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### Course: UP Class 11 Physics > Unit 8

Lesson 5: Conservation of mechanical energy# Law of conservation of energy

Sal discusses how energy can't be created or destroyed in an isolated system, and works an example of how energy is transformed when a ball falls toward the Earth.

## Want to join the conversation?

- So theoretically, the spring-earth-ball system will go on forever?(12 votes)
- Yes as long as it's just a ball falls perpendicular to a surface with no dissipating forces (like air resistance) to take that energy away.

In the real world, this would be impossible without something putting work to the ball to replace the lost energy.(9 votes)

- What about black holes? The gravitational pull of black holes is so immense that even light cannot escape it. What happens to light inside the black hole? Is it transformed into another type of energy, absorbed as heat, or just... gone?(5 votes)
- currently we do not know what happens to light inside a black hole, but we know that that energy is conserved because black holes emit Hawking radiation which returns the energy that is absorbed. Which is also why they eventually break apart(4 votes)

- What about when you drop something more dense like a bowling ball does the potential energy convert onto kinetic energy?(3 votes)
- The law of conservation of energy holds true regardless of density. So yes, a bowling ball will also have its initial potential energy converted into kinetic energy.(5 votes)

- At10:05when discussing the ball falling onto the spring, shouldn't the ball still have some gravitational potential energy, since the ball is still at some appreciable height above the ground? The video says that all of the mechanical energy is due to Kinetic energy at the moment of impact.(4 votes)
- I think it can be neglected because the video was (is??) focusing on the kinetic energy then(1 vote)

- At5:50, can the energy be also dissipated in form of sound energy?(3 votes)
- Yes that is also possible. Energy is needed for a sound wave to propagate through a medium.(3 votes)

- W = F*D.

So with a force of 5N, mass of 1 and acceleration of 5m/s, for one meter I have a work equals to 5J.

The kinetic energy is then half the mass times the velocity square; half the mass is 0.5 which times 5^2(25) is 12.5J.

How did a work of 5 joules equals a kinetic energy of 12.5 joules? Should it not conserve the five joules of energy inserted on it?

Since the first lesson I have this same problem, but now nearing the end I still could not find what I am missing or how to relate kinetic energy to anything else.(2 votes)- You're right that with those givens (f=5N, m=1kg, a=5m/s², Δx=1m) you get a work of 5J.

There are kind of two mistakes with your kinetic energy there, though. The first is that it seems you're assuming but not directly stating that the initial velocity (which changes when the work is applied) is zero. Remember, the final kinetic energy will depend on both the initial kinetic energy (found from the initial velocity) and how much the kinetic energy changes (found from the work done).

Second, I think you made a mistake in calculating the final velocity for the final kinetic energy. In this situation, you can use a kinematic equation to solve for the final velocity. We'll say that the initial velocity*is*actually v0=0m/s and our other relevant givens are a=5m/s² and Δx=1m. We want to find the final velocity (v), so we can use the equation v²=v0²+2aΔx to get v=√(v0²+2aΔx).

Plugging in the given values gets us v=√(0²+2*5*1)=√10m/s. Putting that into the kinetic energy equation, we get Ek=½mv²=½(1)(√10)²=**5J**. The final kinetic energy is the same as the work done as long as the initial velocity is 0m/s.

To get a more general answer, we can substitute the expression for the final velocity (√(v0²+2aΔx)) into the kinetic energy equation Ek=½mv² to get Ek=½m√(v0²+2aΔx)². This simplifies to get Ek=½m(v0²+2aΔx)

Ek= ½mv0²+½m2aΔx

Ek = ½mv0² + maΔx

Ek = Ek0 + W, which makes sense because it's basically the work-energy theorem.

Does that help a bit?(4 votes)

- It's been said that the universe is energy is this statement true being that matter is the "stuff"of the universe and all matter has energy? To me it makes sense to say yes but I need a definitive answer(2 votes)
- You have to consider that just because someone ate tofu doesn't mean they're now tofu, though people believe that matter and energy are the different interactions by the same something. (Take my answer with a grain of salt though. I don't know too much about relativity. I sure hope someone with expertise can help.)(3 votes)

- A closed system is where energy does not go in or out.

An open system is where energy can leave and come in.

Does the law of conservation of energy apply to open systems?

Thanks :D(2 votes)- In a sense it does. If you keep work in you equation you can still use a form of it. Like Ug + W = K. In that example the gravitational energy plus the work done on the system equals the final kinetic energy. You would use something like that in an open system(2 votes)

- How is
**Initial K**+**Initial U**=**K**+**U**? I quite don't get it. In6:02(1 vote)- This is just another way of stating energy is conserved—the initial kinetic energy plus the initial potential energy is equal to the final sum of the two.

Consider a bike rolling from a hill. At the top of the hill, it has 0 kinetic energy, but plenty of potential energy. At the bottom of the hill, it will have zero potential energy, but plenty of kinetic energy. The relationship simply states that the sums are equal.(2 votes)

- Then why doesn't perpetually motion devices work?(1 vote)
- The entropy in the system will always increase. The longer the machine runs the more energy will be converted into heat until the system can no longer sustain its motion.(2 votes)

## Video transcript

- [Instructor] Get excited because in this video
we're going to talk about one of the most important
laws in all of science. And that is the law of
conservation of energy. You'd be amazed by how
much of the universe we can infer based on the law
of conservation of energy. And you'll be amazed by
how many holes you can poke in science fiction plots based on the law of conservation of energy. Let's just start with the language that you might typically see. And then we'll try to understand
it a little bit deeper. So it tells us that the total energy of an isolated system is constant. Energy is neither created nor destroyed. It could only be transformed
from one form to another or transferred from one system to another; I pretty much underlined the whole thing because it's so important. Now to understand this,
let's just think about the types of energy that we have studied. We have studied things
like kinetic energy, which is the energy due
to an object's motion. We have talked about potential energy, which you could view as energy
due to an object's position. And that would be the case of
mechanical potential energy. If you were to combine
these two types of energy, together, they're known
as mechanical energy. So let me put a little box around it. That is mechanical energy. And when you're first learning physics, these are the types of
energies that we focus on. But there are other types of energy. There's thermal energy. You have nuclear energy. You have chemical energy. And so these aren't the only forms. So when we talk about the law
of conservation of energy, things like kinetic energy
could be transformed into chemical energy. But we're not gonna talk about
those other types of energies in this video. So to start to appreciate this, let's first think about
how mechanical energy can be conserved. So you could almost view this as a law of conversation of mechanical energy. But then we're gonna make things a little bit more complicated and see if we can trip ourselves up and see if we can somehow defy the law of conservation of energy. And be very skeptical of anyone who claims that they can defy the law of conservation of energy. So let's start with a system that contains all of the Earth in a ball. So let's just call this
the Earth-ball system. And when you're dealing with the law of conservation of energy, it's important to specify your system. And we're going to assume
that it's an isolated system, that it's not interacting much with other outside systems, the things like the sun or whatever else. And so we have, I've drawn the Earth here, in this kind of grass flat thing. And then let me draw my ball. And let's say my ball
is held above the Earth, just like that. So while the ball is stationary, and we'll assume that there's no air here. So while the ball is stationary like that, then we have all potential energy, we could call it gravitational
potential energy. So it's all, and the symbol for potential energy we tend to use is U. And we could say this is
gravitational potential energy. And we could say there's
no kinetic energy. No kinetic energy. If we thought about a broader system, if we talk about the solar
system or something like that, then the Earth is orbiting around the sun, the sun is orbiting around
the central of the galaxy. But that's why we're specifying, this is the Earth-ball system. But what would happen if I
were to let go of the ball? And especially what is the
energy profile of the ball right as it touches the
ground right over here? And I'm assuming it's just
going to hit the ground and just not bounce in any way, that would complicate things. Well, in that situation, all of a sudden you have no gravitational
potential energy, but right as it touches it,
not when it's stationary, right as it touches it, it's gonna have a lot of kinetic energy. So all kinetic energy. And so what we saw, what we see here, is that that potential
energy all gets transferred into kinetic energy right as
that ball touches the ground. Now I know what you're thinking. But what about right after that moment? If that ball, especially
if doesn't bounce, if it just sits there, it looks like we have no energy anymore. It looks like energy has been destroyed. So my question to you is
where did that energy go? Pause the video and try to think about it. So some of you might
say, "Hey, once the ball "has just, it's at rest there, "well, maybe we found a case "where we have defied the law
of conservation of energy." And, remember, I told you, be skeptical if (chuckles) anyone ever tells you that. Where the energy has gone,
it's been dissipated. It has gone into heat. It would have been converted
into thermal energy. So the ball and the
ground would actually get that much warmer because
that kinetic energy, right as it touches the ground would be turned into thermal energy. So one again, we have not defied the law of conservation of energy. Now another thing you might say is, "Well, okay, imagine a
world that there is air." So let me draw a bunch of air particles right over here. And we know that if as a ball falls down it goes through the air, you could consider that air resistance. Some people would call that
the friction due to air. Well, that would slow that ball down. So maybe it would not have
as much kinetic energy when it gets down here. And so it seems like
energy would be destroyed in that situation. And, once again, I would tell you, no, the energy has not been destroyed. As the ball falls down, it's
going to heat up the ball and the air around it. And so, once again, that air resistance, that is a dissipative force. It's going to result in the
generation of thermal energy. And if we wanted to write this in terms of equations,
there's a couple of ways to write this. We could write, if we're
just writing the law of conservation of mechanical energy, and we're not talking
about dissipative forces, we could say that the
initial kinetic energy plus the initial potential
energy is going to be equal to, is gonna be equal
to your final kinetic energy, your final kinetic energy, plus
your final potential energy. Now another way to write
this exact same thing is to say that the
change in kinetic energy plus the change in potential energy is going to be equal to zero, assuming we don't have
any dissipative forces, and assuming that we're not converting into some of these other forms of energy, like chemical energy, or thermal energy. But if you wanted to
include dissipative forces, dissipative forces do something called non-conservative work. They do actually negative work, because the force of, say, friction, is always acting opposite
in the direction of motion. So to factor that in, we
could rewrite these equations. We could write that your
initial kinetic energy, plus your initial potential energy, plus any work done by
non-conservative forces, this would be like air
resistance or friction. And this would be negative right over here if we're talking about, say, friction. That is going to be equal to
your final kinetic energy, plus your final potential energy. Or this one right over here, we could write your change, change in kinetic energy, plus
change in potential energy is going to be equal to the
work done by dissipative forces. And, remember, if we're
talking about friction, dissipative forces, this right over here is going to be negative. Another way that you
could've thought about this is we could've put in thermal energy, we could say that our
change in kinetic energy, plus our change in potential energy, plus our change in thermal energy is going to be equal to zero. Or you can include the work done by a dissipative force. And so, for example,
if you saw a situation where your total change
in mechanical energy right over here was negative, you're not defying the law
of conservation of energy. It's not the energy was destroyed, it's that you had this negative work done by those dissipative forces. And where did that energy go? It gets converted to thermal energy. So let's do a couple of other examples just to appreciate this. So let's do a Earth pendulum system here. So here, that's the Earth. And then I have some type of tower. And let's say I have a pendulum here. I have a pendulum. And at the low point, the ball just goes right over there, then it goes back up to that point. And let's say the difference in height between this point right over here, and this point where
it is right over there, it is equal to H. And let's say this is the highest point that the end of the pendulum will get to. So at this point we are
maximum potential energy. And right as it's about
to change direction, we have no kinetic energy, it's gonna be stationary
for just a moment. But then the pendulum's
going to swing back. And when it gets right over here, all of that potential energy is gonna be converted to kinetic energy, assuming we don't have
any dissipative forces like friction, slash, air resistance. And then all that kinetic
energy gets converted back into potential energy. Another example that we could look at that would complicate
this a little bit more is to think about an
Earth spring ball system. So that's the Earth. And let's say there's a
spring right over here. And we have a ball that starts stationary. So up here it's all potential energy, gravitational potential energy. So all gravitational potential energy. Assuming we have no air
resistance, we let go. Right before it touches the spring when we have maximum velocity, here it is going to be all kinetic energy, all kinetic energy, but then it's going to
compress the string. And assuming we don't have
any thermal energy generated, you actually will always,
in the real world, have some thermal energy generated, but if the spring gets
compressed and at some point the ball is over here, well, now, you have some of that
energy that's been converted into spring potential energy. Or sometimes it's called
elastic potential energy, because by virtue of
compressing that spring, this thing's going to rebound and it could be converted
back into kinetic energy and/or gravitational potential energy.