- #1

Leo Liu

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- Homework Statement
- A gyroscope wheel consists of a uniform disk of mass M and radius R that is

spinning at a large angular rotation rate ωs. The gyroscope wheel is mounted onto

a ball-and-socket pivot by a rod of length D that has negligible mass, allowing the

gyroscope to precess over a wide range of directions. Constant gravitational

acceleration g acts downward. For this problem, ignore both friction and

nutational motion; i.e, assume the gyroscope only precesses uniformly. For all

parts, express your solution as a vector (magnitude and direction) with components

in the coordinate system shown above.

(a) [5 pts] Calculate the total angular momentum vector of the uniformly

precessing gyroscope in the orientation show above; i.e., the total of the spin and

precession angular momentum vectors.

- Relevant Equations
- Euler's equations

The rate of precession of this gyro ##\Omega## can be found by solving ##\tau_1=DMg=I_s\omega_s\Omega##. But when I apply Euler's equations to this problem, it fails.

I first set the frame in the way shown in the diagram above.

Then I wrote the first equation:

$$\tau_1=\bcancel{I_1\dot\omega_1}+(I_3-I_2)\omega_2\omega_3$$

The first term on the right side of the equation is 0 because the question says that we should ignore nutation.

After applying perpendicular axis theorem and parallel axis theorem, we get

$$I_3=1/4MR^2$$

Therefore,

$$I_3-I_2=-1/4MR^2$$

The equation then becomes

$$Dg=1/4MR^2\omega_s\Omega$$

whose solution is different to the solution produced by the torque-angular momentum equation.

Could someone point out the mistake(s) in the solution which uses Euler's equations?

@etotheipi help me if you please.

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