Physics - Dynamics - Momentum

As discussed on the Noether Theorem page, the conservation of momentum is derived from the symmetry of space, this page defines momentum and explains how it is conserved.

Conservation Law Symmetry
linear momentum The laws of physics are the same regardless of our position when we do an experiment.
angular momentum The laws of physics are the same regardless of the direction that we are facing.

linear momentum

Linear momentum is here denoted by the symbol p, for a solid body:

p = mv

where:
symbol
description
type
units
p linear momentum vector kg m/s
m mass scalar kg
v linear velocity vector m/s

p and v are vectors, that is, they have direction and magnitude. Since they are related by a scalar then the direction of p and v is the same.

In the general case, in the 3D world, p and v are 3 dimensional vectors. If we are constrained to a plane then we can use 2D vectors, if we a constrained to a line (i.e. direction is already defined) then we can use scalars.

If a solid body has both linear and angular movement then all the points on the body have a different velocity, in this case the linear velocity is taken to be the linear velocity

Conservation of linear momentum

So far we are talking about a single solid body or particle. We can define the total linear momentum of a set of solid bodies as the sum of all the individual momentums. We just take the vector addition (add each element separately) of all the momentums to give the total momentum for the system. For any closed system this total momentum is constant. Individual bodies within the system may exchange momentum but the total always remains the same. The following pages explain how momentum can be exchanged between bodies by means of forces.

angular momentum

Angular momentum is here denoted by the symbol L, for a solid body:

L = [I]w 

where:
symbol
description
type
units
L angular momentum bivector kg m2/s
[I] inertia tensor (moment of inertia) tensor kg m2
w angular velocity bivector s-1

So [I] is the analog of mass for rotation, this is a tensor, which means that its form depends on the number of dimensions:

Therefore, in the 3 dimensional case, this adds a lot more complexity than the linear case, for instance:

These factors mean that we have to be very careful and rigorous when doing calculations involving angular momentum.

Conservation of angular momentum

In a similar way to linear momentum, we can define the total angular momentum of a set of solid bodies as the sum of all the individual angular momentums. We just take the vector addition (add each element separately) of all the momentums (L is a vector) to give the total momentum for the system, but to do this we must measure the angular momentum relative to the same point and be consistent. For any closed system this total angular momentum is constant. This is conserved completely independently of linear momentum, it is a separate concept. it is therefore not meaningful to talk about converting linear momentum to angular momentum and visa-versa. It is possible for linear motion to be converted to rotational motion and visa-versa, but in any closed system, the total linear momentum and the total angular momentum are both conserved independently.

Total angular momentum of a system of particles.

If a particle can't rotate then can a system of particles have angular momentum?

Centre of mass and first moment of mass

The centre of mass is the point in a solid body which the body would naturally rotate around without external forces.

Centre of mass

This is the point at which we can consider the total mass of the object acts for the purposes of gravity and inertia.

This allows us to ignore the internal structure and forces of the object and to concentrate this all to one point. This assumes the object is solid and that the internal forces within the object are not large enough to distort it.

So the centre-of-mass is like the average position of the mass.

The centre-of-mass of a number of particles relative to an arbitrary point is:

sum(pi * mi)/mt

where:
symbol
description
type
units
pi relative position of particle i vector m
mi mass of particle i scalar kg
mt total mass scalar kg

For a continuous object this becomes the first moment of mass:

xcm = intergalm dx /masstotal

Angular momentum in 2D and second moment of mass

In 2D [I] is a scalar, so we will just use I.

I = sum(pi * mi2)/mt

where:
symbol
description
type
units
pi relative position of particle i vector m
mi mass of particle i scalar kg
mt total mass scalar kg

For a continuous object this becomes the first moment of mass:

I = intergalm2 dx /masstotal

Angular momentum in 3D

In 3D [I] is a 3x3 matrix

Angular Momentum of particles(p)

The instantaneous angular momentum of a particle is the product of its mass and its angular velocity.

angular momentum

h = r x p = r x mv =mw

where:
symbol
description
type
units
h the instantaneous angular momentum about PC bivector kg m2/s
x the cross product operator (see here for definition)    
r the instantaneous position of the particle relative to point. pv-pc vector m
p the instantaneous linear momentum of the particle = mv vector kg m/s
m the mass of the particle. scalar kg

As with rotation velocity, the angular momentum of a point is not an absolute value, but it depends on which point that the rotation is measured about.

As discussed here, angular momentum of any closed system is conserved. However the particle under consideration here, may have external forces acting on it, so it is not a closed system. When we go on to consider a rigid body rotating about its centre of mass, then the angular momentum of the whole system (about any point) is conserved, this is the sum of the angular momentums of the particles that make up the body. So as it rotates, the instantaneous angular momentum of each particle that makes up the body, will be continuously varying, but the total angular momentum is constant.


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