The vector nature of Angular Momentum

In summary, angular momentum is described as a vector quantity because it is dependent on both the axis of rotation and the rate of rotation of a rotating object. This allows for a more complete understanding of its effects and interactions with other forces and objects.
  • #1
hms.tech
247
0
Here is an animation from Wikipedia : http://en.wikipedia.org/wiki/File:Torque_animation.gif

The angular momentum is given by the Cross product of r and p
We can see that the direction would be perpendicular to the direction of rotation of the particle (as shown in the animation)

I don't think this really makes sense, how is the vector nature of angular momentum justified ?

How can one get an intuitive sense about the direction of angular momentum ?
 
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  • #2
I think that direction of angular momentum is only a convention. Someone thought of that rule, it seemed practical since intensity of cross product of r and p really defines its amplitude. I am not sure, but think you could define another rule and get same physical results.

Such thing is with phasor diagrams, you choose phase of one quantity to be zero, and according to that do everything else.

This is only my opinion.
 
  • #3
Having angular momentum as a vector comes in handy when you want to explain gyroscopic precession.

Maybe someone with more knowledge than me can give better examples :)
 
Last edited:
  • #4
The vector comes into play when you're trying to solve for conservation of angular momentum. The vectors must all add to zero.
 
  • #5
hms.tech said:
We can see that the direction would be perpendicular to the direction of rotation of the particle (as shown in the animation)
It is parallel to the axis of rotation
hms.tech said:
how is the vector nature of angular momentum justified ?
You need to express magnitude and axis of rotation.
 
  • #6
hms.tech said:
Here is an animation from Wikipedia : http://en.wikipedia.org/wiki/File:Torque_animation.gif

The angular momentum is given by the Cross product of r and p
We can see that the direction would be perpendicular to the direction of rotation of the particle (as shown in the animation)

I don't think this really makes sense, how is the vector nature of angular momentum justified ?

How can one get an intuitive sense about the direction of angular momentum ?

You use the right hand rule. For A x B, point your four fingers along A, then rotate your hand until they point along B. Your thumb sticking up tells you the direction of the cross product. The cross product of A and B is always perpendicular to both, just like your thumb sticking up is perpendicular to your four fingers.

If you want to think about it some more, you might ask, why the right hand rule, why not a left hand rule? You could use either, and everything would still make sense. That's because the angular momentum vector is not truly a vector, it is a "pseudovector", one that depends on which hand you use. Well, the laws of physics don't depend on which hand you use, and true vectors don't depend on which hand you use, so true vectors are, in a sense, more "real" than pseudovectors.

For calculation purposes, pseudovectors are nice, just three components that transform almost like a vector. But when you want to do theoretical work, you might not want to deal with the artificiality of pseudovectors. The bottom line is that pseudovectors are better represented by antisymmetric 3x3 matrices (antisymmetric tensors). Instead of a pseudovector [x,y,z] you use [tex]\left[\begin{matrix} 0 & z & -y \\-z & 0 & x \\ y & -x & 0 \end{matrix}\right][/tex] This tensor transforms the same way no matter what, no worry about which hand you need to use, and its better for theoretical work, its the "real thing", unlike the more concise pseudotensor.
 
  • #7
hms.tech said:
Here is an animation from Wikipedia : http://en.wikipedia.org/wiki/File:Torque_animation.gif

The angular momentum is given by the Cross product of r and p
We can see that the direction would be perpendicular to the direction of rotation of the particle (as shown in the animation)

I don't think this really makes sense, how is the vector nature of angular momentum justified ?

How can one get an intuitive sense about the direction of angular momentum ?

It's a convenient way of describing torque that happens to work well with the rest of mathematics. Nothing much deeper than that I think.

I guess a good thing to think about is how else you would do it?
 
  • #8
angular momentum and torque while having vectorial nature are pseudovectors i.e. under an inversion they don't get reflected.
L=r×p
replacing r by -r ,gives inversion.In p=m dr/dt,it becomes -.so overall L does not change direction.
edit:eek:ops,someone else also written it.
 
  • #9
hms.tech said:
I don't think this really makes sense, how is the vector nature of angular momentum justified ?

The effect of a rotating object's angular momentum and the way it will interact with other objects and forces depends just as much on the direction of its axis of rotation as its MI and rate of rotation - so it can only be described fully using a vector. This is the same as for linear momentum (also a vector), where the mass, direction of travel and speed (velocity) are needed for full description.
 

1. What is Angular Momentum?

Angular Momentum is a physical quantity that describes the rotational motion of an object. It is a vector quantity, meaning it has both magnitude and direction.

2. How is Angular Momentum calculated?

Angular Momentum is calculated by multiplying the moment of inertia (a measure of an object's resistance to rotational motion) by the angular velocity (the rate at which an object rotates).

3. What is the difference between Angular Momentum and Linear Momentum?

Angular Momentum is a measure of rotational motion, while Linear Momentum is a measure of linear motion. Angular Momentum is a vector quantity, while Linear Momentum is a scalar quantity.

4. How is the vector nature of Angular Momentum represented?

The vector nature of Angular Momentum is represented by a vector arrow, with the direction of the arrow indicating the axis of rotation and the length of the arrow representing the magnitude of the Angular Momentum.

5. What are the units of Angular Momentum?

Angular Momentum has units of kilogram meters squared per second (kg*m2/s) in the SI system or gram centimeters squared per second (g*cm2/s) in the CGS system.

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