6.2: Effects of Torque
Gyroscopic Precession
Back in Section 1.6 and Section 1.7 , we discussed circular motion at constant speed as motion that occurs because a net force pulling an object toward a central point causes the object's velocity vector to only change direction, and not magnitude. At the time, we hadn't yet discussed momentum, but clearly we can now replace "velocity vector" in the previous sentence with "momentum vector." We can write Newton's second law ( Equation 4.1.4 ) in terms of the changing magnitude and direction of the momentum:
Circular motion at a constant speed would exhibit no change in the magnitude of momentum – the first term in Equation 6.2.1 is zero – while all of the force would go into changing the direction of momentum. As we saw back in Section 1.6 , the two terms in Equation 6.2.1 are always perpendicular to each other, which means that the net force on an object going in a circle at a constant speed is always perpendicular to the momentum vector.
None of this is new to us, but as we have been doing for the last two chapters, we will now look at the rotational equivalent of this behavior. Switching Equation 6.2.1 to the rotational equivalent gives:
We are already aware of how a net torque can change the magnitude of an object's angular momentum – speeding up and slowing down rotation is something we have already looked at in detail. But what if we insist that the magnitude remain constant (the object maintains the same rotational inertia and keeps spinning at a constant rate), while only the the direction of motion changes? That is, what if the first term in Equation 6.2.2 is zero, while the second term is not? How can we construct a physical system that behaves this way? Answering this last question will require quite a lot of facility with the right hand rule, but here goes...
We start with a rotating object. We'll use as our model a bike wheel turning around an axle. The angular momentum vector will point along the axis of the wheel according to the right hand rule. Now we need a net torque that points perpendicular to the angular momentum. We can achieve this by placing on end of the wheel's axle on a support and allowing the weight of the wheel to pull it down as the support pushes up.