### Video Transcript

In this video, we’re talking about
mechanical power. This is a term that we use fairly
often. We might talk about the power of a
weightlifter or a bolt of lightning or the power in somebody’s voice. We’re going to see that in the
world of physics, this term mechanical power has a very specific meaning.

We define this term mechanical
power this way. It’s equal to the amount of energy
transferred in a process divided by the time the process takes. 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. When we talk about power in
physics, this is what we mean by that term. It’s energy transferred per unit
time.

To get a bit of a sense for what
this equation means, imagine that a person is standing at the foot of a
staircase. Now if the person climbs up that
staircase, we can say that they’ve used some energy to do it. That’s because in climbing up,
they’re working against the force of gravity, which is pulling them down. So, it takes energy to go from the
bottom of the staircase to the top, to this new higher elevation. And that amount of energy has to do
with the height of the staircase, how tall the stairs are.

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. Now imagine that they climb back
down to the bottom and then climb the stairs a second time. But in this instance, instead of
taking 10 seconds to get to the top, they take 20, twice as long. We can see that in both cases the
person uses the same amount of energy to go from the bottom to the top. That’s because the height of the
staircase doesn’t change, and the person’s mass doesn’t change. 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. But what is different between the
10-second and 20-second times to climb is the power output by this person.

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.

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. But the difference is in the power
used. That is in how long it takes to
expend that much energy.

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. And, in fact, it is possible to
make a substitution into this equation that makes it even more useful. That substitution involves this
term here, the energy. 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. Another way to say that same thing
is to say that the climber needed to do work. They need to exert a force over
some distance in order to get to the top.

The two terms work and energy, we
can abbreviate them 𝑊 and 𝐸, are very closely related to one another. Both of them are expressed in the
same scientific units joules. And energy is actually defined in
terms of something’s capacity to do work. We can say that whenever something
expends energy or use it up, it’s doing work in some way. This connection between work and
energy is so close that sometimes we’ll see the equation for power written like
this. Power is equal to work done divided
by time.

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. But here we’re using work and
energy interchangeably, as they typically are.

Now we said earlier that when it
comes to units, the units of work are joules, symbolized using a capital J. And then, along with that, we know
that the base unit of time is seconds. And so, that then brings us to the
base unit of power. 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. This is the unit of power. And watt is abbreviated with a
capital W.

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. Both are capital W’s. Thankfully though, the context
usually makes it clear which one we’re working with. So, it’s often not an issue.

So, if ever anyone walks up to you
and says watt is the unit of power, the correct answer is yes. Now watt is indeed the SI base unit
for power, but you may have heard of another unit called horsepower. 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.

In Watt’s day, it was horses that
supplied most of the power for mechanical and industrial processes. 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. So, he came up with a unit of power
based on the most common source of power at that time. 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. So, that’s one horsepower.

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. So, whenever we have power given in
units of horsepower, this is what it refers to. 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. Knowing all this about mechanical
power, let’s now get a bit of practice through an example.

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? A) Power equals time divided by
work. B) Power equals work times
time. C) Power equals work minus
time. D) Power equals work divided by
time. E) Power equals work plus time.

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. We’re working then with these three
particular quantities, power, work, and time. 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.

Starting from the bottom, the SI
base unit of time is the second, abbreviated s. The unit for work is the joule,
abbreviated capital J. And, notice, that this is the same
as the unit for energy. And then, lastly, the SI unit for
power is the watt, abbreviated capital W.

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. That’s part of what makes it an
equation. 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.

As we consider answer options C and
E, we see that the right-hand side will not have those units for these two
choices. For these two choices, what we have
on the right-hand side is a mixture of units. The work has units of joules and
the time has units of seconds. In each of these cases then, we’re
not able to perform the operation on the right-hand side. We’re not comparing quantities of a
similar type to one another. From a units perspective then, we
can say that options C and E are out of the running. The final unit on the right-hand
side of these expressions can’t possibly be watts. And so, it can’t be a correct
formula.

That leaves A, B, and D. At this point, it’ll be helpful to
us to recall the definition of power. Power is defined as energy
transferred per unit time. In other words, it’s an amount of
energy in units of joules divided by an amount of time in units of seconds. And note that this energy transfer
can be happening thanks to the work done by some entity. We could equally well define power
as work done per unit time.

So, the unit of work, as we saw, is
the joule. And the base unit of time is the
second. And our definition for power is
telling us that power is an amount of energy in joules per amount of time in
seconds. 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.

Let’s start out by testing answer
option A. On the right-hand side of this
equation, we have a time in units of seconds divided by work, which has units of
joules. That’s not an overall unit of
joules per second. So, we’ll cross out option A. Moving on to option B, this
right-hand side has work done in units of joules multiplied by time in units of
seconds. The unit here is a joule times
second rather than a joule per second, so option B is also off the table.

Lastly, we get to option D, where
we have work done in joules divided by time in seconds. Here, we have a match for the units
we were looking for, a match for the units of power. We can say then that a joule per
second is equal to a watt, the unit of power. And this makes option D the correct
choice. Power is equal to work divided by
time.

Let’s take a look now at one more
example involving mechanical power.

Suppose that 260 joules of work is
done in 40 seconds. What power is required to do this
amount of work?

Alright, so, in this case, we have
this given amount of work done presumably over some process. To get a sense for an example of
what might be going on, consider this 25-kilogram block sitting on the ground. 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. 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. We would have a process which is
roughly comparable to the process described here.

In any case, what we’re after is
the power that’s needed to do this amount of work in this amount of time. And to get started figuring that
out, let’s recall the mathematical equation for power in terms of work and time. 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.

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.

Now notice that the units we have
in this expression are joules per second. These are the base units of energy
and time, respectively. 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. In other words, the answer we’ll
get in calculating this fraction will have units of watts. When we compute this result, we
find an answer of 6.5 watts. That’s the mechanical power needed
to do this amount of work in this amount of time.

Let’s take a moment now to
summarize what we’ve learnt about mechanical power. 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. Writing this out as an equation
looks like this. Power 𝑃 is equal to energy 𝐸
divided by time 𝑡.

We also saw that, in many cases,
work and energy are interchangeable terms. And as a result, we modified our
equation for power so that it reads work 𝑊 divided by time 𝑡. Furthermore, the unit of work,
which is equal to the unit of energy, is the joule. The base unit of time is the
second. And the SI base unit of power is
the watt.

We noted that watt is often
abbreviated with a capital W. So, we need to be careful not to
confuse it with the abbreviation for work capital 𝑊. 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.