# Component of angular momentum perpendicular to rotation axis

• Soren4
In summary, the conversation discusses a rigid body rotating about a fixed axis, with an angular velocity that can vary in magnitude but not in direction. The focus is on finding the angular momentum vector and its components parallel and perpendicular to the z axis, and how they vary in time and with the application of a torque. The formula for the component perpendicular to the z axis is provided and it is questioned whether its magnitude is proportional to the angular velocity. The relationship between angular acceleration and the variation of angular momentum components is also discussed.

## Homework Statement

Consider the rigid body in the picture, rotating about a fixed axis $z$ not passing through a principal axis of inertia, with an angular velocity $\Omega$ that can vary in magnitude but not in direction. Find the angular momentum vector and its component parallel to $z$ axis ($\vec {L_z }$) and perpendicular to it ($\vec {L_n }$). How do this two vectors vary in time? And if a torque is exerted? Explain how is the angular acceleration related to the variation of the components of angular momentum.

(I start by saying that I'm totally ok with the calculation of angular momentum and the properties of its component parallel to the $z$ axis ($\vec {L_z }$). My difficulties are in understanding what are the properties of $\vec {L_n }$.)

## Homework Equations

From the picture we have that $| \vec {L_ {n, i}} | = m_i r_i R_i \Omega cos \theta_i \implies | \vec {L_n} | = \Omega \sum m_i r_i R_i cos \theta_i$ (1)

## The Attempt at a Solution

Firstly, does it follow from (1) that $| \vec {L_n} | \propto | \vec {\Omega} |$ (2)?

If this is true, suppose to apply a torque perpendicular to the $z$ axis and parallel to $\vec {L_n}$, so that the magnitude of this vector increases. Follows from (2) that there should be an angular acceleration $\vec {\alpha}$, although we are in the absence of a torque with an axial component.
This would go against the fact that $I_z \vec {\alpha} = \vec { M_z}$ (Where $I_z$ is the moment of inertia with respect to the $z$ axis and $M_z$ is the axial component of the exerted torque). How is this possible?

If (2) is not true, then what can be said about the variation of \vec {L_n} in time? Does it change if a torque is applied perpendicular to the z axis?Finally, if a torque is applied with an axial component, how is the angular acceleration related to the variation of the components of angular momentum?

## 1. What is the definition of angular momentum perpendicular to the rotation axis?

Angular momentum perpendicular to the rotation axis, also known as the orbital angular momentum, is a measure of an object's rotational motion around an axis that is perpendicular to the plane of its orbit. It is calculated by multiplying the object's mass, velocity, and the distance from the axis of rotation.

## 2. How is the direction of angular momentum perpendicular to the rotation axis determined?

The direction of angular momentum perpendicular to the rotation axis is determined by the right-hand rule. If the fingers of your right hand curl in the direction of the rotation, then your thumb will point in the direction of the angular momentum.

## 3. How does angular momentum perpendicular to the rotation axis affect an object's stability?

Angular momentum perpendicular to the rotation axis is a vital component in determining an object's stability. A spinning object with a large angular momentum perpendicular to the rotation axis will have a higher resistance to changes in its orientation, making it more stable. This is why spinning tops and gyroscopes are able to maintain their balance.

## 4. Can the magnitude of angular momentum perpendicular to the rotation axis change?

Yes, the magnitude of angular momentum perpendicular to the rotation axis can change. This can occur when external torques, such as friction or gravity, act on the object, causing it to speed up or slow down its rotation. However, the total angular momentum of an isolated system remains constant.

## 5. What is the role of angular momentum perpendicular to the rotation axis in celestial mechanics?

In celestial mechanics, angular momentum perpendicular to the rotation axis plays a crucial role in understanding the motion of planets, moons, and other celestial bodies. It helps to predict their orbits and explains phenomena such as precession and nutation. It also plays a significant role in explaining the conservation of energy in these systems.