Week 6 - Notes: Linear and Angular Momentum#
We’ve talked about the central conservation laws of classical mechanics:
Conservation of energy - in a process, if energy is conserved, the total energy of the system is the same before and after the process. More strongly, in a closed system, the total energy is constant for any process (
).Conservation of linear momentum - in a process, if momentum is conserved, the total momentum of the system is the same before and after the process. More strongly, in a closed system, the total vector momentum is constant for any process (
).Conservation of angular momentum - in a process, if angular momentum is conserved, the total angular momentum of the system is the same before and after the process. More strongly, in a closed system, the total vector angular momentum is constant for any process (
).
We’ve worked with the conservation of energy a lot because it’s a fundamental concept in physics and it lends itself to a scalar equation analysis. This can be quite a bit simpler in many cases, but an energy only view of the world can be limiting.
Linear Momentum#
As we move into the formal study of linear momentum, we will start with a reminder of the definition of momentum, and the mathematical form of the conservation of momentum.
Linear momentum is a vector quantity defined as the product of an object’s mass and its velocity. It is denoted by the symbol
where
where
As you can calculate, the relativistic momentum reduces to the classical momentum when the velocity is much less than the speed of light. As
Linear Momentum and Newton’s Second Law#
You have seen in our discussion of Newton’s Second Law that the net force on a system is equal to the mass of the system times the acceleration of the system. This can be written as:
However, this definition and our thinking here with it is a bit limited. What about systems of objects that are interacting with each other? What about deformable systems? What happens if something is shedding mass, like a rocket or jet?
Newton’s definition from the Principia is a bit more general. He defines the force in terms of the rate of change of the body’s momentum:
We can extend that definition to a system of objects, where the net force on the system is equal to the rate of change of the total momentum of the system:
The second step might not be obvious, but by working through a few examples we can see how this is a more useful and general definition of force.
Forces internal to a system zero out#
Consider an abstract system of
The total force on the system is given by the sum of all the masses times the acceleration of each particle:
where the last term is the net force on the
Here these internal forces are pairwise interactions between the particle
where the sum is over all particles that are not
Cool, what happens to the internal force equation when we sum over all particles in the system?
Concrete Examples#
We have a generic setup, let’s see what happens when we apply this to a few examples: 2 particles, 3 particles, and then N particles.
Two Particles#
With two particles the sum is easy to write out fully.
By Newton’s Third Law, the force of particle 1 on particle 2 is equal and opposite to the force of particle 2 on particle 1. The internal forces cancel out and the net force on the system is the sum of the external forces.
So here the internal forces sum to zero.
Three Particles#
We can write this out in a similar way.
We can group these terms by Newton’s Third Law pairs.
Every interaction on body
N Particles#
Clearly, there seems to be a pattern here. Namely that the internal forces are always zero. We can write out the sum for
And where we make a switch in the sum terms, so we can counting the force from each interaction in each term in the sum to make it clear why the internal forces sum to zero.
Internal forces will always appear as third law pairs, so the internal interactions will always sum to zero. This is a very powerful result.
For a given system, only external forces can change the momentum.
Mathematical Form of Conservation of Linear Momentum#
Let’s look back at the system momentum,
If we take the time derivative of the system momentum, and assume we have point particles, so the masses are not changing,
The net force on the system is given by,
Should there be no external forces, then,
And thus there is no change momentum of the system,
So if the system has no external forces, the total momentum of the system is conserved. We can propose a discrete extension to this form above where
And thus, it’s easy to see:
If there are external forces, then we also have a prediction equation for how the energy will change in a small time step
so that,
Angular Momentum#
Angular momentum is a complex and rich quantity that has deep connections to the shape and structure of a system. The “configuration” or how it is distributed in space can have a big impact on the dynamics of a system. Our study of classical angular momentum will be a stepping stone to our study of quantum angular momentum and the spin of particles.
Quantum Mechanical Spin
Spin is a quantum mechanical property that is not related to the rotation of a particle, but it is a form of angular momentum, and it’s essential to the structure of the universe - it tells us if we have fermionic or bosonic particles, it is what gives us the Pauli Exclusion Principle, and it is what gives us the Zeeman Effect and the Stark Effect.
For the moment, we will limit ourselves to classical angular momentum and we will focus on the abstract case of a single particle. As we work through the semester, we will revisit angular momentum and introduce how to work with distributions of mass and extended objects.
Definition of Angular Momentum#
For a particle with a momentum
This is a quantity that depends on the location of the particle relative origin of coordinates. This means you have some latitude in choosing the origin of coordinates, and you can choose the origin to simplify the problem.
This also means the angular momentum is a vector quantity, and it points in the direction perpendicular to the plane defined by the position and momentum vectors.
When is Angular Momentum Conserved?#
We can ask this by computing the time derivative of the angular momentum,
We did a calculation like this on a homework where we computed
Let’s apply it here:
If we assume that
We group the terms in the time derivative of the angular momentum,
The first term the cross product of the velocity with itself
So the time derivative of the angular momentum is the net torque on the system!
If the net torque on the system is zero, then the angular momentum is conserved, and it is a constant of the motion.
If there’s a net torque, we have a discrete update equation for the angular momentum,
Are we sure there are no internal torques that matter?#
We can ask the same question we asked about internal forces. Are there internal torques that matter? As before, let us define the total force on particle
We assume there are no external forces, and so the net force on the system is the sum of the internal forces,
For given object, we observe an angular momentum
If the total angular momentum of the system is the sum of the angular momenta of the particles,
then the time derivative of the total angular momentum is,
Recall that
But note that
So if the internal forces are parallel to the separation between the particles, then the internal torques sum to zero, and the total angular momentum of the system is conserved. So things like the gravitational force, the electric force, and spring forces are all internal forces that do not contribute to the net torque on the system.
And thus,