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In this video, we’re talking about mechanical power.
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This is a term that we use fairly often.
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We might talk about the power of a weightlifter or a bolt of lightning or the power in somebody’s voice.
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We’re going to see that in the world of physics, this term mechanical power has a very specific meaning.
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We define this term mechanical power this way.
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It’s equal to the amount of energy transferred in a process divided by the time the process takes.
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If we write this all out as an equation, we would say that power capital 𝑃 is equal to energy transferred 𝐸 divided by the time taken for that transfer to happen 𝑡.
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When we talk about power in physics, this is what we mean by that term.
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It’s energy transferred per unit time.
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To get a bit of a sense for what this equation means, imagine that a person is standing at the foot of a staircase.
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Now if the person climbs up that staircase, we can say that they’ve used some energy to do it.
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That’s because in climbing up, they’re working against the force of gravity, which is pulling them down.
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So, it takes energy to go from the bottom of the staircase to the top, to this new higher elevation.
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And that amount of energy has to do with the height of the staircase, how tall the stairs are.
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Now let’s say that in order to climb the staircase, this person took 10 seconds to go from the bottom all the way to the top step.
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Now imagine that they climb back down to the bottom and then climb the stairs a second time.
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But in this instance, instead of taking 10 seconds to get to the top, they take 20, twice as long.
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We can see that in both cases the person uses the same amount of energy to go from the bottom to the top.
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That’s because the height of the staircase doesn’t change, and the person’s mass doesn’t change.
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Bringing that much mass to the top of the staircase this tall requires the same amount of energy, regardless of how long the ascent takes.
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But what is different between the 10-second and 20-second times to climb is the power output by this person.
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Looking back at our equation for power, since this person used the same amount of energy to climb to the top in both cases, but the second time up took twice as long as the first time, we can say that in comparing the first climb, which took 10 seconds, to the second climb, which took 20, both climbs required the same amount of energy, but the second climb took one-half the power as the first.
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So, if you climb stairs very slowly, or at a normal pace, or you leap up them two or three at a time, in all those cases, the same amount of energy is used to get from the bottom to the top.
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But the difference is in the power used.
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That is in how long it takes to expend that much energy.
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We can see from this equation for power that if we know two of these three variables, power, energy, and time, we’re able to use this equation to solve for the third one.
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And, in fact, it is possible to make a substitution into this equation that makes it even more useful.
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That substitution involves this term here, the energy.
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Thinking back to our stair climber, we said that that climber needed to exert some amount of energy to get to the top of the stair.
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Another way to say that same thing is to say that the climber needed to do work.
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They need to exert a force over some distance in order to get to the top.
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The two terms work and energy, we can abbreviate them 𝑊 and 𝐸, are very closely related to one another.
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Both of them are expressed in the same scientific units joules.
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And energy is actually defined in terms of something’s capacity to do work.
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We can say that whenever something expends energy or use it up, it’s doing work in some way.
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This connection between work and energy is so close that sometimes we’ll see the equation for power written like this.
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Power is equal to work done divided by time.
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Once again, then, we see that power is equal to the time rate of change of some quantity, in this case, work, and then, previously energy.
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But here we’re using work and energy interchangeably, as they typically are.
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Now we said earlier that when it comes to units, the units of work are joules, symbolized using a capital J.
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And then, along with that, we know that the base unit of time is seconds.
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And so, that then brings us to the base unit of power.
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If there is a process where one joule of work is done, or equivalently one joule of energy is transferred, in a time of one second, then that amount of energy transferred in that amount of time is said to be equal to one watt.
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This is the unit of power.
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And watt is abbreviated with a capital W.
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Notice then that we have to be a little bit careful not to confuse the abbreviation for the unit of power with the symbol for work.
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Both are capital W’s.
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Thankfully though, the context usually makes it clear which one we’re working with.
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So, it’s often not an issue.
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So, if ever anyone walks up to you and says watt is the unit of power, the correct answer is yes.
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Now watt is indeed the SI base unit for power, but you may have heard of another unit called horsepower.
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Now, interestingly, the horsepower, which is a real unit for power just not within the SI system, was invented by the man who ended up having the SI unit for power named after him James Watt.
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In Watt’s day, it was horses that supplied most of the power for mechanical and industrial processes.
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As Watt, who was an engineer, worked to develop and refine steam engines, he wanted some practical way to assess the power output of his engines.
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So, he came up with a unit of power based on the most common source of power at that time.
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Watt defined one horsepower as the amount of power that’s needed to lift a 550-pound mass a distance of one foot in a time of one second.
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So, that’s one horsepower.
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And if we compare a single horsepower to the unit that came to be named after Watt, we find that one horsepower is approximately equal to 750 watts.
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So, whenever we have power given in units of horsepower, this is what it refers to.
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But the great majority of the time, because we work within the SI, or international system of units, the unit of power that we’ll encounter is the watt, where one watt is equal to one joule of energy transferred over a time of one second.
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Knowing all this about mechanical power, let’s now get a bit of practice through an example.
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Which of the following formulas correctly shows the relationship between the time taken to do an amount of work and the power supplied to do the work?
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A) Power equals time divided by work.
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B) Power equals work times time.
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C) Power equals work minus time.
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D) Power equals work divided by time.
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E) Power equals work plus time.
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We’re told that these five options are all candidates for the correct formula showing the relationship between the time taken to do an amount of work and the power supplied to do it.
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We’re working then with these three particular quantities, power, work, and time.
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Now even if we don’t have the exact relationship between these three terms committed to memory, if we’re able to recall the units of these three terms, we’ll be able to narrow down our answer options.
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Starting from the bottom, the SI base unit of time is the second, abbreviated s.
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The unit for work is the joule, abbreviated capital J.
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And, notice, that this is the same as the unit for energy.
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And then, lastly, the SI unit for power is the watt, abbreviated capital W.
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As we consider our answer options, we know that whichever one is correct will have the same units on either side of the equality sign.
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That’s part of what makes it an equation.
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And since we know that the SI unit of power is the watt, we realize that all five of our options will have that unit on the left-hand side of the equality, which means that whichever answer option is correct must have that same unit, the watt, on the right-hand side.
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As we consider answer options C and E, we see that the right-hand side will not have those units for these two choices.
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For these two choices, what we have on the right-hand side is a mixture of units.
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The work has units of joules and the time has units of seconds.
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In each of these cases then, we’re not able to perform the operation on the right-hand side.
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We’re not comparing quantities of a similar type to one another.
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From a units perspective then, we can say that options C and E are out of the running.
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The final unit on the right-hand side of these expressions can’t possibly be watts.
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And so, it can’t be a correct formula.
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That leaves A, B, and D.
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At this point, it’ll be helpful to us to recall the definition of power.
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Power is defined as energy transferred per unit time.
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In other words, it’s an amount of energy in units of joules divided by an amount of time in units of seconds.
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And note that this energy transfer can be happening thanks to the work done by some entity.
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We could equally well define power as work done per unit time.
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So, the unit of work, as we saw, is the joule.
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And the base unit of time is the second.
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And our definition for power is telling us that power is an amount of energy in joules per amount of time in seconds.
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To answer our question then, we’ll go through answer options A, B, and D and see which one has units of joules per second on the right-hand side.
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Let’s start out by testing answer option A.
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On the right-hand side of this equation, we have a time in units of seconds divided by work, which has units of joules.
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That’s not an overall unit of joules per second.
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So, we’ll cross out option A.
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Moving on to option B, this right-hand side has work done in units of joules multiplied by time in units of seconds.
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The unit here is a joule times second rather than a joule per second, so option B is also off the table.
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Lastly, we get to option D, where we have work done in joules divided by time in seconds.
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Here, we have a match for the units we were looking for, a match for the units of power.
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We can say then that a joule per second is equal to a watt, the unit of power.
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And this makes option D the correct choice.
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Power is equal to work divided by time.
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Let’s take a look now at one more example involving mechanical power.
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Suppose that 260 joules of work is done in 40 seconds.
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What power is required to do this amount of work?
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Alright, so, in this case, we have this given amount of work done presumably over some process.
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To get a sense for an example of what might be going on, consider this 25-kilogram block sitting on the ground.
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Now if we were to take this block and lift it up a vertical distance of one meter, and if it took us 40 seconds to move the block across that distance of one meter, then that gives an idea of the power involved in this process.
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260 joules is about how much work it would take to lift a block this size a height one meter and then doing it over a time span of 40 seconds, quite a long time actually.
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We would have a process which is roughly comparable to the process described here.
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In any case, what we’re after is the power that’s needed to do this amount of work in this amount of time.
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And to get started figuring that out, let’s recall the mathematical equation for power in terms of work and time.
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The power 𝑃 required to do a certain amount of work 𝑊 in a certain amount of time 𝑡 is equal to that amount of work divided by the amount of time.
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So, when we go to calculate the power needed, we’ll call it 𝑃, in this particular process, we know that’s equal to the work done, which is measured in joules, and there are 260 of those joules of work, divided by the time the process takes, which is given as 40 seconds.
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Now notice that the units we have in this expression are joules per second.
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These are the base units of energy and time, respectively.
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And if we were to divide one joule of energy by one second of time, then that would equal one watt, abbreviated capital W, of power.
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In other words, the answer we’ll get in calculating this fraction will have units of watts.
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When we compute this result, we find an answer of 6.5 watts.
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That’s the mechanical power needed to do this amount of work in this amount of time.
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Let’s take a moment now to summarize what we’ve learnt about mechanical power.
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The first thing we saw is that mechanical power is equal to the energy transferred in a process divided by the time that process takes.
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Writing this out as an equation looks like this.
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Power 𝑃 is equal to energy 𝐸 divided by time 𝑡.
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We also saw that, in many cases, work and energy are interchangeable terms.
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And as a result, we modified our equation for power so that it reads work 𝑊 divided by time 𝑡.
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Furthermore, the unit of work, which is equal to the unit of energy, is the joule.
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The base unit of time is the second.
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And the SI base unit of power is the watt.
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We noted that watt is often abbreviated with a capital W.
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So, we need to be careful not to confuse it with the abbreviation for work capital 𝑊.
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And lastly, tying all the units together, we saw that one watt of power is equal to one joule of energy divided by one second of time.