### Ask a Physicist: Conservation of Mass Violation...In a Bowl of Couscous?

Cal, from Italy, wants to know:

#### "When I add hot water to couscous in a bowl, and then zero out the scale it sits on… …it magically starts increasing in weight over time as it absorbs the hot water! I can understand it increasing in volume, but not in weight. How does this happen?"

Cal,
I love questions like this! It's like a puzzle, where sometimes there's an opportunity to use physics and logic to peer into the inner workings of things and figure out a solution from thousands of miles away. It's a magical, second-sight kind of feeling.

I'll be honest, my first reaction to your question was disbelief. “Oh, there’s probably lag time on his scale’s display and it’s just registering the last of the water he added.” After all, conservation of mass is a pretty fundamental tenet of physics—we’ve never seen it violated in a lab before, so the odds that such a thing should be observed for the first time in a bowl of couscous struck me as pretty slim.

 For our readers unacquainted with North African cuisine, couscous is a grainy, starchy dish which is made from wheat and apparently exempt from the normal laws of spacetime. Think of a cross between quinoa and oatmeal. Image Credit: Khonsali, via Wikipedia (CC BY-SA 3.0)
But as soon as I decided to give you the benefit of the doubt, trusting that your observation was something that needed to be explained rather than rationalized away, I understood instantly what was happening!

When you put something on a kitchen scale, its mass applies a force to the weighing plate. This force is given by the simple formula:
...where m is the object’s mass and g is the free-fall acceleration of an object near Earth’s surface, 9.81 m/s2.

This force on the weighing plate is counteracted by a spring inside the scale. Now, there are different kinds of springs—in some, the coils are naturally all scrunched up, like a slinky, and it takes force to pull them apart. In others, like a bedspring or a shock absorber, the coils are naturally far apart, and it takes force to push them together. The one inside the scale is the bedspring kind of spring.

When you place an object on the weighing plate and the force of gravity pulls it down, it compresses the spring by a certain amount, determined with a formula called Hooke’s Law. How much the spring is compressed depends on the magnitude of the force, but also on something called the spring constant, so the formula looks like this:

...where x is how far the spring gets compressed, and k is the spring constant. The spring constant depends on the material that the spring is made of, how thick it is, and a number of other factors—this number is usually determined experimentally, by applying a known amount of force to the spring and seeing how much it compresses, then using that data to solve for k.

So placing an object on the scale creates a balance of forces, between gravity pulling it down and the spring pushing it back up. Since the object is at equilibrium (not accelerating), we know that the forces pulling it down and the forces pushing it up must be equal to each other. We can express this equivalence mathematically, using the two formulas above:

And this is how the scale determines an object’s weight. It measures how far the spring has been pushed down, does some quick math using the spring constant and the force of gravity (both of which are programmed in by the manufacturer) and then solves this equation for the only unknown variable, m, which it spits out to you on its display!
So how does this translate to magic couscous? Well, I mentioned earlier that the spring constant depends on a number of different things—one of them is temperature! When metal heats up, it becomes easier to bend and deform; this is why blacksmiths have to get a hunk of metal glowing-hot before they can shape it.

When you add your boiling water to a bowl of couscous on the scale, the bowl heats up, and transfers some of this heat to the weighing plate of the scale. If the scale’s weighing plate isn’t properly insulated from the rest of it, this heat can transfer to the spring, and when the metal gets hot, its spring constant drops! Slowly, as the spring’s k value changes, the same amount of force creates greater compression in the spring. The scale isn’t equipped to handle this—it assumes the spring constant is…well…constant. (A reasonable assumption given the name, wouldn’t you think?) The scale sees the spring compressing more and more, and concludes that the weight on it is increasing, leading to the appearance of magic, mass-increasing couscous!

Not to get preachy, but if there’s a lesson to be learned here, it’s this: It’s important to understand how your instruments work, so you can be sure you’re actually measuring what you think you’re measuring. About five years ago (almost to the day, actually) OPERA announced that they had detected neutrinos moving faster than the speed of light, when in reality a cloudy fiber-optic cable had been messing with their signal.

Stephen Skolnick

P.S. After further correspondence with the asker, in the interest of proper science, we have determined that the effect arises with or without couscous, but does not occur when room-temperature or cold water is used, confirming the source of the effect.

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