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We're going to talk about a whole new concept, which is the concept of momentum.

We've all heard the expression, "We have a lot of momentum going."

Well, in physics, momentum is a vector, and it is a product between the mass of a particle and its velocity.

And so the units would be kilograms times meters per second.

F = ma.

That equals m dv/dt.

That equals d(mv)/dt and that equals, therefore, dp/dt.

So what you see — the force is dp/dt, and what that means is if a particle changes its momentum, a force must have acted upon it.

It also means if a force acts on a particle, it will change its momentum.

Let us now envision that we have a whole set of particles which are interacting with each other, and the interaction could be gravitational interaction, could be electrical interaction, but they're interacting with each other.

Zillions of them — a whole star cluster.

I pick one here, which I call mi, and I pick another here, which I call mj.

And there is an external force on them, because they happen to be exposed to forces from the outside world, and so on this one is a force Fi external, and on this one is Fj external.

But they're interacting with each other, either attracting or repelling, and so in addition to these external force, there is a force that j feels due to the presence of i.

And let's suppose they are attracting each other.

This would be F that j experience in the presence of i.

Actions equals minus reactions, according to Newton's Third Law, so this force, Fij, will be exactly the same as Fji, except in the opposite direction.

We call these internal forces.

That's the interaction between the particles.

If these were the only two forces, then the net force on this object, on this particle i, would be... would be a force in this direction.

This would be F net.

Now, you can do the same here.

This would be F net on particle j, and this would be F net on particle i.

But since there are zillions of objects here, there are many of these interacting forces, and so I can't tell you what the net force will be.

The net force is ultimately the sum of the external force and all the internal forces.

What is the total momentum of these zillions of particles?

Well, that's the sum of the individual momenta.

So that is p1 plus p2... pi... and you have to add them all up.

I take the derivative of this — dp total/dt.

That is p1/dt.

Well, that's the force on number one.

It's the total force on number one.

So it is F1, but it is the net force on F1.

And F.. . and dp/dt for this particle equals F2 net force, and the force on particle number i equals Fi net, and so on.

And so when we add up all these forces, obviously that is the total force on the entire system.

Now comes the miracle.

The miracle is that all these internal forces eat each other up.

This ji cancels this one if you look at the system as a whole — not if you look at the individual particles, but on the system as a whole, they all cancel each other out.

And so the total force on the system is simply the total force external.

And all the internal forces — you can forget about it.

And this means, then, that we come to a key conclusion — that dp total/dt... in fact, I have it written down there.

It's so important that I want you to look at it this whole lecture.

Look at this.

We've all heard the expression, "We have a lot of momentum going."

Well, in physics, momentum is a vector, and it is a product between the mass of a particle and its velocity.

And so the units would be kilograms times meters per second.

F = ma.

That equals m dv/dt.

That equals d(mv)/dt and that equals, therefore, dp/dt.

So what you see — the force is dp/dt, and what that means is if a particle changes its momentum, a force must have acted upon it.

It also means if a force acts on a particle, it will change its momentum.

Let us now envision that we have a whole set of particles which are interacting with each other, and the interaction could be gravitational interaction, could be electrical interaction, but they're interacting with each other.

Zillions of them — a whole star cluster.

I pick one here, which I call mi, and I pick another here, which I call mj.

And there is an external force on them, because they happen to be exposed to forces from the outside world, and so on this one is a force Fi external, and on this one is Fj external.

But they're interacting with each other, either attracting or repelling, and so in addition to these external force, there is a force that j feels due to the presence of i.

And let's suppose they are attracting each other.

This would be F that j experience in the presence of i.

Actions equals minus reactions, according to Newton's Third Law, so this force, Fij, will be exactly the same as Fji, except in the opposite direction.

We call these internal forces.

That's the interaction between the particles.

If these were the only two forces, then the net force on this object, on this particle i, would be... would be a force in this direction.

This would be F net.

Now, you can do the same here.

This would be F net on particle j, and this would be F net on particle i.

But since there are zillions of objects here, there are many of these interacting forces, and so I can't tell you what the net force will be.

The net force is ultimately the sum of the external force and all the internal forces.

What is the total momentum of these zillions of particles?

Well, that's the sum of the individual momenta.

So that is p1 plus p2... pi... and you have to add them all up.

I take the derivative of this — dp total/dt.

That is p1/dt.

Well, that's the force on number one.

It's the total force on number one.

So it is F1, but it is the net force on F1.

And F.. . and dp/dt for this particle equals F2 net force, and the force on particle number i equals Fi net, and so on.

And so when we add up all these forces, obviously that is the total force on the entire system.

Now comes the miracle.

The miracle is that all these internal forces eat each other up.

This ji cancels this one if you look at the system as a whole — not if you look at the individual particles, but on the system as a whole, they all cancel each other out.

And so the total force on the system is simply the total force external.

And all the internal forces — you can forget about it.

And this means, then, that we come to a key conclusion — that dp total/dt... in fact, I have it written down there.

It's so important that I want you to look at it this whole lecture.

Look at this.

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