- 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.
