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Soda Can Stability

When enjoying a can of coke, at what point is it the most stable if you put it down on a table?
It’s most stable when its center of gravity is at as low a point as possible. In this way, if it gets inadvertently knocked, it requires the largest amount of torque to cause it to topple, and the lower the center of mass (we can interchange ‘center of mass’ and ‘center of gravity’ because we are in a uniform gravitational field), means that it can withstand a higher range of inclination angles and still have the reactive force to its weight pass through the base of the can; it’s for similar reasons that race cars have low centers of gravity to help them stay stable under high cornering loads.
This is a classic problem, made famous by none other than Martin Gardner. There’s a little more to it than first meets the eye.
As a first approximation, a can is a symmetrical cylindrical shell, and we can approximate the soda inside the can as a similarly shaped cylindrical slug of soda. When the can is full, the center of mass of the combined can and soda is in the middle of the can. So far so good. Now we open the can, and start to consume some of the goodness inside. As we imbibe the soda, it is consumed, and the cylindrical slug of soda gets smaller; the center of mass of the combined soda and can lowers. However, when the can is empty, and all the soda has been consumed, the center of mass will have returned back to the middle of the empty can!
Clearly the center of mass got lower as the soda was consumed, until it reached some minimum, then started to raise again until the can was empty. We want to find what this minimum is.

Real Can

Whilst we’ll solve this generically, it’s nice to use real data, so here are some average figures for a can of soda. A full can of soda (can and drink) has a mass of approximately 384g. An empty soda can has a mass of approximately 15g. We’ll define the mass of the can to be MC, the mass of soda to be MS, and the mass of just the soda in a full can to be MFS. At the start:
MFS + MC = 384g
MC = 15g, MFS = 368g

Center of Mass

The overall center of mass of the combined can and slug of soda can be calculated by taking the masses of the individual components and multiplying them by their positions. Then, we add these together and divide that by the total mass. Because of symmetry, we’ll only concern ourselves with one dimension, and use the base of the can as our reference line.
We’ll define x as the height of the center of mass about this reference line. We want to find out when this is at a minimum (you are can probably tell we’re going to use Calculus).
Here H is the height of the can, and h is the current height of the soda. The center of mass of each of these two components, individually, is in the centroid of each component, and since we’re only worried about one dimension, is halfway up the height of each!
The mass of the soda is directly proportional to the height of the soda left (it’s a uniform cross sectional area), so we can use a simple ratio to represent the mass of the soda based on the height of soda:
We can make the math a lot simpler for ourselves by normalizing the can, and setting H=1 (so that x will just be the ratio [0-1] of the position of the center of mass).
Let’s take this normalization a step further and define a new ratio R, which represents the ratio of the mass of the soda in a full can to that of the can. (For our real world example, R = 369g/15g = 24.6). This is an efficiency ratio; how much soda can you hold for each unit mass of can?
Plugging this in we get:
To find the turning point for this function we differentiate, with respect to h, and set this to zero. (Confirming the second differential is positive we can show that this point is a minimum).
Finding the only (real) root of this we see that:
This tells us the height of the soda when the center of gravity is the lowest, but just what is the center of gravity position x at this condition?
We can find this by substituting the value of h back into the equation for x:
This is the same value!
This gives a fascinating result: The can is the most stable when its center of mass corresponds to the top of the liquid!
The can is most stable when the center of mass is right on the surface of the liquid.

Common Sense

Hang on a minute, this actually makes complete sense, and we could have actually dispensed with the Calculus with just some creative thinking. If the center of mass really was at the top of the surface of the liquid, then adding any more liquid to the can would raise the center of mass above the current position (as this added mass would be added above the current center of mass). Conversely, removing any more liquid from the can would similarly raise the center of mass, as there is now less soda below the balance point. Either way you move you raise the center of mass, so that position has to be the minimum.
This fascinating result has another consequence; it really doesn’t matter the shape of the container! We’d assumed symmetry in our calculations but we’ve just shown with simple logic that the minimum always occurs when the center of mass corresponds with the level of the fluid. Neat!

What does the curve look like?

Using real the real world data described above, here is a plot of the position of the center of gravity x (as a ratio of the Height of the can) to the height of the fluid (plotted also as the ratio against the height of the fluid in the can h).
As expected, the height of the center of mass starts and ends at 0.5 (centroid of the can), and drops to a minimum.
You can see the minimum point in the curve and using the value or R=24.6 we can calculate that, in this case, this corresponds to a soda height of 0.165
For this particular can, it will be most stable when there is only 16.5% of the soda remaining (almost exactly a sixth of the soda remaining). Open your can and take a big gulp before setting it down!
Confirming the minimum occurs when x and h correspond, here is the same graph plotted with the line x=h, and you can see this intersects with the curve at the minimum point.

Change in can weight

As the can becomes heavier with respect to the contained soda, it’s mass becomes more significant in the position of the center of mass, which moves upwards, and the curve becomes flatter showing the diminishing influence the soda.

Optimal Can Dimensions

Have you ever wondered why cans are the shapes they are, and what the optimal dimensions they should take? This article does to some calculus to determine the optimal dimensions.