

The period of oscillation $T$ of an object oscillating is given as follows:

$T = 2\pi \sqrt {\frac{m}{k}}$

Rearrange the expression for the mass of the object as follows:

$m = {\left( {\frac{T}{{2\pi }}} \right)^2}k$

Substitute $0.848{\rm{ s}}$ for $T$ and $606{\rm{ N/m}}$ for $k$ as follows:

$\begin{array}{c}\\m = {\left\{ {\frac{{0.848{\rm{ s}}}}{{2\left( {3.14} \right)}}} \right\}^2}\left( {606{\rm{ N/m}}} \right)\\\\ = 11.05{\rm{ kg}}\\\end{array}$

The expression for the mass of the object is given as follows:

$m' = {\left( {\frac{{T'}}{{2\pi }}} \right)^2}k$

Substitute ${\rm{2}}{\rm{.26 s}}$ for $T'$ and $606{\rm{ N/m}}$ for $k$ as follows:

$\begin{array}{c}\\m' = {\left\{ {\frac{{{\rm{2}}{\rm{.26 s}}}}{{2\left( {3.14} \right)}}} \right\}^2}\left( {606{\rm{ N/m}}} \right)\\\\ = 78.40{\rm{ kg}}\\\end{array}$

So, the mass of the astronaut is given as follows:

$M = m' - m$

Substitute $78.40{\rm{ kg}}$ for $m'$ and ${\rm{11}}{\rm{.05 kg}}$ for $m$ as follows:

$\begin{array}{c}\\M = 78.40{\rm{ kg}} - 11.05{\rm{ kg}}\\\\ = 67.35{\rm{ kg}}\\\end{array}$

Ans:

The mass of the astronaut sitting on the chair is $67.35{\rm{ kg}}$ .