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Sunday, March 3, 2024

Wait, Is That Backpack … Floating?

This looks crazy. I mean, it’s a cool invention—the idea of the HoverGlide pack is to reduce the awkward bouncing and jolting you get with ordinary backpacks. But watching it makes my brain freak out a little. It kind of messes with your sense of how things should move.

The trick, I think, is that the bag is suspended on some kind of elastic cord. When your body moves up, it increases the force pulling up on the pack. But since the force increases only gradually with the elastic suspension (compared to the quick jerk you’d get from a stiff harness), it takes time for the upward velocity of the load to increase. By the time it starts moving up, your body is already coming back down.

That's the short explanation. But there’s more going on here than meets the eye. Let’s unpack the physics of this thing!

Why Do Backpacks Bounce?

First of all, why is this even needed? If you’ve ever tried to hustle with a heavily laden pack (late for class again, hmm?), you know how it jostles and jolts, all out of rhythm with your motion. But why is that? Shouldn’t it just move up and down with you?

After all, once you leave the ground, there is a gravitational force on both you and the pack. Since the gravitational force on an object depends on its mass, and the acceleration depends on the same mass, objects with different masses should have the same acceleration. This is why a bowling ball and a baseball dropped from a height will hit the ground at the same time. It seems like the backpack and person should also have the same acceleration and "fall" together.

But here's the deal: The connection of the straps to your shoulders acts sort of like a spring. There’s some give in it. Things in real life aren't perfectly rigid. Everything bends and squishes a little bit when there is an applied force. In fact, we can model this squishiness as if it were an actual spring—which is nice, because springs are easy to model!

The key to springs is that the force they exert is proportional to how much they’re compressed or stretched. This is called Hooke's law, and the proportionality constant relating force and compression is called the spring constant, k. You can think of it as the stiffness of the spring.

So here’s my setup: I have two squishy objects, a human and a backpack—think of them as cubes of Jell-O. The human will jump up into the air, and the pack, which sits on top of the human, will have an upward-pushing spring force on it. This repulsive force can be calculated from the change in distance between their centers as they compress. (In the program I actually treat them as rigid but allow their edges to overlap—same effect).

There are three forces acting on the human during the jump. (1) The gravitational force, which equals the product of the human’s mass and the gravitational field (on Earth, g = 9.8 newtons per kilogram). (2) The spring force. Since there is a spring interaction between the backpack and the human, it pushes on both objects with an equal magnitude but opposite direction. (That’s just how forces work. You can call it Newton's Third Law if it makes you happy.) (3) The upward-pushing ground force, initiated by the human cube, that lets it accelerate off the ground. Of course, once the human loses contact with the ground, this force goes away.

But does this make the backpack get off the human during the jump? How about a Python model to see how everything works? Here's what I have. This is just a GIF, but you can see the code here on trinket.io. (Feel free to enter different assumptions and rerun it to see how things change.)

Here you can see that the pack does indeed jump up. If you think about it, it makes sense. During the jump phase of the motion, the human-pack gets compressed, causing a larger spring force on the pack. This larger upward force is needed to accelerate the pack for the jump.

That's all fine and everything until the human loses contact with the ground. At that point, there are only two forces on the human—the downward gravitational force and the downward spring force from the pack. That yields a downward acceleration greater than what it would be from gravity alone. At the same time, the pack is still being pushed up by the spring force. These two things together cause a vertical separation between the pack and the human.

OK, just to make sure everything sinks in, how about a quick homework question? What would happen if you used a backpack with a very low mass? Would this change the separation distance between the human and the pack? (You can use the code above to find the answer.)

How to Beat the Bounce

When we started you probably thought, "OK, the HoverGlide has a springlike shock absorber built into it—end of story." But now we’ve seen that even a regular backpack has a kind of springlike connection in the harness. So the key isn’t having a spring per se, it’s having the right spring.

Specifically, HoverGlide uses a spring with a much lower spring constant k. Because of that, the pack takes a much longer time to accelerate upward, because the spring exerts a smaller force. By the time the human comes back down, the pack really hasn't even moved. Since the pack barely moves, you don't get that jarring impact when it collides with your shoulder.

Here’s a demonstration of this effect with a 1-kg mass hanging from a rubber band.

See how my hand is moving much more than the hanging weight? That's the idea. This is actually a very interesting physics problem. Every introductory textbook looks at something similar but more boring—a mass oscillating on a spring. But what if the attachment point for the top of the spring is also oscillating? I call this the "jiggle spring."

You know what's super great about this problem? There are two ways to model it. The first uses Lagrangian mechanics to determine the motion by looking at the energy and the constraints on the system. That's all I'm going to say about Lagrangian. It’s the best solution, but it would take a bit of explanation to get there.

Instead, I'm going to do what you know I like to do—a numerical calculation using Python. The main idea of a numerical calculation is to break the problem into very small steps in time. At each step, I make some simplifying assumptions to calculate the new position and momentum. (If you want a more detailed tutorial, here is how you would model an oscillating mass with a stable hanging point, not a jiggle spring.)

So here is a jiggle-spring model in Python. The big difference between a simple oscillating spring and a jiggle spring is that I have to move the top mount point with some set frequency (the code is actually very similar). To show you the difference in jiggling, I have two masses. The one on the left has a higher jiggle frequency than the one on the right. (But you can change these values in the code if you dare. Just click the "pencil" icon to edit the code. I dare you!)

Notice that with a higher jiggle frequency, the hanging mass barely moves. I think that's awesome. Actually, you can sort of think of this as a resonance problem. The mass on the spring can oscillate on its own with a natural frequency that depends on the mass of the object and the stiffness of the spring.

So, to design a HoverGlide pack you would need to carefully choose the stiffness of the elastic cord. Too low a spring constant and the pack will hang down, making you look silly dragging a bag on the ground. Too high and you have a normal backpack—and then what's the point?

One other thing to consider: If you pick the exact wrong value of stiffness, the running action of a human would resonate the pack and cause it to oscillate out of control. That would actually be kind of funny.

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