What If ‘Planetary Alignment’ Really Could Make Brooms Balance?

The broom challenge is back! This fun little trick returns every few years on social media. It’s supposed to show a gravitational alignment (whatever that is) among the planets that allows a broom to stand up by itself. It’s super special and rare. At least that’s what I hear from Caltech high-energy physicists and NASA engineers (to quote My Cousin Vinny).

In my ranking of social media “challenges,” this one is better than the ice bucket challenge and the cinnamon challenge but not as good as the invisible box challenge. And by the way, I wrote about it back in 2012. To quote myself, “All of this has happened before, and all of it will happen again.” It’s sort of silly, but hey, it’s harmless, and it makes people happy. Go ahead and try it!

Will it work tomorrow? What about in six months, when Earth is on the other side of the sun? (Remember it takes 12 months to make a complete orbit.) See, the broom challenge is an invitation to do some great home science experiments. Don’t just believe whatever “NASA” tells you on TikTok—challenge the challenge. That’s what science is all about.

It Works! … But Why?

What you’ll find out is that it doesn’t matter when you try this, because it has nothing to do with the planets and everything to do with some down-to-earth truths about brooms: First, the bristles flex a little bit and act like a spring. So if you get it almost, but not quite perfectly, balanced, the bent bristles will push it back toward an equilibrium point.

But the more important thing has to do with the broom’s shape and center of mass. Think about things that stand up, like a four-legged table. It will stay upright so long as its center of mass is positioned between its support points. Even a two-legged human: When you stand up from your desk, your center of mass is between your feet, so you don’t tip over.

The broom has a bunch of support points—all those bristles. If you position its center of mass within the footprint of the brush, it can remain upright. And crucially, the broom’s center of mass is low, probably just a few centimeters above the brush. Because of that, the handle can lean quite a bit without moving the center of mass very much. It’s very forgiving. Try this with a billiard cue and you’ll see why it’s not a billiard cue challenge.

OK, but what about gravity? Does it have anything to do with the broom challenge? Yes, the broom is interacting gravitationally with the Earth. But that’s the only gravitational interaction of any significance. To see why, let’s do a quick review of gravity.

Basically, there is a gravitational interaction between any objects that have mass—so, that’s just about everything. This gravitational force depends on the product of the two masses and the distance between them. We can model this interaction with the following equation:

Illustration: Rhett Allain

In this expression, m₁ and m₂ are the two masses, r is the distance from the center of one to the center of the other, and G is the universal gravitational constant (with a value of 6.67 x 10^-11 N×m²/kg²). This equation gives the magnitude of the gravitational force. The direction of the force (since force is a vector) is always in a direction along a line connecting the two objects—it’s an attractive force.

As you can see, with r squared in the denominator, the gravitational force decreases very fast with distance. The fact of the matter is, the other planets in our solar system are much too far away to have any effect on broomsticks (or tides or anything else) here on Earth.

What If There Was a Planet Close Enough?

But this raises an interesting question: Could the broom challenge work the way people on social media think it does if there were another planet much closer to us? Maybe even a ginormous asteroid? What kind of force would this require? Let’s model it to find out!

First, we’ll eliminate all the broomy things that make this trick work IRL. No brush at the bottom, no advantageous low center of gravity. For simplicity, I’m going to assume a massless rod with two equal masses at each end. (Don’t worry, we make these kind of weird brooms in physics lab.)

The broom has to stay on the ground, so the net force on the bottom mass has to be downward. But to stay upright, there must also be a net upward force on the top mass. If I draw the forces on these two masses, it might look like this:

Illustration: Rhett Allain

I tilted the broom to the side a little bit so you could see that this upward force on the top mass would straighten it up. Now for the tough part. How close to Earth would a planetlike thing have to be such that the net forces on these two masses pull in different directions?

Suppose there is a giant space rock with mass M_p at a distance h above the ground. That means each mass m on the “broom” has two gravitational forces acting on it—one due to Earth (with a downward value of m×g) and one due to this other orb. The gravitational force from Earth is constant, but if the other planet is close enough, it can pull more on the top mass than the bottom one. Maybe this picture and these equations will make things clear:

Illustration: Rhett Allain

It’s not obvious how to make this work, so let’s just pick a value for the size of the planet. I’ll assume it has a mass of 10²⁰ kg. (Yes, that’s small for a planet—about a thousandth the mass of the moon—but I want to get it close to Earth.) Now we can plot the gravitational forces on the top and bottom of the broom depending on how far away the planet is. In the graph below, a negative force means a mass will be pulled down toward Earth; a positive force means it will be pulled up toward the other planet.

Two things to notice: First, the forces on the top and bottom are pretty much the same—at this scale, the two lines are on top of each other. (Hover your cursor over the curve to see the exact values.) Second, the net force is close to zero with a planet altitude of about 26 kilometers. (Here is the Python code I used to make that graph, if you want to try different assumptions.)

If I zoom in by rerunning this calculation with a smaller variation in altitude, I can see where the forces on the top and bottom of the broom have different signs. Here’s what that looks like:

Check it out: At an altitude of 26.089 km (2.6089e+4 meters on the horizontal axis), there’s a downward (negative) force on the bottom of the broom and an upward (positive) force on the top of the broom. Boom. That did it. Balancing broom.

Oh, but wait. Now we have this massive object above the Earth’s surface. I assume that it’s in orbit, but at an altitude of 26 kilometers it’s in the atmosphere. This means it would be plowing through the air, causing massive winds and other strange disturbances. Also, things on the surface of the Earth would barely fall. Objects and people over 1 meter tall would be pulled up toward this gravity invader. Life would suck. But at least you would be right about that balancing broom.

What If ‘Planetary Alignment’ Really Could Make Brooms Balance?

Author: sgadmin