Finding the precession of a gyro using Euler's equations of rotation

In summary, the conversation discusses the rate of precession of a gyro, which can be found by solving an equation involving torque and angular momentum. However, when applying Euler's equations to the problem, the solution is different. The conversation also discusses possible mistakes in the solution, such as using the parallel axis theorem when it is not needed. The final conclusion is that there are likely some other mistakes in the work, as the coefficient in the equation is incorrect.
  • #1
Leo Liu
353
156
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
1611541343380.png

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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  • #2
I refrained from responding because I only have part of the answer.
In the Euler form, the torque is about the centre of the disc, no? So no need for the parallel axis theorem.
But that still leaves a discrepancy: a magnitude of ##\frac 12MR^2## versus ##\frac 14MR^2##. I can believe the sign should be reversed, making both positive.
 
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Likes Leo Liu
  • #3
haruspex said:
I refrained from responding because I only have part of the answer.
In the Euler form, the torque is about the centre of the disc, no? So no need for the parallel axis theorem.
Oh, I see where the problem is. I didn't quite understand the conditions under which Euler's equations hold so I made this mistake. Thank you!

That said, there are certainly some other mistakes in my work, as the coefficient is 1/4 instead of 1/2. Really weird.
 

Related to Finding the precession of a gyro using Euler's equations of rotation

1. What is precession and how is it related to gyroscopes?

Precession is the phenomenon where the axis of rotation of a gyroscope changes direction over time when it is subjected to an external force or torque. This is due to the conservation of angular momentum, which causes the gyroscope to rotate around a different axis than its initial axis of rotation.

2. What are Euler's equations of rotation and how do they relate to gyroscopic precession?

Euler's equations of rotation are a set of three differential equations that describe the motion of a rigid body in three-dimensional space. These equations are used to calculate the precession of a gyroscope by taking into account the angular velocity, angular momentum, and external forces acting on the gyroscope.

3. How is the precession of a gyroscope measured?

The precession of a gyroscope can be measured by attaching a pointer to the gyroscope's axis of rotation and observing the direction in which the pointer moves over time. The rate of precession can also be measured by using sensors to track the angular velocity of the gyroscope.

4. What factors affect the precession of a gyroscope?

The precession of a gyroscope can be affected by various factors such as the initial angular velocity, the mass and shape of the gyroscope, the external torque applied, and the presence of any external forces such as friction or air resistance.

5. How is the precession of a gyroscope used in practical applications?

Gyroscopic precession has numerous practical applications, including navigation systems, spacecraft attitude control, and stabilization of vehicles and ships. It is also used in precision instruments such as gyrocompasses, gyroscopes, and gyroscopic sensors.

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