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## Work energy theorem

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# Work-energy theorem

## Video transcript

- [Instructor] We've talked
about work and energy for quite a while now. But it's time to answer the big question, why did we introduce these concepts? Because as we see, this will
help us look at the world in a completely different way. So let's take an example. Imagine you take a
bowling ball in your hand, and let's say you throw it. What's gonna happen? Well, that ball will speed
up as it leaves your hand. Now if you ask the question,
why did the ball speed up? What is the answer? Well we can say, well, when
you are throwing that ball, you are pushing on it,
so you put a force on it. And whenever you put
a force on any object, that object accelerates,
and therefore it speeds up, and that's why the ball became faster. And this is in accordance
to Newton's second law, which says F equals ma, which means force will accelerate a body. Of course if there are
more than one forces, then we have to calculate the net force, but we'll talk about that later. Let's consider only one force as of now. So you're putting a force,
and that accelerates the body. That's one way to think about it. Now let's think in terms of work done. You see, because you are pushing on it, and you are displacing that body, you are doing work on that ball, right? And when you're doing work, look at what happened to the ball. Initially it had no speed. It was at rest, so it
had no kinetic energy. But then, after doing work, notice it has gained kinetic energy. So we can now say that by doing work, we have added kinetic
energy to our bowling ball. And so by doing work, whenever
work is done on an object, kinetic energy gets added to that object. This is a new way of looking
at the same situation. Now you can also do work and remove kinetic energy if you want. For example, imagine a
ball is thrown at you, and you try to catch it. Now when you try to catch
a ball like this, again, your hand will move back a little bit. Now think about what happened. When you try to catch that ball, your hand will start pushing it. But this time it pushes it
in the opposite direction. And that's why it's
slowing it down, right? But now think about the work done. Because your are pushing it in the opposite direction
to the displacement, work done is negative. And now, when you're doing negative work, look at what happened
to the kinetic energy. The kinetic energy is reduced. So we can now say when
you're doing negative work, you are removing the kinetic
energy from the body. So negative work removes
kinetic energy from the body. And this is, crudely speaking, what we call the work-energy theorem. It basically says when you do work, you either add, or you remove the kinetic energy from the body. If you don't do work at all, then the kinetic energy will not change. Of course, we'll derive
the mathematical version a little bit later, but before we do that, let's look at some more examples. Let's say now you take that bowling ball and you drop it from some height. Now as it falls down,
it's going to speed up. Again, let's look at work. In this particular case, gravity is pushing down on that ball, and the ball is moving
in that same direction. So gravity's doing positive work. And because gravity's doing positive work, it's adding kinetic energy to that ball, and that's why the ball is speeding up. Similarly, if you take that same ball now and throw it up, let's say you threw that ball up. Now, as the ball goes up,
you might know it slows down. Again, what's going on? Well, again, gravity's
pushing down on that ball. Remember, gravity always
puts a force downwards. But this time the ball is going
in the opposite direction. So gravity's doing negative work. And as a result, gravity's
removing kinetic energy, and that's why the ball is slowing down. This is pretty cool, right? Thinking in terms of work and energy. Now, things get even more interesting when there are more than one
forces acting on the body. So, if there are more
than one forces acting, according to Newton's second law, we need to now calculate the net force. Meaning, we have to add
up all those forces. Well, of course, if the forces
are in the same direction, we add them, and if they're in the opposite direction, we subtract them. But anyways, you calculate what the total force on that ball is, and that will tell you what would be the acceleration of our body, right? Now similarly, if there
are more than one forces acting on the body, you
calculate the total work done by all the forces, and now that decides what happens to the kinetic energy. So if that total work done by all the forces becomes positive, kinetic energy will be increased. Kinetic energy gets added. If that total work done
ends up becoming negative, that means kinetic energy will be removed. And if the total of them is zero, which is totally possible, some forces can do positive work, others might do negative work, and we'll look at some examples, the total work done can be zero, and in that case, kinetic
energy will not change at all. Again, let's look at a
couple of examples of that. Let's say there is a
cupboard a very rough floor, very heavy cupboard, and you want to move it,
so you start pushing on it. Let's say as you push on it, you slowly displace it from here to here. Now after displacing, after you pushed it, the cupboard is still at rest. So let's think about the kinetic energy. Initially the kinetic energy
was zero, it was at rest. After pushing also, the
kinetic energy's zero. So the kinetic energy
didn't change at all. So it was neither added nor removed, that means the total work done
on this cupboard was zero. Why? Clearly you pushed on it, you did work, then why is the work done zero? That's because there
are more than one forces acting over here. So, you are pushing on it, and you are doing positive work, but because the floor is rough, friction is pushing on that cupboard in the opposite direction,
doing negative work. So, you are doing positive work, friction is doing negative work so it just so happens that
the two work dones cancel out. Total work done ends up becoming zero. And that's why the kinetic
energy didn't change at all. Another way I like to think about this is, when you are doing positive work, I like to think that you are adding kinetic energy to the cupboard, but at the same time, friction is removing that
kinetic energy from the cupboard, and that's why the kinetic
energy didn't change at all. Okay, let's do one more. Say I take this very cute puppy, which is initially at rest, it's at rest right now. I push on it and I raise it up slowly, and finally, again it's at rest. So even here, the kinetic
energy did not change. That means the total work
done on this puppy is zero. But, why? Clearly I am pushing on it,
and I am making it move, so why is the total work done zero? Can you pause the video
and think about this one? Well, again, I am pushing on it, and I am doing positive work, and I am trying to add
kinetic energy to this puppy. But there's one more force
acting on this puppy, which one? Gravity. Gravity is pushing down on that puppy, and as a result it's doing negative work, because it's in the opposite
direction of the displacement, and so gravity is removing kinetic energy from the puppy at the same time. So I'm trying to add kinetic energy, gravity is removing the kinetic energy, and that's why there is no change in the kinetic energy at all. So my positive work is canceled out by gravity's negative work, making the total work done zero. So now that we have some idea
of the work-energy theorem, let's go ahead and derive the
mathematical equation for it. This will be useful for problem solving. All right, so I need to connect work and kinetic energy, right? So again, let's start with just one force. If there's one force acting on the body, then the work done by that force, the work done would be, the force, multiplied by the displacement. Let's call the displacement as s. Okay, what do I do next? Well, I want to somehow bring
velocities in to the picture, because I want kinetic energy, right? And I can do that by
substituting F equals ma. 'Cause that's how acceleration comes, and then from acceleration,
I can bring velocities. So, if I substitute F equals ma, so I'll get m times a times s. But I don't want acceleration, and I don't want displacement. I want velocities. So, can you think of a connection between acceleration,
displacement and velocities? Initial velocity and final velocity? Yeah, go back to equations of motion. Think about this. Great idea to pause the video and see if you can remember an equation. Well, there's an equation v squared equals u squared plus two as. This is perfect because
this will help us connect as and right it in terms of v and u. So I want to get rid of as, isn't it. So let me isolate this on one side, and put everything else on the other. So if I rearrange this, we'll
get v squared minus u squared, divided by two. That equals a times s. Right? So now I can substitute that over here. So that will give me m
times as is v squared minus u squared, divided by two. Oops, let's use the same color. U squared divided by two. And now if I open up that bracket, see what I end up with. Left hand side is work, and that equals, mv squared by two, what is mv squared by two? That's the kinetic energy,
final kinetic energy. So we can call that as
the final kinetic energy, minus, you get mu squared by two, what is mu squared by two? That will be the kinetic energy initially, before I started pushing it. And this is the work-energy
theorem mathematically. It's saying the same thing
that we discussed already. If you do positive work, notice the final kinetic energy
will be more than initial. That means you're adding kinetic energy. If you're doing negative work, you are removing kinetic energy. This equation is also telling that if you do hundred joules
of work, as an example, then your kinetic energy will
increase by hundred joules, so you will add hundred
joules of kinetic energy. Similarly, if you do hundred
joules of negative work, then you'll remove hundred
joules of kinetic energy. So you can kind of now see
what work really represents. When you do work on a body,
what happens to that body? Its kinetic energy changes. So whatever work you do, that much kinetic energy
gets added or removed. And again, if there are
more than one forces acting on the body, what happens? Now you just have to
calculate the work done by all the forces. So we can say this will
be the total work done, or we can say this is the net work done. So the total work done on a body will decide how much kinetic energy gets added or removed. And we will see in future videos that certain kinds of problems can be solved much faster
by using this equation rather than F equals ma. Okay, so, to summarize, what
did we learn in this video? We learned the work-energy theorem, which is basically saying that when you do positive work on a body, you add that much kinetic
energy to the body, and if you do negative work on that body, you remove that much kinetic
energy from the body. Of course if there are
more than one forces acting on the body, then you have to calculate
the total work done, and that decides how much kinetic energy gets added or removed.