Angular momentum is a hard concept to teach in introductory physics courses. This is partly because it is a quantity known as a *vector* that has both a magnitude and a direction, and adding and subtracting and multiplying vectors is more complicated than with scalars, like mass. Other things like force and velocity are also vectors but in those cases the directionality at least is intuitively obvious. The direction of the force vector is in the direction along which the force acts and the direction of the velocity vector is the direction in which the object is moving.

But when it comes to the angular velocity of a rotating object, which is also a vector, the direction of the vector is given by something called the right-hand rule. To visualize this, curl the fingers of your right hand into a fist in the configuration that we call ‘thumbs up’. Then orient your hand so that the curling of the four fingers corresponds to the rotational direction of the object. Then the thumb will point in the direction of the angular velocity vector. So the hands of a clock, for example, rotate clockwise (duh!) but the angular velocity vector of the hands points directly *into* the plane of the face of the clock.

Why this weird system? It is partly a result of necessity and partly a result of convention. But it works extremely well.

Finding the angular momentum is a little more complicated. For an object that is symmetrical about the axis of rotation, which is what I will consider here, the angular momentum vector is parallel to the angular velocity vector. This is true in the case of a wheel rotating about the axle passing though its center.

Things get a little more complicated when we relate the *change* in angular momentum to the cause of that change, known as the torque. Take a look at this nice demonstration of the relationship between the direction of the torque and that of the change in angular momentum of a spinning wheel.

According to the laws of physics, the direction of the torque has to be in the same direction as the *change* in the angular momentum. Since a vector has both magnitude and direction, a vector can change just by changing direction without changing its magnitude, and that is what is happening here. The torque is causing the axis of the wheel to rotate. Since the direction of the angular momentum (and of the torque) is not intuitive, the direction of the change in it is even less so. It all works out nicely but it takes a while for students to really grasp it.

I have done a similar version of this demonstration using a spinning wheel and standing on a rotating turntable but I like this better for two reasons.

One is that he uses a motor to spin the wheel to a high rpm producing a much larger angular momentum. The other is that hanging the wheel at one end from the rope makes the direction of the torque easier to visualize and makes the effect more dramatic because it is visually arresting to see a heavy wheel suspended from a cord attached at only one end of the axle remain with its axis horizontal. This only happens if the wheel is spinning and it causes the axis of the wheel to rotate.

timberwoof says

Very nice.

There’s a thicket of disagreement about angular momentum in discussions about motorcycle steering. A properly trained motorcyclist will, when wanting to turn right, push the right handle bar away, as though steering to the left. A combination of forces (the changing angular momentum and the tire outtracking below the motorcycle’s center of mass) will cause the motorcycle to lean to the right. Now the motorcycle is on its curved sidewalls, so it turns right.

As the professor said, none of this is intuitive.

mnb0 says

Excellent indeed.

ollie says

Thank you so much. This is not intuitive, but it is impressive to see it demonstrated.

Peter B says

My high school angular momentum demonstration works like this*. Get the HS (American football) quarterback on a swivel stool. He is handed a pair of 5-pound weights and is told to hold them with arms spread out. He practices pulling the weights to his chest. With the weights held out two of his friend spin him fairly fast – perhaps as much as 30 RPM. Then on command he pulls the weights to his chest.

His spin rate increases. I hope he dies not embarrassed himself by falling off the stool.

*I do not teach high school physics. But I do have a simple way to solve complicated position, distance, velocity and acceleration problems.

Curt Cameron says

That video is a nice demonstration

thatit happens.I was a pretty good physics student in college, and I came away with a good intuitive feel for most of it, especially the Newtonian kind. However, this gyroscopic precession always flummoxed me. I could do the math and get the right answer with the vector cross product, but I never grasped it intuitively.

One day someone explained it to me in terms of a tetherball circling the point where its string is attached, and if you apply an impulse upwards at one point of its orbit, the orbit won’t then tilt up with that point as the highest point of the orbit, but instead you’ve given the ball a little upward speed at that point, and the circle will tilt sideways to the point where you applied the force, and the orbit’s tilt will then be 90 degrees past where you applied the force.

Now a rigid disc isn’t a tetherball exactly, but that explanation does begin to make sense. I’ve accepted that that’s as close as I’m ever going to get as an intuitive explanation for it.