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Friday, February 16, 2024

# Wait a Second, That Table Has No Legs …

Maybe you’ve seen one of these “floating” tables on the internet. They look crazy because at first glance, it seems like the table is standing up on strings instead of on solid legs. Which is impossible, right? I mean, you can pull strings to make something happen, but everyone agrees that pushing on a string is futile. So why doesn’t it collapse?

Of course it’s not magic, it’s just physics. This structure is an example of a tensegrity system—a term coined by Buckminster Fuller—which means that its integrity, or stability, comes from balancing elements under tension.

Here’s one that I made out Lego blocks Yes, I can even put a book on top of it.

If you look closely and think about it, you’ll start to see what’s going on here. Whereas an ordinary table stays up because the tabletop pushes down with the weight of gravity on some rigid legs, this one is held together by a balance of forces pulling in different directions. Those strings on the left are actually pulling up!

Let's figure out exactly how this magic table works, then I’ll show you how to make one of your own to astound and amaze your shelter-in-place mates.

Two Conditions of Equilibrium

If an object is at rest (meaning that it’s not accelerating), we say it’s in a state of equilibrium. This means that the following two conditions must be true:

The first equation says that the total force on the object (Fnet ) must add up to the zero vector. Yes, force is a vector (meaning that it’s defined in more than one dimension), as indicated by the arrow over the symbol. Same for the zero vector, which just means that the total force has to be zero in all directions.

The second equation is a little more complicated. It says that the total torque (τnet ) about some point o (whatever point you want) must add up to the zero vector. These two zero vectors are different in that they have different units—newtons for force and newton-meters for torque.

Torque is complicated, but here you can just think of it as a "twisting" force. The value of a torque depends on the value of the force applied and where it’s applied. Here is a simple example. Suppose you are pulling on the handle of a wrench to tighten a bolt, like this:

This would produce a torque (around the bolt) in the clockwise direction with a magnitude of:

Here F is the force applied, r is the distance from the axis of rotation, and θ is the angle at which you pull. (If you pull straight down here, sin 90° = 1 and this simplifies to τ = Fr.) So, there you have it. That's torque. If an object is in equilibrium, then the sum of the clockwise twisting torques must be equal to the counterclockwise torques.

How It Works

Now, let's see how this idea of equilibrium works with the floating table. Here is a simplified side view of the structure, along with a separate diagram of the forces just on the top part.

You can see three forces acting on the table. The first is the downward-pulling gravitational force (mg). Although the gravitational force interacts with all parts of the tabletop, it turns out it's equivalent to having just one force located at the center of gravity (derivation here).

The next force is labeled T1. This is the upward-pulling tension from the blue bracket. The upward tension in this string in the middle is what holds the whole thing up. Finally, there is another tension, labeled T2. This is a downward-pulling force. Yes, you have to pull down here to keep the table upright; otherwise it would tip over to the left.

(Really, there is another downward-pulling string on the right side that you can't see in this view, but we can just combine the two for the analysis.)

Now, we want the top piece to be stationary, so we can put these forces into our equilibrium equations. Since these three forces are all in the vertical (y) direction, we can ignore the horizontal (x) dimension. That simplifies things. Here are the total forces in the y direction:

Really, this doesn't tell us much. All it says is that the upward-pulling tension has to be equal to the two downward forces (gravity and the other tension).

But what about the sum of the torques? If the object is in equilibrium, you can pick any point on the object to calculate the torque. I'm going to pick point o, where the upward-pulling string is attached. And I’ll say clockwise torques are negative values and counterclockwise are positive.

To get the torque resulting from each force, remember that τ = Fr. But since the distance (r) for T1 is zero, this tension results in zero torque.

So now, with only two other forces, the only way for their torques to offset is for one to pull clockwise and the other to pull counterclockwise. T2 is pulling down on the right side, which creates a negative torque around point o of T2 r2. But the gravitational force mg also pulls down—we can’t change that. That means the center of gravity of the top platform has to be on the other side of the central support string. So here’s our equilibrium torque equation:

That’s the key to the whole thing: The center of gravity of the “floating” tabletop and the downward force T2 need to be on opposite sides of the central suspension string. It’s actually not that complicated, right?

Now that you understand how it works, you can build one yourself. In this video, I'll show how to do it with just the kind of ordinary Lego pieces you probably have at home.

In theory, you could also build a floating table with only the upward pulling string in the middle, if the center of gravity was exactly above the point where the string is connected. But it would be unstable. With just a tiny push, the center of gravity would shift to the side and the whole thing would topple over.

Super-Size Me

Could you stack whatever you want on the top of this table? Nope—there’s a limit to the maximum tension in the string (and in that little support hook). As you add mass on top, the downward pulling string might have to increase in tension to prevent it from tipping over. Then the upward pulling string has to compensate for the added load as well as the extra tension pulling down to balance it. If this force is more than the string can handle, that's it—it will break and crash.

What about a super-sized floating table that could support a car? Would that be possible? Yup. You’d just need to make sure both the platform and the cables are strong enough to exert enough tension without breaking. It would be pretty cool to see.