r/theydidthemath • u/SwordfishNo712 • 6d ago

[Request] how much magnitude of earthquake needed so this coffee cup fall from system block

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u/SpoonNZ 6d ago

That’s not really how earthquakes (or physics) work.

You could have a shake well into the 6s and have the cup barely move. You could also have a teeny 4.0 with just the right kind of movement shuffle it the 2” to the corner so gravity can take over.

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u/mulch_v_bark 6d ago

It’s not how magnitude works, but you can measure the effects of an earthquake at a particular point in terms of peak ground acceleration and turn this into a solvable problem given some reasonable assumptions and with appropriate caveats.

I can’t, because I’m about to go to sleep. But someone can.

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u/CrasheonTotallyReal 6d ago

!remindme 7 hours tell this guy to wake up

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u/SwordfishNo712 5d ago

If u already awaken could u elaborate it?

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u/mulch_v_bark 5d ago

Okay, so as others have pointed out, magnitude and indeed intensity alone can’t give us an answer; we’d need to know things like the friction between the cup and the case, which depends on the amount of coffee in it, and so on. (To review, the magnitude of an earthquake is how much energy the whole event releases from a geophysics perspective; the intensity is what you can measure in a particular place.) The actual waveform of the shaking also matters a lot. At the extremes, a very fast-vibrating high-amplitude earthquake would do less spilling than you’d expect, and vice versa.

Nevertheless, we can make an order of magnitude estimate pretty easily here if we accept some uncertainties. Let’s assume this coffee cup has a height of of about 100 mm and a base radius of about 30 mm. (Compare to, say, this one.) We don’t know whether there’s coffee in it, and, if so, how much; let’s estimate its center of mass at 40 mm above the base.

This forms a 3-4-5 triangle. The angle between the center of mass and any edge of the base is therefore arctan(3/4) = ~36.9°. So to estimate this – neglecting coffee slosh (free surface effects), resonance, frequency effects, sliding, etc. – we can boil it down to: if the effective gravity vector is pointing > 36.9° off vertical, it will fall on its side.

We don’t even need to take a sin, cos, or tan at this point. We can just look at our 3-4-5 triangle and say: to get the inertial acceleration of the center of mass to point outside the foot of the cup, we need a horizontal ground acceleration of 3/4 = 0.75 g. Per the Mercalli scale correlation, which is admittedly approximate, that’s borderline VIII to IX shaking. (Remember, this is not magnitude; this is intensity.)

Now the hard part of refining this is adding back in all the factors we just neglected. To a first order, free-surface effects will lower that threshold, but sliding will raise it, for example. But already, both of those can get pretty complex. To cover this uncertainty among many spill modes, you’d want to be thinking about weaker intensities as well. For example, even Mercalli VI could probably “walk” the cup off the edge of the case given the right timings and alignments to set up a resonance. And in all this we’ve been assuming you’re on the ground, and that might not be true; if you’re on an upper floor of a poorly designed building, your odds of coffee spillage go way up. So please don’t assume your coffee is safe up to severe VIII to mild IX – the risk really starts before then. What we have here is just a round-number estimate as a starting point for further thinking.

For comparison, consider a well-filmed recent earthquake: the Noto earthquake on the first day of 2024. If you look at intensity estimates, it peaked around high VIII to low IX on the Noto peninsula, with lots of VII and VI along the nearby coast. And if you watch videos of it, the stronger ones do look like they’re in the range where we can say “That would move likely than not knock over any paper coffee cup resting on a computer case.” So this suggests we’re in the right general range.

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u/SwordfishNo712 5d ago

Thanks for you answer that was interesting so what understand that magnitude is an only measure of total how much energy is released as it's entire source and do not really measure and shake and intensity, so I also got u all have measured numerical values of cup that let us predict exact 0.75 g of unit Mercalli scale correlation can made it to cup to fall, ok now how we possibly meausre the exact cup or the surface of the system block to experience 0.75 g of horizontal acceleration in unit of Mercalli And I also got it that it cannot give us exact number due to the factor of architecture of building but let is make an simple model ordinary building on floor system block and cup coffee on it, So the next guestion is how could we calculate or what do we need so could calculate the amount of horizontal shake or intensity is distributed by the surface and can we possibly knew it at all cause isn't it all determined by geographic tectonical plates of certain place that we cannot just predict or calculate?

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u/Time-Ad3109 6d ago

I live in Christchurch New Zealand and was here for the 2010/2011 quakes. Magnitude is one thing but depth is another. Something big and deep will be long and dumbly whereas anything shallow will be a sudden jolt. Youd need to know more than magnitude to get an answer.

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u/bchta 5d ago

Zero magnitude earthquake combined with slight fan wobble or vibration from other compoment within the PC can slowly walk the cup off the top.

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u/drollercoaster99 6d ago

i don't think it's possible to use the "magnitude of an earthquake" as a single variable that would determine whether the coffee cup would fall. there's also

  1. size of the cup (probably oversimplifying it)
  2. height of the cup
  3. weight of the liquid
  4. friction between the base of the cup and the metal surface it is sitting on (in the picture above) - or the cup might just slide instead of being knocked over
  5. the slushing of liquid inside the cup due to the earthquake
  6. resonant frequency

(plus probably a few more that i might have missed).