- [Teacher] Not only are there many different kinds
of energies, but both objects,
and systems of objects, can have energy.
Once they have that energy, they can transfer it
to another system or object
or that energy could transform to a different type
of energy inside that system.
When energy gets transferred, we call that work,
and the amount of work that's done is the amount
of energy that was transferred.
You often hear people say, "Energy is conserved,"
which really just means that you can't create
or destroy energy.
You can simply transfer it between objects or systems.
So, what are all the different types of energy?
There's kinetic energy which is the energy
due to something moving, and the formula's
one-half the mass times the speed squared.
There's gravitational potential energy
which is the energy something has due to its height,
and the formula's the mass
times the magnitude of acceleration due to gravity
times the height of the object.
Height above what?
Height above whatever you're choosing
as the H equals zero reference line.
Is that cheating?
No, because all that really matters is the change
in gravitational potential energy,
not the actual value itself.
There's also spring potential energy
which has to do with a compressed or stretched spring,
and the formula's one-half times the spring constant
times x, which is not the length of the spring.
X is the amount that the spring has been compressed
or stretched.
These three types of kinetic and potential energy
constitute what we call mechanical energy.
Mechanical energy's another word for the kinetic energy
plus gravitational potential energy
plus spring potential energy in a system,
and it's important to know that mechanical energy
does not include thermal energy.
Thermal energy's the heat energy generated
by dissipative forces like friction and air resistance,
and you can find the amount of thermal energy generated
by taking the size of the dissipated force
times the distance through which that force was acting.
The unit of energy is Joules and energy is not a vector.
But maybe the most important thing to remember about energy
is if there's no external work done on a system,
then there's no change in the energy of that system.
In other words, if there's no external work done
on a system, the initial energy of that system
will equal the final energy of that system,
which is the way you solve
many conservation of energy problems.
So, what's an example problem involving energy look like?
Let's say a box started with an initial speed
and slides from one platform up to another platform.
We'll assume that frictional forces
and air resistance are negligible.
And for the system that's consisting
of the mass and the Earth,
what's happening to the total mechanical energy
in this system?
So, you've got to pay special attention
to what is in your system.
Since my system includes the mass,
which is going to be moving,
my system's gonna have kinetic energy,
and since my system has two objects
that are interacting gravitationally,
the mass and the Earth, my system's also going
to have gravitational potential energy.
So, when I asked about the total mechanical energy
of the system, that's really just code
for the total kinetic and potential energy of the system.
So, as this mass slides up to a higher point
on the ramp, the gravitational potential energy increases,
but the mass is gonna slow down,
so the kinetic energy's gonna decrease.
However, since the Earth and the mass are in our system,
and there's no dissipative forces,
there's no external work done on our system.
Yes, the Earth is doing work on the box,
but the Earth is part of our system
so it can't do external work,
and that means energy just gets transferred
from one form to another within our system,
and the total mechanical energy, here,
is gonna remain the same for the entire trip.
Now, what if we asked this same question
but we consider a system that consists only of the box.
In that case, our system has a box that's moving,
so it'll have kinetic energy.
But, our system no longer includes two objects
interacting gravitationally so our system will have no
gravitational potential energy.
What happens to the total mechanical energy in this case?
Well, the only energy that I've got in my system, now,
is kinetic energy and since that kinetic energy decreased,
the total mechanical energy of the box,
as a system, decreases.
How does it decrease?
It decreases because now the Earth is outside
of our system and the work that it is doing
on the box is external work
and it's taking away energy from the box.
What does work mean?
In physics, work is the amount of energy transferred
from one system, or object, to another.
In other words, if a person lifted a box
and gave it 10 Joules of gravitational potential energy,
we'd say that person did positive 10 Joules of work
on the box since that person gave the box
10 Joules of energy.
But since the box took 10 Joules of energy
from that person, we'd say that the box
did negative 10 Joules of work on the person
since the box took 10 Joules of energy.
So, you can find the work done
if you can determine the amount of energy
that was transferred.
But, there's an alternative formula to find the work done.
If something's having work done to it,
there's got to be a force on that object,
and that object has to be displaced.
So, if you take the force on the object
times the displacement of the object,
and multiply by the cosine of the angle
between the force and the displacement,
you'll also get the work done.
In other words, one way to find the work done
is by finding the amount of energy
that was transferred.
But, another way to find the work done is
by taking the magnitude of force exerted on an object
times the displacement of the object
and then times cosine of the angle
between the displacement and the force.
Since work is a transfer of energy,
it also has units of Joules.
And even though work is not a vector,
it can be positive or negative.
If the force on an object has a component
in the direction of motion,
that force will do positive work on the object
and give the object energy.
If the force on the object has a component
in the opposite direction of the motion,
the work done by that force would be negative
and it would take away the object's energy.
And, if the force on an object is perpendicular
to the motion of the object,
that force does zero work on the object.
It neither gives the object energy
nor takes away the object's energy.
So, what's an example problem involving work look like?
Let's say a box of mass M slides down
a frictionless ramp of height, H, and angle two-theta,
as seen in this diagram, here,
and a separate box of mass two-M slides down
another frictionless ramp of height, H,
and angle theta, as seen in this diagram, here.
And, we want to know how work done on the object
by the Earth compares for each case?
The easiest way to find the work done here
is by finding the change in energy.
The box will gain an amount of kinetic energy
equal to the amount of potential energy that it loses.
So, the work done by the Earth is just gonna equal
positive m-g-h.
Both heights are the same.
So, the H's are equivalent, but one box has twice the mass.
So, the work done by gravity on the mass, two-M,
is gonna be twice as great
as the work done on the mass one-M.
What's the work energy principle mean?
The work energy principle states
that the total work, or the net work done on an object,
is gonna be equal to the change in kinetic energy
of that object.
So, if you add up all the work done by all forces
on an object, that's got to be equal to the change
in the kinetic energy of that object.
In other words, one-half m-v-final, squared,
minus one-half m-v-initial, squared.
So, this is a really handy way to find how the speed
of an object changes if you can determine the net work
on an object.
In other words, if there's multiple forces on an object,
and you can find the work done by each of those forces,
you can determine how much kinetic energy
that object gained or lost.
So, what's an example of the work energy principle?
Let's say a four kilogram box started
with a velocity of six meters per second to the left.
Some net amount of work is done on that box,
and it's now moving with a velocity of four meters
per second to the right.
We want to know what was the amount of new work done
on the box?
Without even solving it, we can say
since this object's slowed down, energy was taken from it,
so the amount of net work had to be negative
which means it's either B or D.
To figure out which one exactly,
we could use the work energy principle
which says that the net work done is equal to
the change in kinetic energy.
So, if we take the final kinetic energy,
which is one-half times four kilograms
times the final speed squared,
and we subtract the initial kinetic energy,
one-half times four kilograms times the initial speed,
squared, six meters per second,
we get negative 40 Joules of net work.
If you get a force versus position graph,
the area under that graph will represent the work done.
So, when you see F versus x,
you should think area equals work.
But, be careful, area above the x-axis is gonna count
as positive work done, and area underneath the x-axis
is gonna count as negative work done,
and make sure the x-axis really is position.
If you get a force versus time graph,
the area's impulse, not work.
So, what would an example of work as area look like?
Let's say a box started at x equals zero
with a velocity of five meters per second to the right
and a net horizontal force on the box is given
by the graph below.
We want to know, at what position other than x equals zero,
will the box, again, have a velocity
of five meters per second to the right?
Well, since the box will end with the same speed
that it began with, the change in kinetic energy is gonna
equal zero.
But, that means the net work would also equal zero,
since the net work is equal to the change
in kinetic energy.
So, if the box starts at x equals zero,
how far do we have to go in order for us
to have no net work done.
Between zero and three meters,
the work done is gonna be negative,
and the area of this triangle is gonna be
one-half the base times the height
which is one-half times three meters
time negative six Newtons
which is negative nine Joules of work done,
and the area under this triangle,
between three and five seconds,
would again be one-half base times height,
which is one-half times two meters
times height of four Newtons
which is positive four Joules of work done.
So, by the time that the box has made it to five meters,
there's been a total amount of work done
of negative nine plus four,
which is negative five Joules of work.
But, we want no net work done.
So, we're gonna have to keep going
until this positive area contribution is gonna
equal the negative area.
In other words, if I can make it
so that all of this negative area is equal
to all of the positive area,
my net work's gonna equal zero.
My negative area is negative nine.
My positive area, so far, is positive four.
If I continue on to the six meter mark,
I've pick up another positive four Joules of work
since the height of this rectangle is four
and the width is one meter,
which means we're almost there.
Four plus four is eight.
I'd only need to pick up one more Joule,
so I can't go all the way to seven meters.
I'd only need to go one more fourth of a meter
to pick up one more Joule
so that one plus four plus four is equal
to negative nine.
So, the net work would equal zero
somewhere between x equals six and x equals seven
which would ensure that the change
in kinetic energy is zero and we would end
with the same speed that we began with.
What does power mean?
In physics power is the amount of work done per time,
which can also be thought of
as the amount of energy transferred per time.
In other words, the amount of Joules per second
that are transferred,
and the name given to a Joule per second is a Watt.
So, you can solve for the power
by finding the work divided by the time
or the change in energy divided by the time.
And you can increase the amount of power
by increasing the work done
or decreasing the amount of time it takes
for that work to be done.
And just like energy and work, power is not a vector.
So, what's an example problem involving power look like?
Let's say a box of mass M slid all the way
down a frictionless ramp of height H
and angle two-theta as seen in this diagram,
and a separate mass M slides all the way
down a frictionless ramp of height H
and angle theta as seen in that diagram,
and we want to know how the average power developed
by the force of gravity on the boxes compares
for each incline?
So, we use the formula for power,
power's the work done per time.
The work done on these boxes is gonna equal the change
in kinetic energy of these boxes
which would equal the change in potential energy
of the boxes, but the mass of the boxes are the same,
the gravitational acceleration is the same,
and the height they fall from is the same.
So, the work done on the boxes are equal,
but the time it takes for these boxes
to slide down the ramp is not equal.
The mass on the steeper ramp will reach the bottom faster
which means it has a higher rate of power being done
compared to the mass on the less steep ramp.
So, even though the same amount of work is being done,
the rate at which that work is being done is greater
for the steeper ramp compared to the more shallow ramp.