# The Bathroom Scale in Orbit

When being on the International Space Station (ISS), you cannot just go grocery shopping if the food is running out – the reasons for that are mainly of logistical nature, logically. The food has to be shipped into orbit with expensive spacecrafts, which leads to space meals being cured and only available to a limited extent. Admittedly, the food on the ISS is surprisingly delicious, as many astronauts tell, nevertheless you are not at risk of getting fat in Earth’s orbit and not fitting in your space suit anymore. Instead, astronauts rather have to take care of not losing too much weight: Muscles, which aren’t heavily used in weightlessness, fade away quite fast and the density of bones provably decreases.

Astronaut Don Pettit toys with his food – at 28,000 kilometers per hour.
(Credit: NASA)

So people onboard the space station have to weigh themselves regularly in order to keep their body mass under control. They do this by floating into the station’s bathroom and strapping themselves onto the bathroom scale, which is red and white striped and was brought up into the station in 2001. If they realize that they don’t have their desired weight, they just tighten or loosen the straps which are pressing them against the scale. As soon as the perfect weight is displayed they send this data down to ground control. By the way, there hasn’t been a single astronaut with weight problems since 2001!

Well, the last paragraph is absolute rubbish, naturally!
What is true, however, is the fact that astronauts have to be informed of their body mass. For example, if they exercise too little, after spending half of a year in weightlessness, they will have a bad time back on Earth with all this gravity pulling them down towards the ground.

# How can anyone determine the body mass in weightlessness?

Just hopping onto a scale is futile in space since astronauts up there are weightless. The only thing measurable would be the tensioning force of the straps.

Therefore, creative people have reached into their bag of physical tricks and basically found a quite simple solution for this problem: They just make use of the ordinary differential equation which describes the undamped harmonic oscillator. This equation directly follows from Newton’s second axiom ($F=m\cdot a$). After rearranging the equation, this axiom gives an expression for the (astronaut’s) mass: $m=F/a$. So mass equals force divided by acceleration.
Now, let’s take a big spring, of which we know the spring force $F$, and tie an astronaut of mass $m$ to one end. If we now measure the accelerations $a$ of the oscillating astronaut, we are able to determine her/his mass. The heavier the astronaut, the slower the oscillations.

Schematic space scale: In the first approximation, the astronaut is treated as a cube.
(Credit: Oleg Alexandrov, via Wikimedia Commons)

This is, in simplified terms, the concept which is actually applied when “weighing” astronauts. In reality, all this looks a bit more complicated, of course, but the basic principle of this method is indeed as described above.

Astronaut Tom Marshburn in steady state. The Space Linear Acceleration Mass Measurement Device (SLAMMD) will later reveal his actual mass.
(Credit: NASA)

It’s not that difficult to put up the differential equations describing the oscillating astronaut on the “spring pendulum”. For this purpose, some approximations are made – for example, it is assumed that the spring force increases in a linear manner with the deflection of the mass (Hooke’s law). This results in the Space Linear Acceleration Mass Measurement Device (SLAMMD) only working for masses between 43 and 109 kilograms (90-240 pounds).
All we want in this article is to understand the basic principle of the SLAMMD. So why not taking some more approximations? (Physicists really like to do this! 😉 ) Let’s suppose that the apparatus itself doesn’t have any mass – only the clinging astronaut has. Also, we will ignore every kind of friction. These approximations simplify our equations while not warping the basic statements.

As mentioned earlier, the starting equation is the one describing the undamped harmonic oscillator (for those of you who want to know).
A possible solution of it is:

$x(t) = A\cdot \cos\left(\sqrt{\frac{k}{m}}\cdot t\right)$

The $x(t)$ just gives you the deflection from zero position at a given time $t$. $A$ is the deflection from zero position at the moment when the pendulum is “wound up” for the very first time. The cosine in the above equation immediately indicates that the mass (i. e., the astronaut) will begin to oscillate after being released. How fast does the cosine “oscillate”? Well, this is determined by the radical term $\sqrt{\frac{k}{m}}$ in its argument, which is, again, determined by the spring constant $k$ and the mass $m$. The details shouldn’t bother you here. The only essential thing is that all occurring quantities and variables are known – except the astronaut mass $m$. This is why it’s possible to find out the astronaut’s mass by measuring the oscillation behavior and subsequently rearranging of the equation. Doing this isn’t that difficult – however, there’s a dedicated software on an ISS laptop just for this purpose.

This software plots the $x(t)$ curve and determines the astronaut mass from it – with an accuracy up to $\pm$230 grams (0.5 pounds). The slower the astronaut oscillates, the more mass she or he has. This is also nicely apparent from the mathematical solution, where the mass $m$ is in the denominator. If it grows larger, the radical term gets smaller – and since the radical terms represents the oscillation’s (eigen) frequency $\omega_0$, the osciallation gets slower too.

Thanks to Sir Isaac Newton, astronauts onboard the ISS can know how much candies they are allowed to eat.
(Credit: NASA, JAXA)

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• Additional informations on the SLAMMD can be found here.
• If you are into spring pendulum mathematics, you should probably check out this Wikipedia page.