# Orbital Angular Momentum as Generator for Circular Translations

• unscientific
In summary, the angle of rotation is determined by the unit vector ##\vec{n}## pointing in the direction of the axis of rotation, and the angle is rotated by the same amount as the unit vector is rotated.
unscientific

## Homework Statement

Taken from Binney's Text, pg 143.

## The Attempt at a Solution

From equation (7.36): we see that ##\delta a## is in the direction of the angle rotated, ##\vec{x}## is the position vector, and ##\vec{n}## is the unit normal to the plane of rotation.

But at the last line of (7.37), we see that ##\vec{\alpha} = \alpha \vec{n}##. Why is the angle rotated now the same direction as the unit normal to the plane?

I'm not sure if I'm understanding your question. But, I think it's just a matter of definition. The symbol ##\vec{\alpha}## is defined to be ##\alpha \hat{n}##. Note how Binney uses the definition symbol (or equivalence symbol) $\equiv$. This definition allows the rotation operator ##U## to be written more compactly. An angle of rotation has a certain magnitude about a certain axis. In this case ##\alpha## is the angle of rotation and ##\hat{n}## is the direction of the axis of rotation. The notation ##\vec{\alpha} = \alpha \hat{n}## is just a way to express that.

TSny said:
I'm not sure if I'm understanding your question. But, I think it's just a matter of definition. The symbol ##\vec{\alpha}## is defined to be ##\alpha \hat{n}##. Note how Binney uses the definition symbol (or equivalence symbol) $\equiv$. This definition allows the rotation operator ##U## to be written more compactly. An angle of rotation has a certain magnitude about a certain axis. In this case ##\alpha## is the angle of rotation and ##\hat{n}## is the direction of the axis of rotation. The notation ##\vec{\alpha} = \alpha \hat{n}## is just a way to express that.

If ##\alpha## is the angle of rotation and ##\hat{n}## is the direction vector of rotation, then wouldn't ##\delta a = \delta \alpha## x ##x## be into the plane, perpendicular to the plane of rotation?

##\hat{n}## is a unit vector perpendicular to the plane of rotation. It points along the axis of rotation. So, ##\hat{n} \times \vec{x}## lies in the plane of rotation.

Since ##\vec{\alpha} = \alpha \hat{n}##, ##\vec{\alpha}## also lies along the axis of rotation (perpendicular to the plane of rotation). For example, a rotation in the x-y plane of ##\pi## radians could be written ##\vec{\alpha} = \pi \hat{z}##

TSny said:
For example, a rotation in the x-y plane of ##\pi## radians could be written ##\vec{\alpha} = \pi \hat{z}##

That doesn't make sense at all..Usually ##\vec{\alpha}## is the direction vector in which the system moves - I would say it's more like ##\alpha_x \vec{i} + \alpha_y \vec{j}##

Last edited by a moderator:

## 1. What is orbital angular momentum?

Orbital angular momentum refers to the rotational motion of a particle around a central point or axis. It is a vector quantity that describes the magnitude and direction of the rotational motion.

## 2. How is orbital angular momentum related to circular translations?

Orbital angular momentum can be used as a generator for circular translations, meaning it can produce circular motion. This is because the direction of the angular momentum vector is always perpendicular to the plane of rotation, which is a characteristic of circular motion.

## 3. What are some examples of circular translations?

Examples of circular translations include the Earth's orbit around the sun, the rotation of a spinning top, and the motion of a planet's moon around the planet.

## 4. How is orbital angular momentum calculated?

Orbital angular momentum can be calculated by multiplying the linear momentum of an object by the perpendicular distance between the object and the axis of rotation. The direction of the orbital angular momentum vector is determined by the right-hand rule.

## 5. What are the applications of orbital angular momentum as a generator for circular translations?

Orbital angular momentum has many applications in physics and engineering, such as in the design of satellites and spacecraft, the study of planetary orbits, and in the development of gyroscopes for navigation and stabilization purposes.

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