# Precession of a gyroscope

I have arranged a gyroscope at the front of the room in the manner shown above. We'll pretend that the rod connecting all the pieces is massless, and make a few other approximations along the way. We will use units of grams and centimeters for this exercise, so be sure to use g = 980 cm/s^2.

For our purposes,

```     component        mass       dist from axis
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disk         M  = 1700 g       L  = 14.1 cm         R = 12.6 cm

weight 1        m1 =  900 g       d1 = 25.3 cm

weight 2        m2 =   40 g       d2 = 30.4 cm

weight 3        m3 =   50 g       d3 = 19.5 cm
---------------------------------------------------------------------
```

1. What is the moment of inertia of the entire gyroscope around the supporting post?
2. The disk is free to rotate around the rod independent of all the other objects. If I rotate the disk with angular velocity omega in a counterclockwise direction, as seen from the door of the classroom, what is the moment of inertia of the disk around its center?
3. What is the angular momentum of the disk around its center as it rotates with omega? Don't forget to specify a direction.
4. Plug into two particular values of omega, corresponding to 320 RPM and 200 RPM. Compute the angular momentum of the spinning disk for each case.

Now, consider the force of gravity. It pulls all four objects straight down: the three weights and the disk.

1. What is the net torque on the gyroscope around the supporting post? Supply both a magnitude and a direction.
2. Suppose that the disk is spinning, so that the gyroscope has some angular momentum. Write a vector equation which relates three quantities: the initial angular momentum, the torque around the support post, and the new angular momentum after a short time dt.
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3. The initial angular momentum points towards the door of the classroom. If I release the gyroscope so that gravity's torque begins to act on it, in which direction will the angular momentum CHANGE?

After just a very short time, the angular momentum will point in a slightly different direction.

However, the magnitude of the angular momentum of the disk will not change (if we ignore the gradual slowing of the rotation due to friction and small effects).

If we measure the little angle d phi by which the angular momentum vector has changed in radians, then the size of the change dL can be expressed as

1. Use algebra to write an equation which expresses the rate at which the angle phi changes with time, in terms of quantities you know.
2. Compute this precession frequency for the two choices of disk spin rate: 320 RPM and 200 RPM.
3. Compute the precession period for the two choices of disk spin rate: 320 RPM and 200 RPM.