Think about it, the angle between leftward and rightward is not zero, that's actually degrees. So this angle would be right here, or pi radians. And cosine of is going to give you a negative one, so the work done by the force of friction on this penguin is going to be negative f k d. Negative the force of friction, times the distance the penguin slid to the right. But this still doesn't answer Walter's question. Where did the kinetic energy go? Friction may have done negative work on this penguin, but where did that energy end up?
And you probably have a good idea, 'cause when two surfaces rub together, some of that energy of motion is going to get transformed into thermal energy in those two surfaces. In other words, this sheet of ice is going to have a little more thermal energy, it's going to heat up just a little bit. And Walter's feathery coat is going to heat up just a little bit, and they're going to have more thermal energy to end with, than what they started with.
Just like when you rub your hands together vigorously on a cold day to get warm, you're turning some of that kinetic energy into thermal energy that warms up your hands. And you might be like, "Alright, that's all well and good, "but how do we put this all together? And we can add the external work that was done, which we've just figured that out. We know the external work would be the work done by friction, so we'd have a minus, 'cause it was negative work, f k d, and it's negative again because this force is taking energy out of the system.
But some people might object, they might say, "Wait a minute, we just said there was thermal energy "to end with, how come we didn't include that "in our final energy? In other words, Walter, and only his motional energy, his kinetic energy, was the only energy we were keeping track of, that's why we said that, initially, there was just Walter's kinetic energy. And this sheet of ice was external to our system, not part of our system, that's why it exerted a negative external work, removed the energy from the system, and Walter ended up with no kinetic energy.
But there's an alternate way to go about this calculation. You could say, "Alright, instead of just considering "Walter and Walter alone to be part of our system, "let's go ahead and include the ice as part of our system. So an alternate way to solve these problems, is to use this same formula, but now, Walter and the ice are both part of our system.
Our system would still start with the kinetic energy that Walter had at the beginning, that doesn't change. But now there would be no external work, not because force of friction isn't acting, there's still a force of friction, but that's an internal force between objects in our system. So there's no external work done now. That might be a new or confusing idea to some people, so let me just say, if there's forces between objects within your system, then those forces cannot exert external work and they cannot change the total energy of your system.
Only forces exerted on objects within your system from outside of your system, can change the total energy of your system. So when this ice was not part of our system, it was exerting an outside external force on Walter, and the energy of our system changed.
We started with kinetic energy, we ended with no energy. But now that the ice and Walter are part of our system, this force of friction is no longer external. It's internal, exerted between objects within our system, and so it does not exert any external work. It's just going to transform energies between different objects within our system.
So that's why we write this zero here, there'd be no external work done if we choose the ice and Walter as part of our system. And this would have to equal the final energy, and we know where this energy ends up.
It started with kinetic energy, and it ends as this extra thermal energy in the snow, and Walter's feathery coat. So I could write that as e thermal. But I know how much thermal energy was generated, this just has to equal the amount of work done by friction. So even though this work done is not external, it still transfers energy between objects within our system, so when we write that the work was negative f k d down here, we mean that the force of friction took f k d from something and turned it into something else, and that's all we need up here.
We need an expression for thermal energy. But if friction took f k d and turned it into something else, the thing it turned it into was the thermal energy so that value of f k d, that magnitude of the work done, was how much energy ended up as thermal energy.
People might find that confusing, they might be like, "Wait a minute, why do we have this "with a positive here and not a negative? If you take energy from something, you're doing negative work on it. If I gave energy to something, I'd be doing positive work. So this negative sign in the work done, just means that the force of friction took this much energy from something, and turned it into thermal energy. So when we want to write down how much thermal energy did we end with, well, we ended with the amount that we took.
So we took f k d, the thermal energy ended with f k d. And I can still set this equal to the kinetic energy that Walter started with, and I get the same formula I ended up with over here, because I had to. Because we're describing the same universe and the same situation, so no matter what story you tell, you should get the same physics in the end. And we do, but some people prefer one to the other.
Some people like thinking of friction as a negative external work, and not including the energy within the surfaces as part of their system. And some people like including those surfaces as part of their energy system, and just including that thermal energy on the e final side. Which is fine, you can do either, you just can't do both. Either the surface is part of your system, and you include it in your final energy, or the surface is not part of your system, and you include it as external work.
But you can't say it does external work and it gains some final energy over here because it's got to be either part of your system, or not part of your system. So long story short, you can basically just think about the thermal energy generated by friction, as f k d, this is a formula that'll let you solve for the amount of thermal energy generated when two surfaces rub against each other.
And we can take this idea a little further. The force of kinetic friction is going to be equal to the coefficient of kinetic friction times the normal force between the two surfaces.
So we could rewrite this. This thermal energy term can be rewritten as mu k times f n times d. And you might say, "Well that's not all that remarkable, "it just looks even worse than it did before. And we still multiply by the d, but now that we've replaced the normal force with m g, we notice that the mass cancels. And this should blow your mind! This means, no matter what the mass of this penguin is, if he starts with the same speed as some other penguin that's more or or less massive, he'll slide the exact same distance.
Now some people will object, they'll be like, "Wait, a really massive penguin's going to have "a lot of inertia, it really wants to keep moving, "it should slide farther. In other words, the more massive penguin does have more inertia, and has more friction. And the less massive penguin has lass inertia, but it has less friction, so the mass ends up canceling, and all penguins, no matter what their mass are, slide the same amount if they start with the same speed. And this also means that two cars, a really tiny Smart Car and a huge SUV, if they've got the same tires, they'll have the same coefficient of friction, and if they start with the same speed and slam on their brakes, they'll both skid to a stop in the same distance.
Again, a lot of people would think that the really massive SUV has to slide farther, but that massive SUV that has more inertia also has more friction, so it stops in the same distance as the smaller Smart Car, that has less inertia, and less frictional force.
While it is easy to think of friction as a 'bad' thing, friction is needed in order to drive it's what pushes the wheels on our car forward and allows us to stop and turn , or even walk. Friction in engines and machines contributes to energy loss, which is what wears out the parts in a car hence the need for lubricating oil.
Friction is a non-conservative force , meaning energy is transferred to new forms not useful to the system but doesn't disappear from the universe, see conservation of energy. For a more complete description of physics please consult hyperphysics. The University of Colorado has graciously allowed us to use the following Phet simulation. This simulation explores the relationship between macroscopic motion, frictional forces, microscopic kinetic energy and temperature:.
Fossil Fuels. Nuclear Fuels.
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