What is the point of angular momentum?

[nhmllr]

So I'm trying to teach myself torque and angular momentum, and I know how to calculate the vector (direction can be found using the right hand rule, magnitude equals
sin(angle between)*position vector* momentum vector )
but what is the POINT of this value? I've read a few things on it but it's not clicking. I get that it's conserved from an outside point, but so what? If I added up all the vectors and multiplied them by 76, that'd be conserved too. What's so special about this vector?

 

[Pengwuino]

At the most basic level, it allows you to determine how the momentum and the position of a particle are related as a system evolves. You probably have gone over energy conservation so you can use that as an analogy. Imagine a roller coaster 50m high. Now you probably have seen all the examples of someone going down the roller coaster from rest up a hill and back down and up and what have you. In that particular type of problem, you can find a relationship that will always hold relating the current height of the roller coaster and the current speed of the roller coaster.

Angular momentum is the same idea. An object rotating will have a conserved angular momentum that can allow you to relate the objects position and momentum. For example, if you have a planet orbiting a star in a non-circular orbit. If you know the angular momentum of the planet, you can relate how fast it will be traveling at various radii.

One thing to note about Torque is that it will allow you to change the angular momentum of an object (however, of course, if something applies a torque to an object, the object will apply a torque back).

Take a look at this video :

Warning: They're British.

 

[nhmllr]

At the most basic level, it allows you to determine how the momentum and the position of a particle are related as a system evolves. You probably have gone over energy conservation so you can use that as an analogy. Imagine a roller coaster 50m high. Now you probably have seen all the examples of someone going down the roller coaster from rest up a hill and back down and up and what have you. In that particular type of problem, you can find a relationship that will always hold relating the current height of the roller coaster and the current speed of the roller coaster.

Angular momentum is the same idea. An object rotating will have a conserved angular momentum that can allow you to relate the objects position and momentum. For example, if you have a planet orbiting a star in a non-circular orbit. If you know the angular momentum of the planet, you can relate how fast it will be traveling at various radii.

One thing to note about Torque is that it will allow you to change the angular momentum of an object (however, of course, if something applies a torque to an object, the object will apply a torque back).

Take a look at this video :

Warning: They're British.

Okay, so it's not something, "real" (like a force vector, in my specific definition of the term "real") but rather just some incredibly useful value that stays the same. I can dig that.

 

[RedX]

For each independently conserved quantity, you have an equation that contains terms that only contain positions and velocities, not accelerations. So it may be possible to avoid Newton's 2nd law altogether and use the conservation equation(s) as your starting equation. For example in one-dimension with force independent of time, you never need Newton's 2nd law to solve this. This is because energy is conserved. It turns out that knowing angular momentum and energy is conserved, you can solve for something like planetary motion without the need for Newton's 2nd law.

 

[jschmidt]

Okay, so it's not something, "real" (like a force vector, in my specific definition of the term "real") but rather just some incredibly useful value that stays the same. I can dig that.

Well, I look at angular momentum like this: there is linear velocity, and a corresponding linear momentum.

We have the concept of angular velocity (how many rotations, degrees, or rads per unit time that something is rotating/orbiting). Angular velocity is useful because it abstracts out the radius, which will vary (e.g. a rotating disk, at the center, the radius is 0, at the edge, the radius is r_max, and all across its width, the radius takes every intermediate value), and thus, simplifies our formulas/calculations when dealing with rotations/orbits.

When using angular velocity, you use angular momentum instead of linear momentum for solving momentum-related problems, yes?

Angular momentum is real, to the extent that it describes a real physical value - the momentum of a rotating/orbiting object (and if an object has both linear momentum, and angular momentum - e.g. a wheel rolling down a road or ramp - to model the total momentum of the object, you have to take into account *both* types of momentum).

Update: I suppose I should mention, you probably *could* describe angular momentum in terms of the linear momentums of all the particles in an object/system. That math gets horrendously difficult very quickly. Angular momentum is a nice, convenient simplification of all the math that gets us to the solution much quicker, with fewer steps.

 

[vanhees71]

From a practical point of view, angular-momentum conservation provides, as any conservation law, a "first integral" (if you have full rotational symmetry like in the motion of a point particle in a central potential, there are even three conserved quantities, i.e., the three components of angular momentum; this tells you that the trajectory of any such motion is in a plane perpendicular to angular momentum), and this helps a lot to find complete solutions of the equations of motion.

From a more theoretical and fundamental point of view, angular-momentum conservation is a manifestation of rotational symmetry of space (Noether's theorem, applied to invariance of the action under spatial rotations).

 

[AlephZero]

Possibly the most "fundamental" use of it is in consdering the motion of a finte-sized rigid body.

In principle, you could do this by taking account of all the internal and external forces acting on the body. But it is much simpler to use the general result that the situation is always equivalent to

1. A force, acting at the center of mass, whcih is related to the translational acceleration of the body by Newton's laws of motion, and
2. A torque acting about the center of mass, which is related to the angular acceleration.

Torque and angular acceleration don't add anything "new" to Newton's laws of motion, but because of this general result they are very useful concepts in practice.