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.

EXCELLENT!!!…… Period!

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?

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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.

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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!

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

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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!

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