Storm in a Teacup: The Physics of Everyday Life(36)

If you put water in a mug, sit the mug on a flat surface, and give it a bit of a push, the water will start to slosh from side to side. What’s happening is that as you shove it, the mug moves but the water initially gets left behind, so it piles up against the side of the mug you’ve pushed. Then you have a mug that has higher water on one side than the other, so gravity pulls the higher water down, and the water on the other side is pushed upward. For an instant the surface is flat again, but the water has no reason to stop moving. It just carries on going up the other side. Gravity is tugging on it as it goes, but it takes a while to stop the water completely. By the time it’s stopped, the water level is higher on the second side than the first, and then the cycle starts all over again. If the mug is sitting on a flat surface, the sloshing from one side to the other will gradually die away, and equilibrium will be reached. But if you’re walking, things are different.

The cycle is where the problem lies. If you try the shoving test with mugs of a few different sizes, you’ll see that the sloshing happens in the same way for them all, but it happens more quickly in a narrow mug and more slowly in a wider mug. A mostly full mug always sloshes the same number of times each second, however big the initial push was. But that number depends on the mug, and the thing that matters most is the mug radius.

There’s a conflict between the downward force of gravity, which is pulling everything back to equilibrium, and the momentum of the fluid, which is greatest just as it passes through the equilibrium position. In a bigger mug, there’s more fluid and it has farther to go, so the cycle takes longer to turn around. The special frequency that each mug has is known as its natural frequency, the rate it will slosh at if pushed and then left to get back to equilibrium by itself.

I spent a while playing with the mugs in my office. I have one tiny one with a picture of Newton on it that is only 1? inches across. Water in this one sloshes about five times each second. The biggest one is about 4 inches across, and it sloshes about three times each second. This large mug is old and cheap and ugly and I’ve never really liked it, but I still have it because sometimes you just need a lot of tea.

When I come out of the tearoom with my full mug and take a couple of brisk steps down the corridor, I start the sloshing. If I want to get back to my office without spilling the tea, I have to prevent this sloshing from growing. This is the crux of the problem. As I walk, I can’t help rocking the mug slightly. If the pace of that rocking matches the natural frequency of the sloshing, the sloshing will grow. When you push a child on a swing, you push in a regular rhythm that matches the rate of the swing, and so the swing gets bigger. The same happens with the tea. This is called resonance. The closer the external push is to the natural frequency of the sloshing, the more likely it is that tea will be spilled. The problem for all thirsty humans is that it just so happens that most people walk at a pace that is very close to the natural sloshing frequency of the typical mug. The faster you walk, the closer to it you get. It’s almost as if the system were designed to slow me down, but it’s just an inconvenient coincidence.

So it turns out there isn’t really a satisfactory solution. If I use the tiny mug, it sloshes too fast for my walking pace to make the sloshing worse and the tea won’t spill. But I want more than a thimbleful of tea. If I use the larger mug, my brisk walking is very close to its natural frequency, and disaster is just three steps down the corridor. The only solution is to slow down, so that the rocking from the walking is much slower than the sloshing frequency.? I feel better for having tried, but the lesson here is that I can’t beat the time-dependency of the physics.

Anything that swings—oscillates—will have a natural frequency. It’s fixed by the situation, and the relationship between how hard the pull to equilibrium is and how fast things are going when they get there. The child on the swing is just one example, along with a pendulum, a metronome, a rocking chair, and a tuning fork. When you’re carrying a shopping bag and it seems to be swinging at a rate which doesn’t match your steps, that’s because it’s just swinging at its natural frequency. Big bells have deep notes because their size means that they take a long time to squish and stretch and squish again, so they ring with a low frequency. We get a huge amount of information about the size of objects by listening to them, and it’s because we can hear how long they take to vibrate.

These special timescales are really important for us, because we can use them to control the world. If we don’t want the oscillation to grow, we have to make sure that the system isn’t pushed at its natural frequency. That’s the game with the tea. But if we want an oscillation to continue without much effort, we choose to nudge it along at its natural frequency. And it’s not just people who use this. Dogs do, too.

Inca is poised and ready, focused on the tennis ball like a sprinter waiting for the starting gun. As I lift the plastic arm holding the ball, she tenses, and then the ball sails over her head and she’s off, a slim bundle of enthusiasm and seemingly endless energy. Her owner, Campbell, and I chat while Inca rushes happily across the scrubby grass. She doesn’t bring the thrown ball back to us, because she’s already got a second tennis ball in her mouth (apparently this is a “spaniel thing”), but when she reaches it she stands guard until we catch her up and lob the first ball farther ahead. After half an hour of non-stop chasing, she finally sits down, tail cheerily swishing the grass, and looks up at us, panting.