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 (μs).
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 (μk). This is usually lower, because it’s easier to keep something moving than to start it moving.
So now we can quantify the static (Ffs) and kinetic (Ffk) frictional forces:
Now, static friction is what matters for walking. When you push off the ground with your back foot, you need solid traction to propel your body forward. And notice that the relation for Ffs above is an inequality. This says there’s a maximum force you can apply before you lose traction. That’s why in a car, if you accelerate too quickly from a stop, your tires just spin in place.
We could plug in numbers at this point. There are tables of coefficients for all kinds of materials. For rubber shoe soles on asphalt, μs is 0.9. And on flat ground, N equals the gravitational force acting on you—which is your mass (m) times Earth’s gravitational field (g). mg is what you usually call your “weight,” so maybe N is 150 pounds.
Putting this together with our earlier equation, F = ma, you can see that your maximum possible forward acceleration will be greater, the higher the friction coefficient and the normal force:
Now let’s put it on ice! Instead of a nice grippy coefficient of 0.9, rubber shoe soles on ice have a static friction coefficient of only 0.1—practically zero. That’s why you have to walk very, very slowly on ice. Try to lunge over a puddle and you can be confident of landing in the puddle.
Walking on a Slope
Now, what if you want to walk uphill? Let’s go back to the asphalt for a moment. What changes here? Well, the normal force still pushes in a direction perpendicular to the contact surface, but that’s no longer perpendicular to gravity. Here’s a force diagram to show you what’s going on:
In fact, with a tilted surface, the normal force (N) decreases as the incline angle (θ) increases. That’s why you can’t walk up a vertical wall—in case you ever wondered—the normal force falls to zero, which means you have no frictional force at all.
Walking in an Ice Bowl
So now we get to the crux: What if this slope is the inside of an ice bowl? From what you now know, you can see there are two problems: First, the friction coefficient is extremely low. And second, because it’s a spherical bowl, the slope gets steeper the higher you go, which quickly shrinks the normal force. Even just standing still, the maximum angle at which you could remain stationary, without sliding down, would be 5.7 degrees. Trust me, that’s not very steep.
So, you can’t walk out of the bowl. Even less can you run up the wall, because that would immediately take you over your maximum frictional force. If it’s a small bowl, what about jumping out? Well, you know what you need to jump at an angle? Yup, friction.
But it’s not hopeless. We can actually use our knowledge of frictional forces to our advantage. In fact, there are three tricks you could use to get out of this pickle.
Method 1: Don’t Get Stuck
When starting this ice challenge, most people just step on the side of the bowl and slide down to the bottom. Then you’re stuck. Yes, you gain kinetic energy as you slide down the side—but there’s also kinetic friction (even a little bit) which gradually reduces your energy. Here’s what that looks like:
But don’t be a sucker and get stuck; do this instead: As you approach the ice bowl, don’t slow down—speed up! You will again slide down toward the center, but because you started off with some speed you will still be moving at the bottom. This means you can (hopefully) slide up the other side of the bowl and reach the edge before you stop and slide back down. Check it out:
You are free because you never got trapped.
Method 2: Back and Forth
You can walk on ice—you just need to have a very small acceleration. At the bottom of the bowl, the ice is flat, so you can take a few small steps. You’ll soon start slipping back, but instead of giving in at that point, turn around and walk back to the center. With a little bit of friction you can get to the bottom with some speed. Keep going up the opposite side and walk until you lose your grip. Since you started off with a non-zero speed, you should be able to get a little bit higher this time. Repeat this process, back and forth, until you are out:
Method 3: Walking in a Circle
When you’re driving on a highway and you go around a big turn, your traction feels more secure if the road is banked, right? Why is that? Remember, Newton’s second law says the net force is the product of mass and acceleration. But acceleration doesn’t just mean speeding up; changing direction is also an acceleration, pointed toward the center of the circular motion.
And like we said, the normal force is always perpendicular to the surface, so on a banked turn, the road is literally pushing against your tires. The faster you go around, the greater the acceleration and the greater the normal force. And as we saw way up top, the frictional force is proportional to this normal force. So by accelerating on a banked curve, you actually increase the frictional force.
Here’s how you can use this knowledge: Start at the center and move in a tiny little circle. It has to be tiny because the ice is flat. But now you can move into a larger-radius circle with a steeper bank angle. This means you get a greater normal force and thus a greater frictional force. You will basically run in a widening spiral until you get to the edge.
Oh, you know I made an animation for this too. Here it is:
You’ve escaped again. Physics for the win!
The post How to Use Physics to Escape an Ice Bowl appeared first on Wired.
