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The Physics of Going Fast—but Not Too Fast—on a Giant Slide

You might have seen some of the viral videos of this giant steel slide at Belle Isle Park in Michigan, which temporarily closed this summer because riders were sliding so fast they were losing contact with the track. (No one was injured, and the attraction reopened again in late August after park operators made some fixes.) Unlike an ordinary park slide, this one is many times taller, wide enough that six burlap-sack-wearing people can ride at a time, and it’s got ripples, which ensure that each slider will go “Wheeeee!”

This gives us a great opportunity to explore the physics of a giant slide.

Straight and Simple Slide

One of the things we like to do in physics is start with a simple situation and then add stuff to make it more complicated. So before we get to the physics of a slide that has alternating curved sections, let’s start with just a plain straight slide. (These can still be fun, as long as they don’t get too hot.)

As a rider moves down the slide, there are just three forces to consider: the gravitational force, the force perpendicular to the slide (we call this the normal force), and the frictional force. Here’s a diagram that might help:

Illustration: Rhett Allain

Let me start with the simplest, the gravitational force, which is the downward-pulling force due to the interaction between the person and the Earth. The nice thing is that it doesn’t change during a sliding motion. It’s always pulling down with a magnitude equal to the product of the mass of the human (m) and the gravitational field (g = 9.8 newtons per kilogram).

Next we have the normal force (N). This is a contact force that prevents the person from falling through the slide. It’s always perpendicular to the surface of the slide, but its magnitude can change depending on the angle of the slide and the motion of the person. For a given straight slide (like the one above), however, it’s constant in both direction and magnitude.

Finally, we have the frictional force (Ff). This one is a bit more complicated. The frictional force is always parallel to the surface and its direction is against the motion of the person, slowing them down. In the example of a straight slide, that direction is up the slide. (Since the person will be sliding, not stationary, we call this “kinetic friction.”)

Also, the magnitude of this force depends on two things: One is the two surfaces that are interacting, which we call the coefficient of kinetic friction and note as μk. The other is the normal force (N). If you push down on a block sitting on a table, you will increase the normal force, which in turn increases the frictional force. (This is going to be important later.)

We can write the magnitude of this frictional force as the following equation:

Illustration: Rhett Allain

Thanks to all three of these forces, a person will accelerate toward the bottom of the slide. The value of this acceleration depends on both the incline angle of the slide (θ) and the coefficient of kinetic friction.

For a normal playground-type slide, you would like the value of this acceleration to be fairly small, so that by the time the rider gets to the bottom, they aren’t going so fast that something awkward happens—like crashing into other people or tumbling to the ground.

Upward Curved Slide

OK, but that’s boring. And who wants a boring slide? Let’s consider a more interesting ride, like a slide with up and down curves. (Yes, like the one in Detroit.) This makes for a more complicated physics problem, so let’s start off with just the part of the slide that curves upward.

First, we need to talk about the acceleration of an object moving in a circular path. The most common idea about acceleration is that it describes how an object changes speed. An acceleration of 1 meter per second per second (m/s2) says that the speed of an object will change (either increase or decrease) by 1 m/s every second.

If you are moving on a straight slide with an acceleration of 1 m/s2, you could go from a speed of 2 m/s to 3 m/s in just one second. That probably makes sense when thinking about acceleration. But … acceleration is actually a little more complicated than that. Here is a better definition of acceleration:

In this formula, both acceleration (a) and velocity (v) are vectors. (That’s why they get that cool arrow over their symbols.) For these quantities, direction matters. If you have a car driving 20 m/s while going east, this is a different velocity than a car going 20 m/s north, since they are traveling in different directions. This definition of acceleration says that it depends on the change in the vector velocity. (We use the delta symbol (Δ) symbol to denote change.)

So, imagine that the car driving 50 m/s is going east and then turns to go 50 m/s north over a time interval of 1 second. How would you determine the acceleration?

It’s not a super trivial problem, since it requires vector subtraction, but it’s not impossible either. If you have an object moving in a circle of radius (R) with a speed of (v), then the magnitude of the acceleration would be:

Illustration: Rhett Allain

That “c” subscript on the acceleration stands for “centripetal,” which literally means “center-pointing.” The direction of this acceleration is pointing towards the center of the circle. That’s important for our curved slide.

Now let’s consider a person on a slide with an upward curve. Here is a diagram showing the forces with a centripetal acceleration. (I’ve replaced the person with a box because it’s not so messy.) The gray C-shape represents the person’s path along the slide, but you can also think of it as part of a giant imaginary circle, representing the circular path their body would take if they left the slide and kept moving. The dot represents the center of the circle, or the direction in which their acceleration is pointing.

Photograph: Getty Images

Although this might look like the same force diagram as the one for the straight slide, there’s a really big difference. For the straight slide, the person has an acceleration vector pointing down the slide, and only down the slide. For the upward-curving slide, the person is still accelerating down the slide, but since they have a circular motion, there is also an acceleration pointing towards the center of the circle. In this case, this means the direction of the change in velocity vector is upwards. Or at least, up-ish.

Why does that even matter? Well, now the normal force (N) has to push with a greater force than it did on the straight slide to create an acceleration in the direction of the center of circular motion (up-ish). With a greater normal force, you also get a greater frictional force. And on top of all that, this normal force changes with both the position of the person on the slide (because the angle of incline changes as they move along the slide’s curve) and with their speed.

It turns out to be a nontrivial problem, but I solved this for you anyway. Here is a plot of a human’s trajectory on a circular curved slide, and you can see how fast they go with a high (0.6) or low (0.1) coefficient of friction. (Remember, that’s the Greek letter “mu,” or μ, up above.)

Video: Rhett Allain

The blue curve shows the rider’s motion with very low friction (a slippery slide), and the red is with more friction (a stickier one). Clearly the slippery slide is faster—you can see that the rider goes farther in the same amount of time.

Speed is the important aspect of a slide—it’s what people want, and it’s what causes a problem when there’s too much of it.

Downward Curved Slide

If you want a slide with multiple humps, then you need both upward- and downward-curved parts. So let’s look at the motion of a person on a downward-curved part. This is very similar to an upward curve, but I’m going to draw a force diagram anyway:

Illustration: Rhett Allain

Really, the only difference is that this is a downward-curving path with the center of this circular curve below the slide instead of above. (Once again, the gray C-shape represents the path of the rider on the slide and the circular trajectory of their body, and the dot is the center of the circle.) That means the centripetal acceleration is also in the downward direction and towards the center of the circle.

Since the acceleration switched directions, the normal force (N) has to be less than the radial component of the gravitational force, which is pulling the person downward toward the Earth. What happens when the normal force gets smaller?

Remember that in order to get a rider to move in a circular path, there needs to be a net force pointing towards the center of the circle, which is down-ish for a downward-curving slide. Since the frictional force is always tangent to the slide path, this net radial force, which we call the centripetal force, is composed of the normal force (pushing away) and a component of the gravitational force (pulling towards the center).

If the speed of the rider is slow enough, you don’t need a very large centripetal force to move them in a circle. The component of the gravitational force alone could be enough to achieve it. The normal force from the slide could just be some small value pushing away.

If the rider’s speed gets too fast, then the gravitational force alone wouldn’t be enough to produce circular motion. You would need the normal force to also pull toward the center of the circle. But slides don’t do that: They only push away. That means that the sliding human wouldn’t actually move in a circle, but instead along a parabolic path as they leave the surface of the slide and become airborne—at least for a short time, until they crash back into the slide. That’s what happened to the Detroit slide riders.

Let’s model the motion of a person on a downward-curved slide. I’m going to start with a rider at the top of a curve. You can see that at some point the person flies off the track and becomes a free-falling projectile:

Video: Rhett Allain

The person’s speed as they begin their ride is important. If a person starts the downward curve with a high enough speed then they will fly off the track—but the exact value for the speed that will cause the person to come off the track depends on the starting and ending angle of the slide’s curve.

If you want to keep your riders on the slide, you need to increase the coefficient of friction between them and the slide. In the end, the Michigan Department of Natural Resources, which runs Belle Isle Park, posted a Facebook video explaining the updates they have made: “We have scrubbed down the surface and started to spray a little water on the slide between rides to help control the speed,” they wrote. They also urge the riders to lean forward—which a park employee demonstrates in the video.

Why water? Water is actually kind of sticky, so just adding a little bit of it could increase the friction due to its cohesive nature. (Of course, adding enough to create a full-on waterslide might reduce the friction and make the rider even faster—but that would take a lot more water.)

Leaning forward could help ensure that each rider’s weight is on the cloth sack on their legs. The sacks are made of burlap, which is scratchy and provides some friction—and because all the riders have to wear these sacks, that makes for a more consistent, known surface than whatever clothing the riders happen to be wearing. Asking them to lean forward makes sure the burlap is in contact with the slide, not the material that makes up the person’s shirt—which is what would happen if they leaned backwards.

If the park operators want to get even more creative, another option would be to have riders slide while wearing something other than those burlap sacks—maybe something with a little bit of rubber to increase the friction interaction. It’s also possible that a coat of paint could increase the coefficient of friction.

Happy sliding!

 

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