9. Work & Energy
Power
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guys. So up until now, all of our problems have been involving work and energy. And in some problems, you'll have to calculate or find how quickly that energy gets transported or is done to another object. And that's exactly what power is. So, in this video, I'm just going to quickly introduce you to the equation for power. We're going to see what it is and we're going to check out some problems. So the whole equation for power is that the average power is equal to the amount of work that is done or gets done by an object divided by the change in time. It's how quickly that work gets transferred to something else. So the reason this is often called an average power is because we're calculating between two points in time. Now, another way we can write this is by using the relationship between work and energy. Work is just a transfer of energy, so sometimes you might actually see this equation written as E over T. The unit that we'll use for power is called the watt, which is written by the symbol w, which is kind of confusing. So don't confuse this w here with the work that's done. Unfortunately, they both use the same letter, but the watt is really just a joule per second. It's energy divided by time. So we've actually seen this diagram before up here. This is really just a diagram that relates all of the important variables that we've seen so far in the chapter like forces and works and energies. And all we're doing here now is we're just adding another branch. We're just adding another connection between work and power. Power, as we said before, is just how quickly we are doing work. So this is just w over delta t. That's really all there is to it. Let's go ahead and check out some examples here. So we're going to calculate how many energies or how many joules of energy is done by a 100 watt light bulb. So, this is actually just giving us the power. This is p equals 100 and we want to find out how much energy uses in an hour. So, this is actually a time. So, this is in hours and we want it in seconds. So this is just going to be 60 minutes times 60 seconds per minute. So just in case you didn't know this, this is actually 36 100, that's how many seconds are in an hour. And so basically what happens is we have our P average equation, which is equal to the work that's done over time, but it's also related to the amount of energy that something consumes per time as well. So between these two equations work and energy, we're trying to figure out how many joules of energy is done, so we're actually gonna use this relationship right here, we're trying to solve for this E. So what we can do is we can just solve for this E, this E is really just going to be the average power times the time. So this is gonna be 100 watts times 36 100 seconds and you get a energy of 360,000 joules. So a lot of your problems are actually going to be solved using this pretty straightforward equation. So that's how you do those. Alright. So let's go ahead and move on to the second one. Now we're gonna have objects that are moving, and and exerting forces and changing velocities. Right? So, here we have a 1300 kilogram sports car. So, I've got this flat ground like this. We know that the mass of this car is 1300 and it starts from rest, which means that the initial velocity is equal to 0. And what happens is it's accelerating from rest because of the engine of the car. So it's being pushed by some force here. I'm going to call this F engine like this and eventually at some later time, we know that the velocity of the car is not zero anymore, it's actually equal to 40 meters per second. So we have the time frame that this happens in this delta T is equal to 7 and we want to figure out how what is the average power that's delivered by the engine. So think about it like this, right? This is our P average like this. That's ultimately what we want to find. And this is really just because the force that the engine produces is going to move this car along or through some distance. This force acts through some distance like this, this x, which I actually don't know, but it's doing some work as it's pushing it through that distance. That's the whole reason that this car accelerates from 0, which has no kinetic energy and then it finally has an energy velocity of 40. So we've had a transfer of kinetic energy and therefore some work is done. So how do we calculate this power? So our P average is really just going to be the work that's done divided by the change in time or it's the energy divided by time. So we're not going to use this equation because we know that some work has been done and we also know that the change in time is 7 seconds. So all we really have to do here to figure this out is figure out how much work is done by the force of the engine. So how do we do this? How do we calculate work? Well, according to our diagram here, we can always calculate works if we have forces and displacements by using F d cosine theta. And then if there's multiple works, you can just add up all of the works done. So if you don't have a force and you actually can't use either one of these equations to calculate work. So then we're kind of stuck here. How do we actually figure out the work? Well, remember that the all the other way you can solve for the work is by using the kinetic energy theorem. The net work that's done on object is equal to the change in kinetic energy and remember that this change in kinetic energy is actually related to the change in velocity. So here we have a velocity of 0 and at 40, so now we can actually calculate what that change in kinetic energy is. This is Delta KE and this is really just 1 half mvfinal squared minus 1 half mvinitial squared. So all we do here is we have actually have the mass and we have the velocities at both the beginning and end so we can calculate this. We know that this right term here is going to go to 0 because the initial velocity is equal to 0. And so basically, we're just going to plug in this expression in for our work. Alright. So you can actually calculate all of this really in one step. We're gonna take the one half m b squared and we're gonna say that the average power is really just the one half m b squared divided by delta t. So in other words, this is one half of 1300, that's the mass, times the final velocity, which is 40, square that, and then divide it by 7. So just be careful when you plug this in, but when you plug all of this into your calculator, you should get a final power, an average power of 1.49 times 10 to the 5th watts. That's a 149 kilowatts. And if you want to, you can convert that to horsepower, but that's how much power is delivered by the engine during this acceleration. Alright? So, hopefully, that made sense. Thank you for watching, and I'll see you in the next video.
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