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A gyroscope is a a fascinating mechanical device with a lot of technological applications. You can easily make a gyroscope from a wheel with a handle put through the middle.


If you dangle the end of the handle from a string, the wheel doesn't flop over put stays up straight! This might seem a bit odd because the weight tex2html_wrap_inline934 is pointing straight down. Instead it slowly moves around in a circle, that is it precesses. Here's a picture of what's going on when you look down on this from above. The wheel is shown at two different times.


I kid you not! This is quite counter-intuitive but can easily be understood by using the concepts of angular momentum and torque.

The angular momentum of the wheel points along its axis of rotation. Here's the same picture but with the angular momentum vectors tex2html_wrap_inline778 displayed.


Now let's think about what the angular momentum vectors do as a function of time.


Here they are displayed for two times t and t+dt. The difference between them is the vector tex2html_wrap_inline942 . Does this make sense yet? Well what direction is the torque? As shown in the figure, it points in the horizontal plane. We know that


So tex2html_wrap_inline730 points in the same direction as tex2html_wrap_inline942 . That's what we see in the figure. He're we're assuming that the rate of rotation of the wheel tex2html_wrap_inline890 is much greater than the rate of precession tex2html_wrap_inline950 , so that we can properly use tex2html_wrap_inline792 . So let's compute the rate of precession.

If the length from the string to the center of mass of the wheel is called L, then the torque has magnitude


What is dL? It's tex2html_wrap_inline958 . So


But by definition


So we have


The bigger tex2html_wrap_inline890 , the slower the wheel precesses. The larger L, the distance between the center of mass and the axis of rotation, the faster the object precesses.

Let's test how well we understand this

next up previous
Next: example Up: Angular Momemtum and Torque Previous: solution

Joshua Deutsch
Sun Feb 23 15:54:50 PST 1997