Ask a Physicist: Balancing Gravity

Greyson wrote in this week to ask:

What would happen if you put a metal object in between the earth and a magnet that had the same pull as gravity?

The short answer is that it’d be very hard to balance the two forces, and your piece of metal would likely end up either falling to Earth or jumping up to stick to your magnet. In physics, we call this situation an instability—it’s a bit like perching a ball at the top of a hill. As long as it’s undisturbed, it can sit there indefinitely, but small jostles in any direction will cause the ball to roll downhill. At precisely the right point in space, it’s theoretically possible to sit your piece of metal on this “hill” and have it levitate, but it won’t stay there for long.

But what you’re talking about has a more general manifestation, known as the Lagrange Point. It doesn’t have to be a magnet balancing the Earth’s gravitational pull—the moon will work just fine, for our purposes! Imagine standing on the moon and jumping. Ordinarily, you’ll fall back down to the moon’s surface: although its gravitational pull is much weaker than Earth’s, you’re so much closer to the moon that, even though Earth’s pull is what’s keeping the moon in orbit around it, it’s not going to pull you down.

But jump high enough, from the surface of the moon (and we’re talking impossibly high, like 38,000 miles) and you’d hit the Lagrange point between the two, coming up out of the Moon’s potential well, over the “hill” and down into Earth’s.

In this image, it’s the earth and the sun, but L1 still marks the point where the gravitational pull from one balances out the pull of the other, leaving just the necessary force to keep the object in orbit.
Image Credit: Wiki user CMGLee (CC BY-SA 3.0)

Of course, the fact that these bodies are all moving with respect to each other complicates things. If the earth, moon, and sun were all stationary, it’d be a simple task to set the force from one equal to the force from the other and solve backwards to find the distance that satisfies those conditions. But with everything in constant motion, you need to find the point where your test mass still feels enough outward centrifugal “force” to keep it in orbit around the main body at the same rate as the satellite. The math there is a little more complicated, but the upshot is that, for a system like Earth and the moon, you’d have to go 84% of the way from here to there to find that L1 Lagrange point. That distance is going to depend on the relative masses of the objects and the distance between them, which determines the orbital velocity.

Only points L4 and L5 in the above diagram are stable—that is, objects at L1, L2, or L3 will tend to fall out of that orbit as described above if they don’t have some way to accelerate and remain at those points. L4 and L5, where the distance to the two masses is equal, are more stable, and tend to trap massive objects—the Earth/Sun system’s L4 and L5 Lagrange points play host to asteroids and dust clouds. Curiously enough, the L1-L3 points can also be orbited as if there were a massive body there—although these orbits are also unstable, so it requires active management to keep an object there.

The influence of other bodies obviously complicates things as well—for an at-home exercise (and to help get you riled up for Monday’s coming astronomical event), try to imagine where the Earth/Moon L1 point would be during a total lunar eclipse vs. a total solar eclipse. Which would be farther from Earth?

—Stephen Skolnick

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