Theorizing a Zero-G Experiment

[1d20]

I recently saw How the Universe Works, and was impressed by the video of astronaut Don Pettit’s sugar-and-salt experiment. (6 minutes into the show. Put sugar and salt in a plastic bag, and watch as the grains visibly gravitate toward each other in zero-G.) I’m trying to express this process with math, but the numbers I’m coming up with are way too big. Here’s the formula I derived:

Assuming two masses (m) separated by some distance (d) in zero-G conditions, how long does it take for the two masses to meet halfway?

First I take the distance equation and solve for time: $t = \sqrt{2d/a}$

Now I modify this equation to take into account the fact that each mass only has to move half of $d$: $t = \sqrt{d/a}$

Now I use the G-force equation to derive each mass’ acceleration: $a = F/m = Gm^2/d^2 / m = Gm/d^2$

Now I plug the second equation into the first: $t = \sqrt{d/ (Gm/d^2)}$ $= \sqrt{d^3/ Gm}$

Being extremely liberal, I’ll assume each grain is 1 gram and the two are separated by 10 cm. The equation says it takes them 34 hours to drift together, but that's way too long. The video isn’t time-lapse, and it clearly takes only moments for gravity to pull the grains together.

I suspect that I need to do some integration, but I have no practice defining integrals. Obviously acceleration is the variable that has to be integrated, but with respect to what? I’m not sure I can even do set up the proper integration without going back a few steps. Anyway, help is appreciated!

 

[Nabeshin]

You are incorrect in assuming that it is gravity responsible for clumping. (Note: your analysis is incorrect, for it assumes constant acceleration. A more full analysis would involve solving a differential equation, as you note. It's more complex than I want to get into here.) What is important is that as you note the free-fall time for these two objects is really large. Another way of arriving at your result is using dimensional analysis, we have three quantities: G, m (the mass of the particles), and d (their separation). Forming a time out of us leads us to conclude $t \sim \sqrt{\frac{d^3}{G m}}$. The result is the same, but I haven't assumed anything. In reality, we're only missing a factor of pi/2!

At any rate, using my formula for d=1cm, m=1g, and the standard value for G, I get t~ 1hr. You can quibble about these values a bit, but the point is, it's not on the order of a few seconds.

What I suspect IS responsible, is simply the clumping due to friction between bodies when they meet. They are all floating around more or less in random motion, and they happen to collide every so often. When they do, they latch onto each other. Thus the object becomes larger and presents a larger cross section to all the other particles floating around, so it will tend to grow even without this gravitational interaction.

 

[Lsos]

As Nabeshin said it's not gravity that causes them to clump. I suspect it's electrostatic force due to, as again Nabeshin mentioned, the friction between the particles. They rub against each other, they gain different charges, and they attract (I hope my analysis there is correct). Once they get big enough (much MUCH bigger), gravity can take over.

 

[1d20]

That's disappointing. Thanks anyway.

 

[pmacias]

Thank you for sharing your thoughts and calculations on the zero-G experiment shown in How the Universe Works. It is always exciting to see the principles of science in action, especially in a unique environment like zero gravity.

I understand your desire to express this process with mathematical equations, as it can help us better understand and predict the behavior of objects in zero gravity. However, I believe your current formula may not accurately represent the situation in the video.

Firstly, the equation for time you have used assumes that the two masses are accelerating towards each other with a constant acceleration. In reality, the acceleration would decrease as the two masses get closer together, due to the inverse square law of gravity. Therefore, the time it takes for the two masses to meet halfway would be longer than what your equation predicts.

Additionally, the G-force equation you have used is for the force between two masses, not the acceleration of each mass. In this experiment, the acceleration of each mass would be equal to the gravitational acceleration, which is approximately 9.8 m/s^2 on Earth. Therefore, the equation for time would be t = \sqrt{d/ (9.8m/d^2)} = \sqrt{d^3/ 9.8m}.

However, even with this correction, your calculation assumes that the two masses are initially at rest and are only affected by the gravitational force between them. In reality, the two masses would also have some initial velocity, which would affect the time it takes for them to meet in the middle. This may explain why the experiment in the video appears to happen much faster than your calculation predicts.

In order to accurately model this experiment, you would need to consider the initial velocities of the two masses and use equations of motion to calculate the time it takes for them to meet. This would require some integration, but the exact setup would depend on the specific initial conditions of the experiment.

In summary, while it is commendable that you are trying to express this experiment mathematically, I believe there are some factors that need to be considered in order to accurately model it. Integration may be necessary, but the exact setup would require more information about the initial conditions of the experiment. I hope this helps in your further exploration of this interesting topic.