# Precession of Rotating Rigid Body from Two Different Frames

• Andrew Wildridge
In summary, the conversation discusses using Euler's equations to solve for the precession rate of a rapidly spinning gyroscope, first in the body-fixed frame and then in the precessing frame. The equations are derived and the torque is found to be time-varying with angular frequency. The conversation also mentions the importance of considering the orientation of the axes when working with Euler's equations in the body-fixed frame.
Andrew Wildridge

## Homework Statement

A)
Use Euler's equations to solve for the precession rate of a rapidly spinning gyroscope precessing uniformly (no nutation). Use I3 for the moment along the z axis along the axle. Beginning with Euler's equations, derive the coupled equations:
## \dot ω_1 + Ω ω_2 = \frac {N_1} {I_1} ## where ## Ω = \frac {I_3 - I_1} {I_1} ω_3 ##
## \dot ω_2 + Ω ω_1 = \frac {N_2} {I_1} ##
And then solve for the precession rate of the gyroscope. (Remember the technique of adding i-times the second to the first equation, as for Focault’s pendulum.) Because Euler’s equations are in the body frame, the torque is time-varying with angular frequency ##ω_3##. Note that the gyroscope precession rate is very different than the force-free precession rate Ω.

B) Solve the problem of 1) in the precessing frame. The precessing frame is neither the body frame or the fixed frame.

## Homework Equations

Euler's Equations:
$$N_1 = I_1\dot ω_1 - (I_2 - I_3) ω_2 ω_3$$
$$N_2 = I_2\dot ω_2 - (I_3 - I_1) ω_3 ω_1$$
$$N_3 = I_3\dot ω_3 - (I_1 - I_2) ω_1 ω_2$$

Angular Momentum:
L = Iω

Torque:
τ = r x F

## The Attempt at a Solution

A) At first I decided to break it down component by component...We can assume the rod is massless so we do not need to include the moment of the inertia of the rod. Setting the body coordinates to ##x_3## through the axle and out the top of the gyroscope parallel with the angular velocity the top is rotating with. ##x_2## is perpendicular to ##x_3## but in the same plane as the image pointing towards the top left of the image from the fixed point. ##x_1## is into the page. Here is an image with my frame of reference with ##x_3## being blue and ##x_2## being yellow.

From there I have
$$I_1 = I_2$$
$$ω_1 = 0$$
$$ω_2 = \dot φ sinΘ$$
$$ω_3 = ω + \dot φ cosΘ$$
where ##\dot φ## is the rate of precession. I also have all of the time derivatives of the angular velocities are zero. Therefore, my Euler Equations are
##N_3 = 0##
##N_2 = 0##
##N_1 = -(I_1 - I_3)ω_2ω_3 = sinΘ*bmg##

However, in the given problem it wants the time derivative of the angular velocities to still be there and for ##N_2## to be nonzero. Are the time derivatives of the angular velocities nonzero because the entire frame is rotating with the body? I also still have ##ω_1## being zero which I don't know how it cannot be zero as there is no rotation in the x axis. I know that ##N_2## cannot be zero since there needs to be a toque being applied to balance out the torque being applied by the gravity so that ##N_2 = cosΘbmg##.

B)
For this one I'm not even sure if I am setting the frame correctly...My guess is to set the frame on the top of the gyroscope that is precessing and rotating. So I have the same convention as before just with the origin centered around the center of mass. Now I have all of the angular velocities being zero besides ##ω_3## since no other coordinate is rotating relative to this frame. Also all the time derivitives of the angular momentum are zero too since none of them are accelerating. However, when I do this I get zero applied torques which seems kind of right since I am in the frame of reference centered about where the torques would be...But I know the forces need to be balanced then and I don't know how I am going to balance the force due to gravity and I still need a force to cause myself to precess. I believe I am looking at this problem incorrectly but I just don't know where. I'm assuming it has something to do with my angular velocities or perhaps my rate of change of my angular velocities.

For part (A) you are working with body-fixed axes. So, the ##x_1## and ##x_2## axes are attached to the rotating gyroscope and rotate with the gyroscope. You can take the orientation of the axes at ##t = 0## to be as you described, with the ##x_1## axis into the page and the ##x_2## axis along the yellow line. But after a quarter revolution of the gyroscope, the ##x_1## axis will now be in the direction of the yellow line and the ##x_2## axis will point out of the page. At this instant, the torque has zero ##x_1## component and a negative ##x_2## component.

The way you chose the axes, so that the ##x_1## and ##x_2## axes do not spin with the gyroscope, corresponds to the "precessing frame" of part (B).

Ahhhh ok. Give me a few to work with that and ill see what I get.

## 1. What is the concept of precession in a rotating rigid body?

Precession is the phenomenon in which the axis of rotation of a spinning object changes its direction. This change in direction is caused by a torque acting on the object, and results in a circular motion of the object's axis around a fixed point.

## 2. How does the precession of a rigid body differ in two different frames of reference?

In one frame of reference, the precession of a rigid body is described by the motion of its axis relative to a fixed point. In another frame of reference, the precession is described by the motion of the object's axis relative to the first frame. This difference in perspective can lead to different equations and predictions for the precession of a rigid body.

## 3. What are the factors that affect the precession of a rotating rigid body?

The precession of a rotating rigid body is affected by the angular velocity of the object, the moment of inertia, and the magnitude and direction of the torque acting on it. Additionally, the shape and orientation of the object can also affect its precession.

## 4. How is precession of a rotating rigid body used in real-world applications?

The concept of precession of a rotating rigid body is used in various fields such as astronomy, engineering, and physics. For example, precession is used in the design and operation of gyroscopes, which are used in navigation and stabilization systems. It is also used in the study of the Earth's rotation and its effect on the planet's climate and seasons.

## 5. Can the precession of a rotating rigid body be controlled or manipulated?

Yes, the precession of a rotating rigid body can be controlled or manipulated by applying a torque in a specific direction. This is how gyroscopes are able to maintain a fixed orientation in space and resist external forces acting on them. However, the magnitude and direction of the applied torque must be carefully calculated in order to achieve the desired precession motion.

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