Good morning. Today we are going to review the work, energy,
and power portions of the AP Physics C Mechanics curriculum.
♫ Flipping Physics ♫
Bobby, what is the equation for the work done
by a constant force?
- Work done by a constant force equals the dot product
of the force doing the work
and the displacement of the object.
- Or force times displacement times the cosine of the angle
between the force and the displacement using only
the magnitudes of the force and displacement vectors.
- That's what the dot product means.
- Yeah.
- Work is a scalar.
- Let's do a simple example.
If the force acting on an object is 2.7i minus
3.1j newtons
and the displacement of the object is 4.6i meters,
Billy, what is the work done by the force on this object?
- Work equals the dot product of force and displacement,
so it equals the dot product of 2.7i minus 3.1j,
and 4.6i plus 0j.
Multiply the i's together, multiply the j's together
and take the sum.
So it is
2.7
times 4.6
plus the negative 3.1 times zero,
or 12.42,
which is twelve with two significant digits.
Oh, and it's in joules.
- Notice the dot product multiplies the component
of the force which is in the direction of the displacement
of the object, with the displacement of the object.
In this particular case, the component in the direction
of the displacement of the object is 2.7 newtons
multiplied by the displacement of the object of 4.6 meters.
The component normal to the direction
of the displacement in this particular case,
3.1 newtons in the negative y direction, does no work
on the object, again because it is at a 90 degree angle
to the displacement of the object.
And yes, Billy, the units for work are joules.
Bobby, what is a joule?
- A joule is a newton times a meter.
- And a newton is a kilogram meter per second.
- Squared.
- A newton is a kilogram meter per second squared.
- The work done by a force which is not constant
uses a different equation. That work is equal to
the integral from position initial to position final
of the force with respect to position.
This is called a definite integral.
Which simply means the integral has limits.
In this particular case, from the initial position
to the final position.
Bo, the derivative represents
the slope of a function.
What does the integral,
or what we also call the antiderivative, represent?
- An integral, or antiderivative, represents the area
under the curve, and the area under the curve means
the area between the curve and the horizontal axis.
An area above the horizontal axis is positive.
The area below the horizontal axis is negative.
- Notice how we have two different equations for work.
We use one equation for work when the force is constant.
We use a different equation for work when the force
is not constant, and that equation uses an integral.
This is going to happen a lot in this class, where we use
one equation when the item does not vary.
We use a different equation for that same item
when it does vary and that equation uses an integral.
Please pay attention to that.
Now let's talk about
the force caused by a spring: Hooke's Law.
The force of a spring equals negative kx.
Bobby, can you please clarify all of these letters?
- K is the spring constant, and it is a measure of
how much force it takes to compress or expand the spring
per meter. Delta x is the displacement of the spring
from equilibrium position, or rest position,
which is where the spring would be if it were not
being compressed or elongated. The negative has to do with
the direction of the force of the spring. It means the force
of the spring is opposite the direction of the displacement
of the spring, and the units for the spring constant
are usually newton meters.
- Actually, it's newtons per meter.
- The spring constant is in newtons per meter.
Joules are newtons times meters. Thanks.
I always mess that up.
- You are welcome.
- Billy, which of the two work equations do we use
when determining the work done by the force of a spring?
- Well, that would depend on whether the force caused
by a spring is constant or varies.
- The force caused by a spring changes with position,
so it is not constant.
- Right, therefore the work done by a spring equals
the intergral from position initial to position final
of the force of the spring with respect to position.
- Bobby, please determine the work done by a spring.
- Well, the equation for the spring force is the negative
of the spring constant times the displacement from
equilibrium position. The integral of x to the first power
with the respect to x is x squared over two,
so we have negative kx squared over two
from position initial to position final.
Substituting in our limits
gives us way too many negatives, so I'm going to factor out
a negative one,
and one half kx squared is
elastic potential energy, so that is the negative
of elastic potential energy final minus elastic potential
energy initial, or the negative of the change in elastic
potential energy of the spring. The work done by
a spring force equals the negative of the change
in elastic potential energy of the spring.
That's pretty cool.
- Previously we derived to the
net work kinetic energy theorem.
We're not going to do that derivation right now,
however you're more than welcome to review it
by clicking on the link which just appeared.
- [Mr. P.] It was during that derivation
where we defined kinetic energy.
Kinetic energy equals one half times the mass of the object
times the velocity of the object squared.
The AP equation sheet just uses the capital letter K.
I prefer capital KE so that you don't confuse
the spring constant with kinetic energy.
Gravitational potential energy equals the mass of the object
times the acceleration due to gravity times h, the vertical
height above the horizontal zero line.
Please, don't ask me why the symbol for gravitational
potential energy is a capital U. I don't know.
- [Mr. P.] Please remember whenever you use gravitational
potential energy, you have to identify the
horizontal zero line. This is the reference line
for the vertical height h.
The equation on the AP equation sheet is
instead in terms of the change in gravitational potential energy.
Energy can neither be created nor can it be destroyed.
Therefore, in a system which is not isolated.
The change in energy of a system equals the sum of the
energy which is transferred into or out of the system.
Bo, this is important. Could you please review it for us?
- When we have a non-isolated system, which means
a system which is
receiving energy or having energy taken away from it,
the change in energy of the system equals the addition
of all the energy which is transferred into or out
of the system. It actually makes a lot of sense if you think
about it because you cannot create or destroy energy.
The energy has to come from or go somewhere.
- Right. Now, if the system is isolated
then no energy is transferred into or out of the system,
therefore the change in energy of the system must be zero.
The change in energy of the system equals the change
in mechanical energy of the system plus the change
in internal energy of the system which add up to zero.
The change in internal energy of the system equals
the negative of the work done by non conservative forces.
And remember, the work done by a conservative force
does not depend on the path taken by the object.
The work done by a nonconservative force does depend
on the path taken by the object.
Therefore, the work done by nonconservative forces
equals the change in mechanical energy of the system.
Because the only non conservative force that I know of
is friction, I usually write this equation as:
The work due to friction equals the change
in mechanical energy of the system.
And remind me, Bobby, what needs to be true for the work
due to friction to be equal to the change
in mechanical energy of the system?
- There needs to be no energy transferred into
or out of the system.
- The system needs to be isolated.
- Don't confuse the work due to friction equals change
in mechanical energy equation with the net work equals
change in kinetic energy equation.
- They do look similar.
- Net work equals change in mechanical energy
is not the same as work due to non conservative forces
equals change in mechanical energy.
- And if the system is isolated and no work is done
by friction, then the change in mechanical energy
equals zero, which works out to be
conservation of mechanical energy.
Class, remind me, what do you need to identify whenever
you use conservation of mechanical energy or work
due to friction equals change in mechanical energy?
- Initial point.
- Final point.
- Horizontal zero line.
- Remember, just like work, all of these forms
of mechanical energy are in joules.
Now let's talk about power.
Power is the rate at which work is done.
Therefore, average power equals work over change in time.
Instantaneous power equals the derivative of work
with respect to time. I will point out that the equation
for power in terms of work on the AP equation sheet is:
Power equals the derivative of energy with respect to time.
Now, substituting in the equation
for work done by a constant force
gives us the derivative with respect to time
of the dot product of the force and
the displacement of the object. But the force is constant,
so this is really just the derivative of the position
with respect to time, which is velocity.
Therefore, the power delivered by a constant force
on an object in terms of velocity is
the dot product of force and velocity.
Bo, what are the SI units for power?
- Power...
Is in horsepower?
- Horsepower is the English unit for power.
Watts are the SI unit for power.
- Right, and 746 watts equals one horsepower.
- Watts are joules per second. But we don't have to memorize
that there are 746 watts in one horsepower, right?
- You are correct that you do not need to remember
how many watts are in a horsepower.
What you do need to remember is that every derivative is
an integral, or an antiderivative. For example,
power equals the derivative of work with respect to time
can be rearranged to dw equals power times dt.
Taking the definite integral of both sides
gives the change in work equals the integral
from time initial to time final
of power with respect to time.
Which means if you have a graph of the power exerted
on an object with respect to time, the area under
that curve will be equal to the change in work
exerted on that object. Which could be very helpful.
So please remember, every derivative can be
rearranged as an integral, and every integral
can be rearranged as a derivative.
Billy, what is the equation which is not on your
AP equation sheet which has to do with conservative forces?
- There's an equation which has to do with
conservative forces which is not on our equation sheet?
- I did not know that.
- It's a pickle.
- Pickle?
A conservative force equals the negative
of the derivative of the potential energy
associated with that conservative force.
As a side note, you have to know this equation because
much of the time, when the phrase "conservative force"
is used on the AP Exam, this is the equation
you need to use. Capeesh?
- Capiscilo!
- Yeah.
- Ich verstehe.
- For example, for a spring,
the force of a spring equals the negative of the derivative
of the elastic potential energy associated with
that spring with respect to position.
We can substitute in the equation
for elastic potential energy, and Bo,
could you please take that derivative?
- The derivative of x squared with respect to x
is two times x. 1/2 times two is one.
So the force of a spring equals negative k times x.
But we already know that, don't we?
- [Mr. P.] Well Bo, that is my point.
- Right.
- Billy, could you please do the same
thing with the force of gravity?
- The force of gravity equals the negative
of the derivative of gravitational potential energy
with respect to position. However that position
is going to be in the y direction because the force
of gravity is in the y direction.
For gravitational potential energy, we can substitute
in mass times acceleration due to gravity times
vertical height above the horizontal zero line.
Although, let's use y for that instead of an H
to match the variable we used for position.
Mass and acceleration due to gravity are constants,
and the derivative of y with respect to y is just one.
So the force of gravity equals the negative of
mass times acceleration due to gravity. Why is it negative?
- The force of gravity is always down.
That's why it is negative.
- Oh, okay. That makes sense.
- Now let's talk about neutral,
stable, and unstable equilibrium.
This ball is in neutral equilibrium because
the gravitational potential energy of the ball
remains constant, regardless of its position.
So the graph of the gravitational potential energy
of the ball with respect to position is a horizontal line
because the gravitational potential energy is constant.
This water bottle is in stable equilibrium because
its gravitational potential energy increases
as its position moves away from the equilibrium position.
This is because the center of mass of the water bottle
goes up as the position goes away from equilibrium.
In other words, the water bottle naturally returns
back to the equilibrium position when it loses
gravitational potential energy.
This dry erase marker is in unstable equilibrium because
its gravitational potential energy decreases as its position
moves away from the equilibrium position.
This is because the center of mass of the dry erase marker
goes down as the position goes away from equilibrium.
In other words, the dry erase marker naturally moves
away from the equilibrium position when it loses
gravitational potential energy.
That completes my review of work energy and power.
Next, feel free to enjoy my review
of integrals in kinematics for AP Physics C.
Or you can visit my AP Physics C Review webpage.
Thank you very much for learning with me today.
I enjoyed learning with you.