- Good morning.
Today we are going to use calculus
to derive the work-energy theorem.
And I do understand because this is an algebra-based class
that many of you have not taken calculus yet.
However, I consider it important
to sometimes see math which is a little bit more advanced
than what you currently understand.
- Thank you.
♫ Flipping Physics ♫
- The definition of work using calculus is
work equals the integral
from the initial position to the final position
of force with respect to position.
So, this big curve here means the integral.
Below and above the integral are the limits.
So, from position initial to position final of force,
and dx means with respect to x,
or with respect to position.
In other words, work is equal to the integral
from initial position to final position
of force with respect to position.
However, we want to use net work,
which means when we replace work with net work,
we also need to replace force with net force.
Bo, what is Newton's Second Law?
- Net force equals mass times acceleration
where both force and acceleration are vectors.
- Correct.
Which means we can replace the net force
with mass times acceleration.
Notice I've dropped the vector symbol on acceleration.
That is because everything here is in the x direction,
so we do not need that vector symbol.
Mass is a scalar, so we can take it out
from underneath the integral.
Now, we cannot integrate.
We cannot take the integral yet,
because we're trying to take the integral
with respect to position, and acceleration
does not have position in it.
Bobby, what is the equation for acceleration?
- Acceleration equals change in velocity
over change in time.
True. However, because we are using calculus today,
we need to use the calculus definition
of instantaneous acceleration.
Acceleration equals the derivative of velocity
with respect to time.
dv divided by dt means the derivative of velocity
with respect to time.
We can substitute the derivative of velocity
with respect to time in for acceleration.
However, we are still unable to take the integral.
So let's take a closer look at this derivative of velocity
with respect to time.
The derivative of velocity with respect to time
is the same thing as the the derivative of velocity
with respect to position times the derivative of position
with respect to time.
Notice how the dx's cancel out,
and we're left with the derivative of velocity
with respect to time.
Therefore we can substitute in the derivative of velocity
with respect to position
times the derivative of position with respect to time
in for the derivative of velocity with respect to time.
These two dx's cancel one another out,
and we're left with
net work equals mass times the integral
with respect to velocity of the derivative of the position
with respect to time.
And notice what we need to do to our limits,
because we've changed what we're taking the integral
with respect to.
We were taking the integral with respect to position,
so we went from position initial to position final,
but now because we're taking the integral
with respect to velocity, our limits have changed.
We're taking the integral from velocity initial
to velocity final.
And we are almost there.
We can almost take the integral.
Billy, what is the equation for velocity?
- Velocity is change in position over change in time.
However, I bet you are going to say
we need the calculus version of instantaneous velocity,
which is probably the derivative of position
with respect to time.
- Correct.
We need instantaneous velocity
equals the derivative of position
with respect to time, because then
we can substitute velocity in for the derivative of position
as a function of time.
And now we can finally take the integral.
The integral of velocity with respect to velocity
is velocity squared divided by 2,
and we retain those limits from initial velocity
to the final velocity,
and we can now plug in those limits.
Net work equals one half times mass
times velocity final squared minus one half
times mass times velocity initial squared.
Which is where one half times mass times velocity squared
comes from.
We give it a special name, which is what, class?
- [All] Kinetic energy.
- And it becomes net work equals kinetic energy final
minus kinetic energy initial,
which is the change in kinetic energy.
Therefore, the net work done by all the forces
acting on an object causes a change in kinetic energy
of that object.
The work-energy theorem.
Basically, all the forces acting on an object, the net force,
will do work on that object, the net work,
which will cause a change in kinetic energy
of that object, a change in velocity of that object.
The work-energy theorem.
- Hey, wait a minute.
That looks just like the work due to friction
equals change in mechanical energy equation.
- Bo, that is a good point.
Thank you for mentioning that.
Notice how similar these two equations are.
The work due to friction equals the change
in mechanical energy,
and the net work equals change in kinetic energy.
Both of them have work on the left-hand side,
and change in energy on the right-hand side.
And students often confuse the two,
which is why I prefer not call this just
the work-energy theorem, but rather
the net work kinetic energy theorem.
One of our other energy equations
is conservation of mechanical energy.
Conservation of mechanical energy
is true when the work done by friction
is equal to zero and the work done
by a force applied on the object
is equal to zero.
Work due to friction equals change in mechanical energy
is true when the work done by the force applied
on an object is equal to zero.
Now, class, what did we have to assume
when we proved the net work kinetic energy theorem?
- Nothing.
- I don't think we made any assumptions.
- Penguins?
- Correct.
We did not need to make any special assumptions
when proving the net work
equals the change in kinetic energy,
because the net work kinetic energy theorem
is always true.
And unlike other alwayses which aren't quite always true,
this one is always true.
The net work done by all the forces acting on an object
is always equal to the change in kinetic energy
of that object.
Oh, and
because the net work kinetic energy theorem
has kinetic energy final and kinetic energy initial in it,
you must identify the locations of your initial
and final points.
Thank you very much for learning with me today.
I enjoyed learning with you.
- I want my two dollars.