Hard Physics Problem?

Question

You’re sitting on the subway with a cup of coffee on the floor. It never seems to fall, despite the large accelerations and decelerations that the train makes. With what magnitude must the train accelerate (or decelerate) in order to make the cup topple over. Assume that friction is strong enough that the cup does not slide, it just falls. You can think of the cup as an ideal cylinder, full to the brim with coffee that has the same density as water. The dimensions of this perfectly cylindrical cup are diameter = 5cm and height = 12cm.
It will be very difficult to find an exact number for this acceleration, though there is most certainly a distinct number that if the train reaches, the coffee will spill. You do not have to find the exact number, just list steps to solve it.

How does the ratio between the diameter and height affect the cup?
How does friction play into this?
What precision is lost by considering this cup a uniform cylinder?
How does an inclination with the ground affect this situation?

I like where your head's at so far. Imagine that the coffee is contained inside the cup by a lid (which has the same density as the rest of the cup and coffee and maintains the object as a perfect cylinder). My intention with this problem was that no coffee spills, but I like your analysis small amounts of coffee spilling out, so do include it in your response, it clearly shows logical thinking about the physics of the problem. But focus more on the cup toppling.

Answer 1

[PS: read to the end before feeling discouraged]

You have set many conditions:

Do you want the cup to topple over or do you simply want coffee to spill over?

If the cup is filled to the brim, then the only thing keeping the coffee inside is the surface tension of the liquid. This may be affected by the amount (and fat content) of the cream, the sugar, the wax of the paper cup or the dishwashing liquid still present on the glazed surface of the china cup.

Without surface tension, then the slightest acceleration will cause coffee to spill over the brim.

For the cup to topple over, you have to worry about two things:

1. The vectorial addition of the Earth gravity and the subway acceleration. Graph the resultant from the centre of gravity of the "cup+coffee"; if the resultant passes through the bottom of the cup, it will not topple over but if it passes through the side, beyond the edge of the bottom, then the cup will topple.

2. The free surface effect. As we have seen above, the slightlest acceleration should cause some coffee to spill. Once the cup is not longer full to the brim, the coffee will slosh inside, due to the acceleration (and deceleration). This changes the location of the coffee's centre of gravity. The change is always to the disadvantage of stability. You must consider the change in the location of the centre of gravity, due to the free-surface effect before performing the test in number 1 above.


If this is simply a school assignment (i.e., homework) and the course did not cover things like suface tension and free-surface effect, then number 1 is sufficient.

The centre of gravity can be taken to be at the centre of the cylinder. The angle between the vertical and the bottom edge of the cup can be measured. This can help you determine the vector due to the acceleration of the subway that will cause the resultant (acceleration + gravity) to be at that angle. That will be the critical acceleration.

Shortcut: the ratio of acceleration over gravity is the same as the ratio of height (of centre of gravity) to radius of cup.

Acceleration (in m/s^2) / gravity (9.8 m/s^2) =
H (c.of gravity) / radius of cup

Answer 2

It's volume relative to the "push" of the vehicle is what will knock it over - inclination assumed.

Happy Holidays!

Answer 3

you have to use equation