# CLEAR Calculus

## Lab 4: The Precessing Top

A couple of young children, Billy and Janie... in the midst of their carefree childhood... are playing with a top. Billy sets the top in motion at a slight angle from vertical, and Janie asks why the top doesn't fall over. She picks up the top and sets it down at the same angle only not spinning, and the top falls. " See," she says, "when it's not spinning it just falls over!" Billy snidely responds, "That's because of gravity. Don't you know anything about physics?" Janie, in her infinite patience tries to explain once more, " But that doesn't explain why the top doesn't fall when it is spinning." She sets the top in motion again at a slight angle, and they both are shocked when they notice something even stranger. " It's precessing!" shouted Billy. He's right! The axis of the top sweeps out a slow circle! Eager to figure out why this was occurring, they consult Wikipedia and discover Newton's Second Law, $${\bf F}=m{\bf a}$$, where $${\bf F}$$ is a force vector, $$m$$ is the mass of an object, and $${\bf a}$$ is the acceleration vector of that object due to the application of the force $${\bf F}$$. Janie pulls out some scratch paper, and says, " Look! If we define torque to be the cross product of the position vector $${\bf r}$$ with the force vector $${\bf F}$$, and define angular momentum to be mass times the cross product of the position vector $${\bf r}$$ with the velocity vector $${\bf v}$$..." At the same time she writes down

\begin{align*} \text{torque: }\pmb{\tau}&={\bf r}\times{\bf F}\\ \text{angular momentum: }{\bf l}&=m({\bf r}\times{\bf v}) \end{align*}

"Wait!" interrupts Billy, "Why are you defining these weird things?" to which Janie responds " Because look what happens when you differentiate the equation defining angular momentum, then apply Newton's Second Law. You get

$\pmb{\tau}=\frac{d{\bf l}}{dt}{\atop.}{" \atop}$

1. Prove this relation by proceeding as Janie suggests.

"Aha!" exclaims Billy, "This explains the precession of the top!"

2. Using the point where the top touches the floor as the origin, determine the direction in which the angular momentum vector of the top points?
Hint: Each point in the top contributes. Pick a point and draw the velocity and position vectors for that point.

3. In which direction does the force vector due to gravity point?
In which direction does the torque vector point?

4. Use Janie's equation to determine what effect the force of gravity must have on the axis of rotation of the top.

The result you just derived mathematically can seem extremely counterintuitive. The next question helps you picture why the combination of forces that seem like they should move the rotating object one direction actually causes it to move in a different direction.

5. The ring below is spinning with total angular momentum pointing straight up.

1. Draw appropriate velocity vectors at each of the marked points.

2. Now imagine applying a force that would cause the front of a non-spinning ring to rotate down and underneath and cause the back side to rotate up and over the top. Draw corresponding acceleration vectors due to these forces at each of the marked points.

3. Now draw how the velocity vectors at each marked point would look after some amount of each of the corresponding accelerations.

4. Finally, draw a new spinning ring that would have velocity (tangent) vectors pointing in the resulting directions you determined. What does this imply?