Physics: Power

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We've defined the physical quantity work. It's a very useful concept, but it's not always the whole story. Let me just give you an example. Supposing that I was to start at the ground floor of my physics building. It's a big tower. And walk up ten stories. I would have to do a certain amount of work against gravity to go from the bottom to the top. The work done against gravity is mgh. It's the mass of me times gravitational acceleration constant times the height of the building. So a certain fixed amount of work, I would have to do a little bit more work, because I would also be fighting friction. But that's the work that I would have to do against gravity.
Now supposing instead of walking up, the next day I run up the stairs. I'm in a hurry. And it sure feels like there's something totally different going on. And if we were just being sloppy in our use of English, it might seem like I did more work. But no, going from the bottom to the top takes mgh of work against gravity, whether I do it fast or whether I do it slow. So the concept of work is somehow missing this physical sense that it matters how long it takes. What we're worried about here is the rate at which work is being done. And so since that's clearly got some physical sense, we give it a name. And we define power. What I'm talking about that's different when I walk up or run up is not the work done. That's the same in both cases. It's the power, which is required.
Power is defined to be work done divided by the amount of time you took. So in this example, the work is the same, but the amount of time when I run up is much smaller, so the average power is larger. Remember a bar over a quantity just tells you that it's averaged over some interval of time. Power is not a vector. It's a number. And you might worry a little bit about the fact that when I'm running up the stairs, maybe at first I'm running really fast, so I get up the first half in a really short amount of time. And then I'm getting more tired, so that the average power for the first half of the trip is different than for the second half of the trip. It's the usual story. When you've got an average quantity, sometimes that's useful, but you might also be interested in how much power is being expended right now at this instant in time. And just like with velocity and with acceleration, with any quantity that's defined in terms of an average, you can also ask what's the limit as you average over a shorter and shorter time interval? What's the limit as delta T goes to zero? And that's the instantaneous power. It's just the rate of work per second right now. Work done is energy transferred. So sometimes people think of power as how much energy is transferred per second.
What are the units of power? It's a new physical quantity. It's always interesting and useful to think about the units. When you measure power, it's going to be units of work, which were joules, divided by units of time, which is seconds. Now of course a joule is a Newton-meter. And a Newton is a kilogram-meter per second squared. So there are lots of different ways to write it, and whenever you get these complicated combinations of units, we usually just give it a name. So one joule per second is called one watt. One watt of power is one joule per second. And a watt is abbreviated one w, which is a little annoying, because it's the same symbol as the symbol for work. So you've just got to use context to keep this straight.
When you talk about power, you think immediately of the power company. It's a little bit ironic, because the power company isn't really charging you for power. What they're charging you for is the energy that they're delivering to your home, or the work that they're doing in your home. So what's the story here? If average power is work divided by delta T. Then the amount of work that's done is average power times the amount of time. So if you run a 100 watt light bulb that's consuming 100 watts, that's 100 joules every second, how much work the power company has to do, how much energy is being expended, depends on how long that bulb burns for. So when you get your bill at the end of the month, they don't tell you average power. They tell you how much energy did you purchase, or how much work was done in your house. So they're telling you in the bill power times time. And the unit that they will usually write the bill in, it's a unit of energy now, is kilowatts times hour. It's not divided by. It's multiplied by. Power, that's kilowatts, times time, that's a unit of energy. How much energy is it or how much work? 1,000 watts times 3,600 seconds is 3.6 x 10^6 joules, one kilowatt hour. In Colorado we've got pretty cheap power. It's about ten cents for 3.6 megajoules. Sounds big.
It's a lot of energy, 3.6 megajoules. It's kind of nice, when you see numbers like this, to try to get some kind of physical intuition for how much energy is that. We know that when you lift something up, you have to do mgh of work against gravity to lift it up a height h. So how high would you have to lift me, with a mass of 60 kilograms, in order to consume 3.6 x 10^6 joules of energy? And the answer is, if you just set this equal to my mass times g times h, and solve for h, about 6,000 meters. Mount Everest is 8,000 meters. That would be, of course, a perfect conversion of work into lifting me up. That would be neglecting friction and everything. But still it's an impressive amount of energy. And it's sort of impressive how cheap it is.
There's another formula for power. Here I've written it as . And remember what's dW? It's force times infinitesimal displacement divided by dt. That's just our old definition of work, force times displacement. And is change in position with time. That's velocity. This is F dotted with v. So that's a very useful alternative formula. You can use either one as you wish. If I give you work and time, this one is convenient. If I give you force and velocity, this one's convenient. It's just a question of convenience. You can see this dot product gives you a scalar. And this is often handy when you're pushing on something. Think about me walking up the tower. This formula makes some sense. When I walked up rapidly, I said that required a lot of power. When I walked up slowly, it required less power. The work done was the same in both cases.
Let's do one more example. A couple of weeks ago, I took my bike out and I rode up a little mountain behind my house called Flagstaff Mountain. And it took me about 45 minutes to get up Flagstaff. I'm not the super strongest biker in the world. And my little bike odometer said that I had traveled about ten kilometers. And I looked at a map later, and the map indicated that I had gone up about 600 meters. It's a big climb going up Flagstaff. So you could ask the question what was my average power on this trip? Kind of an interesting just a fun question. How much power was I expending to do this ride? A lot of numbers here, so you think should I use force times velocity? I can figure out velocity, at least on average. I know distance divided by time. But I don't know what force I was applying on the bike. So that formula is not so convenient. But I know how much work I have to do, at least just the work against gravity. Let me neglect friction. Bicycles are pretty efficient. Of course there's internal friction in my body that's pretty important. But at least against gravity, I went up a height h. And so I did mgh worth of work against gravity. So I suppose I should say the average power was greater than this, since I'm neglecting friction. And the amount of time was 45 minutes. So I can plug in numbers. 60 kilograms, 9.8 meters per second squared, the h is the vertical rise 600 meters divided by time. And when I plug in the numbers, I get about 130 watts. You know, a bright light bulb. It's also impressive now when you look at a 100-watt bulb to think about how much energy that thing is consuming every second. In fact, when I'm just standing here, just doing internal chemical work against--and friction inside of my body, I'm expending, on average, about 100 to 130 watts just sitting still. So this is sort of the extra work that I had to do against gravity.
Once I know the average power, I can go back to my force times velocity. My average velocity was ten kilometers over 45 minutes. And I could deduce what the force that I had to apply was. It came out to be about 35 or 40 Newtons, not the hugest force in the world. It's not all that steep of a grade. It just goes on for quite a ways.
130 watts, you may have heard of the British system of power. The British unit of power is horsepower. It's the amount of power that some kind of average horse can expend. Nowadays one horsepower is defined to be exactly 746 watts. So 130 is about a fifth of a horsepower. So that's me, I'm about a fifth of a horse when I'm riding my bike.
A strong athlete can put out--you can put out power, but the amount of energy consumed is power times time. So for a brief amount of time, you can put out a lot of power, but if you go over a long amount of time, typically you can't maintain that power level. A good athlete might be able to put out 400 or 500 watts for a length of time like an hour, for a marathon runner or a bicyclist. That's an Olympic athlete. And for a shot-putter, someone who is just putting out power for a very brief amount of time, they're expending energy over a very short amount of time, they might get up to a kilowatt.
So power is a useful concept. It contains physical information that's not contained in work alone. Power is not energy. It's not work. It's certainly not force. The English language kind of mixes up all these words. Power is definitely well defined. It's amount of work done per second. And you've got two ways to calculate it; either work per time, or force times velocity. You look at the problem. See which is more convenient. And it's often a handy quantity to calculate.
Energy
Work, Kinetic Energy, and Power
Power Page [2 of 2]