# AP Physics C: Work, Energy, and Power Review (Mechanics)

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.
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.

1. Philip Y says:

EXCELLENT!!!…… Period!

2. Sebastian says:

If i were riding in car 80mph and threw a pencil up in the air inside the car, shouldn't the pencil fly back immediately just as if i were to do it outside the car?

3. Camila Mendes says:

U r such an amazing person 😀

4. Rehan Rashid says:

5. Camila Mendes says:

Can you tell me about your technique of learning ? U have got such an amazing personality with good humor ! Very less people possess them ! We will always be supporting you ! I told all of my classmates and my relatives about your videos ! I m totally renowned in the class because of your videos ! Everyone's saying "its great " ! Keep uploading videos ! Can you do some more videos relating to nunerical of kinematics

6. Shoaib Rashdi says:

I love the fact that you are so precise. I've got books where the displacement is simply denoted by "d", and that annoys the hell out of me.

7. Jordan Bueno says:

I am in college and these videos help greatly with calculus-based physics. Thanks a lot!

8. Grant Wilkins says:

Will there be a series of videos like this for APPC: E&M in the future? Thank you for all you do!

9. 廓俊傑 says:

Is it possible to purchase a button-down Flipping Physics shirt, as seen in your videos?

10. Daniel Kwon says:

For Ue=1/2kx^2, can this be only used when the situation is elastic?
Also, if so, what does it mean by Elastic Potential Energy?
Thank you!

Comment: Your explanation….if someone does not understand, that is weird
#having_question_does_not_mean_no_understanding
Thank you!

11. Main Line says:

Question: what's different about the marker and the bottle that makes ones center of mass go down and ones center of mass go up, as you said in the video? Thanks for the videos

12. Lightning Speed says:

It was awesome

13. APileofBabies says:

You taught me more than my physics teacher.

14. Gaurav K says:

I did some research and U is purely a technical varibale, not a symbol like F for force. It was chosen because all the other letters were taken up in Thermo!

15. Aki T says:

Your videos and editing are amazing!

16. Jodh Pullela says:

I've got my apc mechanics final tomorrow and this is really helping me out, thanks!

17. Dong Hai STUDIO says:

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