The Physics of Almost Whacking Someone With a Bowling Ball

Every once in awhile a new science show comes on TV. I find some of them pretty good and others not that great. I was pleasantly surprised to find Outrageous Acts of Danger on the Science channel features a reasonable amount of science and makes it interesting. It does this by making otherwise common science demonstration absurdly dangerous.

One recent episode riffed on the classic physics demo in which you hang a heavy ball (bowling balls are common) on a wire and release it near someone's head. The ball swings away in an arc and returns, barely missing the person's face. It helps people understand a harmonic oscillator and conservation of energy (and maybe scares them just a little, too.). The guys at Outrageous Acts of Danger did it with a one-ton ball because a one-ton ball is pretty outrageous. But the physics are the same.

Energy and a Swinging Ball

You don't actually need energy to model the motion of a swinging ball (aka a pendulum), but it makes things easier. As usual in cases like this, I'll start with the work energy principle. (If you need more generic description of energy, this older post should do the trick .) The work energy principle states that the work done on a system is equal to the change in energy for that system. Ah, but what is work? It is the product of a force in the direction of a displacement. You can write it all out like this:

In the second equation, Δr represents the distance the object moves and θ represents the angle between the force and the displacement. But what about the energy? This gets complicated because it depends on the system. If I pick a system consisting of Earth and the ball, I have two kinds of energy: kinetic energy, which depends upon the speed of the ball, and gravitational potential energy, which depends upon the height of the ball.

The kinetic energy term is fairly straightforward. For the potential energy, g is the gravitational field (9.8 N/kg here on Earth) and y is the vertical distance above some point. It doesn't matter what point is because the only thing relevant to the work energy principle is the change in energy. Measure it from the same point each time and the equation works just fine.

For this system, are there any forces doing work? Nope. The tension on the ball provides the only external force, and this force does zero work. As the ball moves, the tension remains perpendicular to the direction of ball. This means that the angle θ is 90 degrees and the cosine of 90 degrees is zero. See? No work done. The change in kinetic energy plus the change in potential energy must be zero.

Let me use an example. Suppose I release this pendulum from the top of its arc. At this point, it has zero kinetic energy because it's not moving. It also has zero gravitational potential energy if the position y = 0 meters (I am doing this because I can and you can't stop me). That means that the total energy at this starting point is zero joules.

As the ball starts to swing, the y value is negative (since the pendulum is lower than where it started). This means it has negative gravitational potential energy. But since the total energy must add up to zero joules, the kinetic energy must be some positive amount and the ball is moving. The lower it goes, the more negative the potential energy and thus the greater the kinetic energy. At the bottom of the swing, the ball is moving at its greatest speed.

As the ball swings back up through its arc, the opposite happens. The kinetic energy decreases as the potential energy increases. However, the ball can never exhibit more than zero joules of total energy because there is no work done on the system. Wait. There is, actually. I left one force out of the explanation: air drag. As the ball moves through the air, the air pushes back against the ball. This negative work on the system decreases the total energy. As the ball completes is arc, it ends up just a bit lower than where it started.

The One Ton Pendulum

Back to Outrageous Acts of Danger. Anyone can make a small pendulum. But what about a truly massive one? That's what makes this demo so cool: a one-ton ball. (I guess that would be 907 kilograms, unless the guys on the show mean one metric ton, which would be 1,000 kg.) Given that mass, this ball will have it greatest amount of kinetic energy as it reaches the bottom of its arc. Let's say that it drops two meters from the top to the bottom of the swing. The kinetic energy in the bottom would exceed 17,000 joules. For the sake of comparison, if you stand up right now, the increase in gravitational potential energy is about 350 joules.

But look past the energy to the danger. Imagine standing with your head near the starting point. Move just 2 centimeters closer and that ball will whack you. It'll be moving slowly, but with that kind of mass, it'll knock your teeth out. Now, I definitely positively absolutely do not recommend putting your face in front of a swinging ball. But if you really want to try this demo to impress your friends, I will offer some advice to minimize the risk of you breaking something.

First, you need a mass on a string. I don't recommend a one-ton wrecking ball. A bowling ball works nicely, or maybe a softball if you want something smaller. You'll need some way of attaching a cable to the ball, and that probably means screwing something into it. That renders it useless for bowling or softball. You've been warned.

Make sure the cable is secure, then hang it from something. A hook in the ceiling works. If you can suspend the ball from at least 3 meters of cable, it looks cooler. You want to hang the ball in such a way that it's just a few inches from the wall at the start of its swing.

This is important, because you're going to have a friend stand against that wall so the ball just touches his (or her) chin or nose at the highest point of its arc. The wall is important (and too often left out), because it ensures the person stays put and doesn't move forward into the path of the ball. This has been known to happen. It isn't pretty.

My high school physics teacher put his own clever twist on this experiment. He'd set everything up like I just explained, but instead of asking a student to stand in front of the wall, he'd just release the ball—but give it an imperceptible push so it started with non-zero energy. The ball would crash into the wall with a thud on its return. Then he'd ask for a volunteer to stand in front of the ball (somehow he always got one) and repeat the experiment without juicing the ball. Of course it would complete its arc and narrowly avoid hitting the kid in the face. It made the experiment more exciting. That might be why I still remember it after all these years.