Boomerang-Cycle Lesson Will Come 'Round Back To Me

There was an article in the Moline Dispatch yesterday on making a cardboard boomerang. I thought I'd write a lesson to explain how boomerang comes back to you.

On the other hand, Nick and I saw a young woman the other day talking on cell phone while riding bike. Nick commented that that looked more dangerous than talking on the phone while driving a car. I'm not sure I disagree, but at least you can steer a bike without your hands.* After that exchange, I decided to write a lesson to explain how to steer a bike with no hands.

Luckily the two are related, based on the same principles of rotational motion, angular momentum, and torque, so I can do both at once! So, here goes (If you already know the bicycle stuff, you can skip to "But what about the boomerang?" below):

Linear motion is pretty straightforward. If a force is exerted on an object, there will be a change in its momentum in the direction of the force. Rotational motion is similar: If a torque is exerted on a rotating object, there will be a change in its angular momentum in the direction of the torque.

The hard part comes with understanding the directions of these rotational quantities. The direction of linear momentum is just the same as the direction of motion, but the direction of angular momentum is not so simple.

Take the bicycle wheel, for example. As you ride forward, the wheel rotates in such a way that the top of the wheel is moving forward, the front of the wheel is moving downward, etc. Different parts of the wheel are moving different directions at the same time, so what is the direction of the angular momentum? To answer that question, curl the fingers of your right hand to mimic this motion (a right-hand rule), so the tips of your fingers are pointing the direction the wheel is turning. Now stick your thumb out. Your thumb will be pointing to your left, and that is the direction of the angular momentum.

Exercise for the reader: How would the wheel be oriented if the angular momentum were pointing directly backward? That's right, if the you turned the wheel hard to the left, and it kept rotating, then it's angular momentum would be pointing toward the back of the bike.

Ok, how about the direction of the torque? Suppose your wheel is rolling straight forward, but now you lean it a little to the left, so gravity will be pulling down on it. What effect will gravity's torque have on its motion?

What direction is gravity's torque around the pivot point, which is the point of contact of the tire with the road? We can take the net effect of gravity on the whole wheel as a force of the weight of the wheel directed downward from the center of mass (basically the center of the wheel). To determine the direction of the torque, we hold the fingers of our right hand out straight, angling them up and to the left, just like the wheel is - along the line from the pivot point (road) to the point of application of the force (center of wheel). Now, without changing the angle of your hand, bend your fingers to point straight down. If your fingers don't bend that way, rotate your hand (but still keep it angling up and left) until you can. Now stick out your thumb, and it will point in the direction of the torque due to gravity on the wheel. Your thumb should be pointing straight backward.

What is the conclusion, then? Well that backward torque is going to change the direction of the angular momentum, adding a little backward component to it. And remember, backward angular momentum comes from a wheel turned to the left, so the effect of gravity's torque is to turn the wheel a little to the left.

That's how you steer a bicycle no-handed: lean to the left, you turn to the left; lean to the right, you turn to the right (you can prove that one yourself).

But what about the boomerang?**
(Follow this link for some images that might make the following description clearer.)

What I didn't know - and maybe you didn't either - is that you don't throw a boomerang horizontally like a frisbee, but instead you hold it almost vertical to throw it. Since it is spinning end-over-end, its angular momentum is pointing to the side, like a bike wheel's.

Where does the torque come from? Lift! The arms of a boomerang are like wings on a plane. Plane wings provide an upward force called lift that keeps the plane from falling down. The boomerangs "wings" are (mostly) vertical rather than horizontal, so "lift" force is (mostly) sideways (to the left, say) rather than straight upward, like it is on a plane wing.

Let's suppose that at one moment in time, one of the boomerang arms is pointing up, while the other is pointing down (approximately). The torque direction can be found just as we did for the bike. For the arm pointing up, point your fingers up (because the point of application of the force of lift is on the arm itself), and bend your fingers to the left (the direction of the lift force). Your thumb will point backwards, just like it did for the wheel, so the boomerang will curve to the left just like the wheel did.

But wait! What about the other arm - the one pointing down? If you do the same thing for the arm pointing down, you find that torque direction is forward. If you have a torque backward and a torque forward, don't they just cancel each other and nothing happens?

Nope. They only cancel if they are the same size. But they aren't. The force of lift on the upper arm is greater than the force of lift on the lower arm, so the backward torque is larger than the forward torque, and the boomerang curves to the left.

Why is the lift force greater on the top arm? Relative motion. The lift force depends on the speed of the wing or arm relative to the air. The boomerang as a whole is moving through the air, so all parts are moving relative to the air. But it is also rotating.

If the angular momentum of the boomerang is to the left, like the wheel's was, then the top arm is moving forward due to the rotation, and its net motion through the air is the boomerang's speed PLUS the forward motion due to rotation. The bottom wing is moving backward due to rotation, so its net motion through the air is the boomerang's speed MINUS the motion due to rotation. The top arm is moving faster relative to the air, thus experiencing more lift force, making the backward torque on the upper arm greater than the forward torque on the lower arm.

And so, when all is said and done, the boomerang curves to the left, and continues to curve to the left, making a big circle, and hopefully coming right back to the thrower.


*_{Never mind the people who try to steer their car with their knees!}
**_{For those sticklers out there, I am technically describing a "returning boomerang," and I'm leaving out some fine details like wind, the vertical component of the lift, and stalling.}