9. Work & Energy
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Hey 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 transport 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, is how quickly that work it's 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 tea. The unit that will use for power is called the watts, 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 want is really just a jewel per second, its 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 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 and check out some examples here. So we're going to calculate how many energies or how many jewels of energy is done by a 100 watt light bulb. So this is actually just giving us the power, this is peak was 100 we want to find out how much energy uses in an hour. So this is actually a time. So this is an hours and we want it in seconds. So this is just gonna be 60 minutes Times 60 seconds per minute. So just in case you didn't know this, this is actually 3600, 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 jewels of energy is done. So we're actually going to use this relationship right here, we're trying to solve for this e. So we can do is we can just solve for this. E. is really just going to be the average power times the time. So this is going to be 100 watts Times 3600 seconds, and you get a energy of 360,000 jewels. So, a lot of your problems are gonna actually gonna be solved using this pretty straightforward equation. Um So that's how you do this. All right, so let's go ahead and move on to the second one. Now, we're gonna have objects that are moving uh and and exerting forces and changing velocities. Right? So here we have a 1300 kg 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 zero. 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 40m/s. So we have the time frame that this happens in this delta T. Is equal to seven. 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 or through some distance. This force acts through some distance like this, this X. Which I actually don't know what it's doing. Some work as it's pushing it through that distance. That's really that's the whole reason that this car accelerates from zero, which has 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 RP 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 seven 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. Cosign 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 network that's done an 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 zero and at 40. So now we can actually calculate what that change in kinetic energy is. This is delta K. E. And this is really just one half M. The final squared minus one half M. The initial 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 zero because the initial velocity is equal to zero. And so basically we're just gonna plug in this expression in for our work. So we can actually just go ahead and calculate this and use in one fell swoop. We can say that the average power is going to be one half M. V squared like this divided by the change in time. So we just plug in all of our numbers. This is gonna be one half of 1300 the final velocity is 40 squared and they're gonna have divided by seven seconds. And what you're gonna get is you're going to get um and then when you plug this in you're going to get 1.4 times 10 to the sixth watts. That's the total amount of power output by the engine in order to accelerate this car and give it some kinetic energy. So that's it for this one. Guys, let me know if you have any questions.
How much power must an elevator motor supply in order to lift a 1000 kg elevator at constant speed a height of 100m in 50 seconds?
Power of Pushing a Box
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Hey everyone, Let's check this out here. So, we have a 15 kg box that is sliding across a frictionless surface and its speed is going to increase. We're told that the initial speed of this block is 10 m per second, but it's accelerating over some distance. And then at some point over here it's final velocity is going to be 30. We're told that the acceleration along this interval here, this A is equal to two. That's basically all we know in the first problem, we want to calculate what is the change in the mechanical energy of the blocks. Mechanical energy. So, the variable that we're looking for is delta M. E. It's the change in mechanical energy. Now, remember for blocks, mechanical energy could be potential or kinetic. So, really what they're asking for here is what is the total change in the potential energy plus the change in the mechanical energy. All right, So, let's consider each one of these terms here. Now, we have a block that's not attached to any springs. There's no springs in this problem and there's also no gravity because we're just going along the horizontal surface. So there's no change in the potential energy. So, really all that happens here is that the bloc's kinetic energy is changing across this interval and because it's going faster right from 10 to 30 and therefore there's a change in the mechanical energy. So that's delta Emmy. This is just gonna equal K final minus K. Initial. So delta Emmy here is just gonna equal the formula for kinetic energy is one half M. V final squared and then one half mv initial squared. Now I have all of those values, I have the masses and velocities. So I can actually figure this out. My delta Emmy. It's just gonna be one half of 15 times V final which is squared minus one half times 15 which is times squared when you go ahead and plug this all into your calculator, you're gonna get 6000 jewels. So one way you can also think about this here is that there's a force that is pushing this thing that's pushing this block, that's why it's causing it to accelerate. We don't know what that force is but that forces doing work on the object and that work is equal to the change in the kinetic energy. So that work that you're inputting, changes the mechanical energy by 6000 jewels. Alright, that's one way you can think about it as well. Let's move on to the second problem. And the second problem, we want to figure out the rate at which energy is transferred. So that's a dead giveaway that this is actually talking about power. Remember that power is equal to the change in energy over the change in time. And because we're not told because we're basically told two points right from initial to final. We don't know what's happening in between. This is an average power. So all we really have to do here is realize that the change in energy is actually just the change in the mechanical energy that we just calculated. So if we want to figure out the power, all we have to do is just have the mechanical energy which we already have. But we also need to know the time. We don't have anything any information about time. So that's what we need actually. I'm sorry. This should be a question mark. We don't have anything about time. So we're gonna have to go solve that in order to get power. So let's go ahead and do that. Right? So if we want time here, let's see. We're also told that the mass of this block, the information we're told is also the initial velocity, the acceleration and the final velocity. These are Kinnah Matic variables. So if I want to figure out t which is also a kingdom attics variable. I just need to do exactly what I did when we talked about one dimensional motion. I need to set up my five variables and I need three out of five. So I don't I don't know t that's actually what I'm gonna be looking for here. So I have venus which is 10 v final which is 30 and I have the acceleration which equals to the last one is delta X. It's the distance over which these things is accelerated. I actually don't know what that is. All I'm told is the initial and final velocities and the acceleration between I don't know what the distance, but luckily I actually have three out of five variables, I have 12 and three to the equation that's going to relate them. And also give me time. It's gonna be the first cinematic equation. This is basically just the final equals v knots plus 80. Change of velocity equals acceleration times time. So you re arrange for this because we really want to solve for this time over here and you'll find that we're gonna get the final minus the initial divided by a equals time. So we gotta plug this in. This is going to be 30 -10 divided by two. And this is going to give me 10 and this is gonna be in seconds. So this number here, this 10 seconds is now what I just plug into the power average equation and I'm done. So this p average is just gonna be the change in the mechanical energy which is joules divided by 10 seconds. And so that's gonna give me an average power of 600 watts. Alright, so hopefully that makes sense guys, let me know if you have any questions. That's it for this one
Power of a Winch on an Incline
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Hey guys, so let's check this one out together. Here we have a block that is 1400 kg that gets pulled up by a winch and cable which is basically just like a cable that's attached to a spinning or cranking motor, that's gonna pull this block up the incline. So this is probably happening like a rock quarry or something like that. I don't know. So I'm gonna call this force of the cable the tension, right, That's tension by a cable. What I want to ultimately figure out is the power that is due to the force of this cable. So how do we figure this out here? Remember that? The power equation is just delta V. Over delta T. But it's also equal to the work that is done, divided by delta T. Remember that power that work and changing energy means the same thing. So we're actually gonna do here is because we're trying to figure out the force of the cable. We're actually gonna use this work equation. Now, what I have is I actually have the delta T. And that's just gonna be the 25 seconds. That's this delta T. Over here. And remember that work is always equal to force times distance. So what I can do here is I can set this up as the tension force times the distance that this block gets pulled up. I'm gonna call this some D. Here times the cosine of the angle between those two vectors, right? So this tea and this d. They both pointed the same direction up the incline, Therefore the angle between them is zero and this just equals one. And now we're just going to buy this by delta T. So in order to figure that power, I'm gonna actually need to figure out the tension force. I'm gonna actually have a number for it. And I also need to figure out the distance that this block gets pulled up. I don't have either of those numbers. So how do we do this first? I'm actually gonna go ahead and list all the things I know about this problem already. Have the mass. I know that the incline angle is 37 degrees and I know that this block is pulled up at constant speed. What that means is that the acceleration is equal to zero. I'm also told that the coefficient of kinetic friction is mu k 0.4. And I have that this delta t. Here is 25 seconds. The only other thing that I know here is that the height of the incline, which is this over here, not R. D. This is I'm gonna call this H is equal to 60.2 m. So let's go ahead and get started here. The first thing I want to do is figure out the tension that's in the cable. And because I know a couple of things about the forces in this problem, it's really just gonna turn into a forces on an inclined plane problem. So remember that there's this force that's pulling this up. But the reason that this thing has pulled at constant velocity is because I have an MG. But this MG has a component down the incline. This is MG. X. And I also have a friction force. So this is gonna be my friction K. That is preventing this thing right? That's actually exerting a force downward again because this thing is being pulled up at constant speed this way. So this is gonna be my V. So if I go ahead and set up in Newton's laws I need to set an F. Equals M. A. In order to figure out this tension here. So I'm gonna have to do that. So F. Equals M. A. Now remember what we know is that the acceleration is equal to zero, it's constant speed. So we actually know that this is just gonna be zero. So this allows me to set up an equation relating on my forces. I'm gonna pick the upward direction to be positive because that's where the block is going. So therefore my attention is gonna be positive and this is gonna be minus MG X minus friction kinetic equals zero. When you move both of these terms over to the other side they built become positive and I'm just gonna go ahead and expand them. Remember that MG. X. Is just equal to MG. Times the sine of data. Remember that friction is on an inclined plane, it's mu K. Times the normal force and the normal force is MG times the cosine. Theta. We've seen this written a bunch of times. So hopefully that's familiar to you. So if I'm sorry, this is actually supposed to be positive as well. Now if you go through these variables, we actually have all of them. We have mass, we have G. We have feta, we also have the coefficient. So I'm just gonna have to put this in. Unfortunately this is gonna be really long. But you can just plug it into your calculator and follow it all. Follow along. So the mass is gonna be 1400 G. Is 9.8. Then we have the sine of 37 plus 0. times 1400 times 9.8 times the cosine of 37. Just make sure your calculator is in degrees mode and what you should get when you plug both of these things in and sort of add them together. Or you can just plug it in. As long expression is 12 6 40 newtons. So that's the first variable. I need I need the tension. So that's done. The next thing I need is I need to figure out what the distance is along the cable is because I don't have what that devalue is. So let's go ahead and do that. So what I'm gonna do here is I'm gonna draw a simplified version of the triangle because there's a lot going on here. So what's happening here is I've got this triangle like this and I've got some of the values, I've got the D. Which is what I actually need, what I what I need, I need that distance. And I also know what the angle of the incline is 37°. And I'm also told with H is 60.2. Now remember for triangles as long as I have one angle and one side, I can figure out any other piece of the triangle. So if I want to relate to the hypotenuse which is D. And then the height which is the opposite side, I'm gonna have to use a sine function on a sine function here. So remember that the sine of the angle is related to the opposite side which is H. Over the hypotenuse. So if I want to figure out what this with with this distance is. And all I have to do is just trade these two these two variables and switch their places. So D. Is equal to H. Over sine theta. So this equals the height which is 60 points two divided by the sine of 37. If you go ahead and work this out, what you're gonna get is a distance of exactly 100 m. So that's how far that this block gets pulled up the incline. So this D. Here equals 100 and now we have everything we need to solve this problem here. So we're done with T. And D. Now we just plug this all in. So the power is just gonna be 12 6 40 times the distance, which is 100 then we're gonna divided by the 25 seconds it takes. And when you go ahead and do that, you're gonna get exactly 50,560 watts. That's how much energy this is. Uh this has to actually, this cable actually has to exert, this motor has to exert in order to pull this block up the incline like this. It's a lot of power. So, you know, you might see this written as 50.5 kW. That's another way you might see that written. Alright guys, so that's it for this one.
Additional resources for Power