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WALTER LEWIN: Today we will talk exclusively about work and energy.
First, let's do one dimensional case.
The work that a force is doing when that force is moving from point A to
point B, one dimensional, here is point A and here is point B, and the
force is along that direction or you go in this
direction or in this direction.
But it's completely one dimensional.
That work is the integral in going from A to B of that force, dx.
If I call that the x-axis.
The unit of work you can see is newton meters.
So work is newton meters, for which we call that joule.
If there's more than one force in this direction, you have to add these
forces in this direction vectorially and then this is the work that the
forces do together.
Work is a scalar.
So this can be larger than 0, it can be 0, or it can be smaller than 0.
If the force and the direction in which it moves, in an opposite
direction, then it is smaller than 0.
If they're in the same direction, either this way or that way, then the
work is larger than 0.
F = ma, so therefore I can also write for this m * dv / dt.
And I can write down for dx, I can write down, v * dt.
I substitute that in there.
So the work in going from A to B is the integral from A to B times the
force which is m * dv / dt d x, which is v * dt.
And look what I can do.
I can eliminate time.
And I can now go to a integral over velocity.
Velocity a to the velocity b.
And I get m times v times dv.
That's a very easy integral.
That is 1/2 * m * v^2, which I have to evaluate between v_A and v_B.
And that is 1/2 m * v_b^2 - 1/2 m * v_a^2.
1/2 * m * v^2 is what we call in physics kinetic energy.
Sometimes we write just a K for that.
It's the energy of motion.
And so the work that is done when a force moves from A to B is the kinetic
energy in point B, you see that here, minus the kinetic energy in point A.
And this is called the work energy theorem.
If the work is positive, then the kinetic energy increases when you go
from A to B. If the work is smaller than 0, then the work-- then the
kinetic energy decreases.
If the work is 0 then there is no change in kinetic energy.