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# Power

IIII haaaave the powerrrr! Power is the rate at which work is done. Created by David SantoPietro.

Video transcript

Check out these weightlifters. The one on the right is
lifting his weight faster, but they're both doing
the same amount of work. The reason I can say
that is because work is the amount of energy
that's transferred. Or to put it a
simpler way, this is the way I like to
think about it, work is equal to
the amount of energy you give something or
take away from something. Both weightlifters are
giving their weights the same amount of
gravitational potential energy. They both lift them two
meters, and the masses are 100 kilograms each. Plug those into the formula for
gravitational potential energy, and you find that the work
done by each weightlifter is 1,960 joules. But the weightlifter
on the right is lifting his weight faster. And there should be
a way to distinguish between what he's doing
and what the other slower weightlifter is doing. We can distinguish
their actions in physics by talking about power. Power measures the
rate at which someone like these weightlifters or
something like an automobile engine does work. To be specific, power is
defined as the work done divided by the time that
it took to do that work. We already said that
both weightlifters are doing 1,960 joules of work. The weightlifter on the
right takes 1 second to lift his weights, and
the weightlifter on the left takes 3 seconds to
lift his weights. If we plug those times into
the definition of power, we'll find that the power
output of the weightlifter on the right during his lift
is 1,960 joules per second. And the power output
of the weightlifter on the left during his lift
is 653 joules per second. A joule per second
is named a watt, after the Scottish
engineer James Watt. And the watt is abbreviated
with a capital W. All right, let's look
at another example. Let's say a 1,000 kilogram
car starts from rest and takes 2 seconds to reach a
speed of 5 meters per second. We can find the power
output by the engine by taking the work
done on the car divided by the time it took
to do that work. To find the work
done on the car, we just need to figure
out how much energy was given to the car. In this case, the car
was given kinetic energy and it took two seconds to
give it that kinetic energy. If we plug in the values
for the mass and the speed, we find the engine had a
power output of 6,250 watts. We should be clear that what
we've really been finding here is the average power
output because we've been looking at the total
work done over a given time interval. If we were to look at
the time intervals that got smaller and smaller, we'd
be getting closer and closer to the power output
at a given moment. And if we were to make our time
interval infinitesimally small, we'd be finding the power
output at that particular point in time. We call this the
instantaneous power. Dealing with
infinitesimals typically requires the use of
calculus, but there are ways of finding
the instantaneous power without having to use calculus. For instance, let's
say you were looking at a car whose
instantaneous power output was 6,250 watts at
every given moment. Since the instantaneous
power never changes, the average power just equals
the instantaneous power, which equals 6,250 watts. In other words, the average
power over any time interval is going to equal the
instantaneous power at any moment. And that means work
per time gives you both the average power and
the instantaneous power in this case. Let's say you weren't so lucky,
and the instantaneous power was changing as
the car progressed. Then, how would you find
the instantaneous power? Well, we know that power
is just the work per time. So something we
can try is to plug in the formula for work, which
looks like FD cosine theta, and then divide by the time. Something that you might
notice is that now we have distance per
time in this formula. So let's isolate the
distance per time. Distance per time
is just the speed. So I can replace d over
t with v in this formula. And if you plug in the
instantaneous speed of the car at a given moment
in time, you'll be finding the
instantaneous power output by the force on the car at
that particular moment in time. So to find the instantaneous
power output by a force, plug in the force on the object
at a particular moment in time, multiply by the
speed of the object at that same moment in time,
then multiply by cosine theta. But be careful here. Theta isn't any old angle. It's the angle between
the force on the object and the velocity of the object. But in many cases, the force
is in the same direction as the velocity, which means
the angle between the force and the velocity is zero. And since cosine of 0
is 1, you don't really need the cosine in
the formula at all. And you find that the
instantaneous power is just the force times the speed. All right. So what does power mean? Power is the rate at
which work is done. What does average power mean? Average power is the work done
divided by the time interval that it took to do that work. What does the
instantaneous power mean? Instantaneous power
is the power output of a force at a
particular moment in time.