Rotation about two axes and angular momentum

In summary, the conversation discusses finding the infinitesimal change in angular momentum along the z-axis at a later time ##t+\Delta t##, assuming the initial angular velocity and axes are fixed in inertial space. One approach involves considering the contribution of rotation about the y-axis, which affects the x-component of angular momentum. The question of whether the x-component remains constant in magnitude and the actual motion of the body are also raised. A similar method is mentioned and the possibility of the z-component of angular momentum being zero or periodic is discussed. It is assumed that the x- and y-components of angular momentum are conserved in later time.
  • #1
Kashmir
465
74
IMG_20210709_103319.JPG

I've a body having initial angular velocity at ## t=0 ## as shown. The axis shown are fixed in inertial space and initially match with the principal axis. I want to find the infinitesimal change at ##t+\Delta t## in the angular momentum along the ##z## axis.

I've seen the following approach which I don't understand:
One contribution to change in ##L_z## is due to rotation about y axis. This causes ##L_x## to rotate and hence a component ##-L_x \Delta{_y}## appears.
IMG_20210709_105348.JPG

How do we know that ##Lx## will remain constant in magnitude? Also the actual motion won't be as is shown, in which the body simply goes around the y-axis while maintaining it's spin ##L_x##

A similar method is used here by Kleppner and Kolenkow here
IMG_20210709_112436.JPG
 
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  • #2
May we expect Lz=0? At least periodic it seems.
 
  • #3
anuttarasammyak said:
May we expect Lz=0? At least periodic it seems.
Initially?
 
  • #4
Yes, and I assume ##\mathbf{L}=(L_x,L_y,0)## is conserved in later time.
 

1. What is rotation about two axes?

Rotation about two axes is a type of rotational motion in which an object or system rotates simultaneously around two different axes. This type of rotation is also known as double rotation or compound rotation.

2. How is angular momentum defined?

Angular momentum is a measure of the rotational motion of an object or system. It is defined as the product of an object's moment of inertia and its angular velocity. In simpler terms, it is the measure of how fast an object is rotating and how much mass is distributed away from the axis of rotation.

3. What is the conservation of angular momentum?

The conservation of angular momentum states that the total angular momentum of a system remains constant if no external torque is acting on the system. This means that in the absence of external forces, the total angular momentum of a system before and after a rotation will be the same.

4. How does rotation about two axes affect an object's stability?

Rotation about two axes can affect an object's stability in different ways depending on the distribution of mass and the axes of rotation. Generally, if the object's mass is evenly distributed and the axes of rotation are aligned, the object will be stable. However, if the axes of rotation are not aligned and the mass is unevenly distributed, the object may experience instability and wobbling.

5. What are some real-life examples of rotation about two axes?

One common real-life example of rotation about two axes is the movement of a gyroscope. Another example is the motion of a spinning top, which rotates around both its vertical and horizontal axes. Additionally, planets in our solar system rotate around both their own axes and the sun, which can be considered rotation about two axes.

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