How to Use Physics to Escape an Ice Bowl

I don’t know who invented this crazy challenge, but the idea is to put someone in a carved-out ice bowl and see if they can get out. Check it out! The bowl is shaped like the inside of a sphere, so the higher up the sides you go, the steeper it gets. If you think an icy sidewalk is slippery, try going uphill on an icy sidewalk.

What do you do when faced with a problem like this? You build a physics model, of course. We’ll start with modeling how people walk on flat ground, and then we’ll apply it to a slippery slope. There are actually three possible escape plans, and I’ve used this model to generate animations so you can see how they work. So, first things first:

How Do People Walk?

When you shuffle from your front door to the mailbox, you probably don’t think about the mechanics involved. You solved that problem when you were a toddler, right? But this is what scientists do: We ask questions that nobody ever stopped to wonder about.

Speaking of which, did you ever wonder why ice is slippery? Believe it or not, we don’t know. The direct reason is that it has a thin, watery layer on the surface. But why? That liquid film exists even below the freezing point. Physicists and chemists have been arguing about this for centuries.

Anyway, to start walking, there needs to be a force in the direction of motion. This is because changing motion is a type of acceleration, and Newton’s second law says the net force on an object equals the product of its mass and its acceleration (F = ma). If there’s an acceleration, there must be a net force.

So what is that force propelling you forward? Well, when you take a step and push off with your back foot, your muscles are applying a backward force on the Earth. And Newton’s third law says every action has an equal and opposite reaction. That means the Earth exerts a forward-pointing force back on you, which we call a frictional force.

The magnitude of this frictional force depends on two things: (1) The specific materials in contact, which is captured in a coefficient (μ)—a number usually between 0 and 1, with lower values being more slippy, less grippy. And (2) how hard these surfaces are pushed together, which we call the normal force (N).

The normal force is kind of a weird concept for physics newbies, so let me explain. Normal means perpendicular to the contact surface. It’s an upward-pushing force that prevents you from plunging through the floor under the force of gravity. If you’re standing on flat ground, these two forces will be equal and opposite, canceling each other out, so there’s no vertical acceleration.

One last note: There are two different types of frictional coefficients. One is where you have two stationary objects, like a beer mug on a bar, and you want to know how hard you can push before you cause it to move. That limit is determined by the static friction coefficient (μ_s).

Then, when the bartender slides your mug down the bar, the frictional resistance—which determines how far it goes—is determined by the kinetic friction coefficient (μ_k). This is usually lower, because it’s easier to keep something moving than to start it moving.

So now we can quantify the static (F_fs) and kinetic (F_fk) frictional forces: