thing that comes to mind? Maybe a cubicle? Or a briefcase?
Or that history exam thats coming up soon? But if youre a physicist, work has a very specific meaning -- one that has very little to do with spreadsheets or the fall of the Roman
Empire. Today, were going to explore that definition
-- and how it connects to one of the most important principles in physics: conservation
of energy. Well also learn what physicists mean when
they talk about another concept that comes up a lot in daily life: power. So lets get towork.
[Theme Music] So far in this course, weve spent most of our time talking about forces, and the way they make things move. And you need to understand forces before you
can understand work. Because work is what happens when you apply
a force over a certain distance, to a system -- a system just being whatever section of the universe you h appen to be talking about at the time. For example, if youre using a rope to drag
a box across the floor, we might say that the box is your system, and the force youre
using to pull on it is an external force.
So, lets say youre pulling on this box-system by dragging it straight behind you, so the rope is parallel to the ground. If you use the rope to pull the box for one
meter, wed say that youre doing work on the box. And the amount of work youre doing is equal
to the force youre using to pull the box, times the distance you moved it. For example, if you pulled the rope -- and
therefore the box -- with a force of 50 Newtons, while you moved it 5 meters, then wed say
that you did 250 Newton-meters of work on the box.
More commonly, however, work is expressed
in units known as Joules. Now, sometimes, the force you apply to an
object wont be in exactly the same direction as the direction in which the object is moving. Like, if you tried to drag the box with your hand higher than the box, so that the rope was at an angle to the floor. In that case, the box would move parallel to the floor, but the force would be at an angle to it.
And in such an instance, youd have to use one the tricks we learned back when we first talked about vectors. Specifically, you must separate the force youre using on the rope into its component parts: one thats parallel to the floor,
and one thats perpendicular to it. To find the part of the force thats parallel to the floor -- that is, the one thats actually pulling the box forward -- you just have to multiply the magnitude of the force by the cosine of the ropes angle to the ground. Youll remember that we typically designate
an angle in a system as theta.
So, to calculate the work you did on the box,
you just multiply the horizontal component -- or F times the cosine of theta -- by the
distance you moved the box. Thats one way physicists often write the
equation for work -- theyll set it equal to force, times distance, times the cosine
of theta. And that equation will fit any scenario that involves a constant force being applied over a certain distance. But what if the force isnt constant? What if, say, you started out pulling hard
on the box, but then you started to get tired, so the amount of force you exerted on the
box got smaller and smaller the farther you dragged it.
To calculate the work you did in that case,
youd have to count up the amount of force you applied over each tiny little bit of distance. And if youve watched our episodes on calculus,
then you know that theres a faster way to add together infinitely tiny increments:
integration. So, to find the work done by a varying force,
you just have to integrate that force relative to the distance the object moved. Which would
look like this.
But force-times-distance is only one of the
ways that physicists measure work. Because, you know how we just said that Joules are
the units of work? Well, Joules are often used as the units for
something else: energy. And work uses the same units as energy,
because work is just a change in energy. It's what happens when an external force is applied
to a system and changes the energy of that system.
In fact, thats one of the ways to define
energy -- its the ability to do work. There are all different kinds of energy, but in this episode, well mainly be talking about two of them: kinetic energy and potential energy.
Kinetic energy is the energy of motion. When the box was resting on the ground, wed
say that it had no kinetic energy. But once you applied a force and it started
moving, it did have kinetic energy.
And the energy of the box changed, which means
that you did work on it. More specifically, the kinetic energy of an object is equal to half of its mass, times its velocity squared. If this looks familiar, thats because it
comes from applying both Newtons second law and the kinematic equations to the idea
that work is equal to force times distance. So, if the box had a mass of 20 kilograms,
and at some point while you were dragging it, it reached a velocity of 4 meters per
second, wed say that its kinetic energy at that moment was 160 Joules.
Then theres potential energy, which actually
isnt what it sounds like. Potential energy isnt potentially energy
-- its potentially work. In other words, potential energy is energy
that could be used to do work. One common type of potential energy is gravitational
potential energy - - basically, the potential energy that comes
from the fact that gravity exists.
If I hold this book a meter above the ground,
wed say that it has gravitational potential energy. Because if you let it go, then gravity is
going to do work on the book. Gravity exerted a force that moves it to the ground. Once the book hits the ground, though, wed
say that its gravitational potential energy is zero, because gravity cant do work on
it anymore.
Calculating gravitational potential energy
is easy enough: its just the force of gravity on the object
-- so, the objects mass times small g -- multiplied by the objects height.
Or mgh for short. Which means that, just by knowing that this
books mass is about a kilogram, and that its a meter above the ground, we can calculate
its potential energy: which is 9.8 Joules. Another type of potential energy that shows
up a lot is spring potential energy. Despite its name, this is not a seasonal thing --
and yes, I really made that joke.
Rather, its the type of potential energy thats
specific to springs! The force of a spring is equal to the distance
by which its either compressed or stretched, times a constant that we write as k. This equation is known as Hookes law, after British physicist Robert Hooke, who came up with it in 1660. Now, the constant, k -- also called the spring
constant -- is different for each spring, and its a measure of the springs stiffness. And the equation makes total sense, if you
think about it: The further you push on the spring, and the
stiffer it is, the harder it will resist.
You even can test this out for yourself by taking apart a clicky pen and playing with the spring inside. By combining Hookes law, with the idea
that work equals force times distance, we can find the potential energy from a spring: its half times k times the distance squared. For example: if you have a spring with a spring
constant of 200 Newtons per meter, and a block is compressing it by half a meter, then the
potential energy of the block would be 25 Joules. So, when something does work on a system,
its energy changes.
But how that energy changes depends on the
system. Some systems can lose energy. These are known
as a non-conservative systems. Now, that doesnt mean that the energy thats
lost is literally disappearing from the universe...
And it doesnt have anything to do with
the systems personal politics, either. It just relates to one of the most
fundamental principles of science: that energy can neither be created or destroyed. But systems can lose energy, like when friction
from the box dragging on the floor generates heat. For non-conservative systems, you can still talk about their kinetic energy or potential energy at any given moment.
But conservative systems let you do much more
than that. A conservative system is one that doesnt
lose energy through work. Say, a simple pendulum. When the pendulum is at the top of its swing,
it stops moving for a brief moment as it changes direction -- meaning that its kinetic energy, at that point, is zero.
But it has lots of potential energy, because
the gravitational force can do work on the pendulum, pulling it down until it reaches
the bottom of its swing. At the bottom of the swing, that potential
energy becomes zero, because gravity cant pull the pendulum down anymore. But now the pendulum has lots of kinetic energy,
because its moving through the swing. And it turns out that, at any given point
in the pendulums motion, its kinetic energy and its potential energy will add up to the
same number.
If its potential energy increases? Its kinetic energy will decrease by the exact same amount, and vice versa. So, now that we know how to define work, we
can use that definition to help explain another common term that physicists have a very specific meaning for: power. Or, more specifically, average power. Average power is defined as work over
time, and its measured in Watts, which is just another way of saying Joules per second.
Basically, its used to measure how much energy is converted from one type to another over time. So, remember that box you were pulling? We figured out that you did 250 Joules of
work on the box when you moved it 5 meters. If it took you 2 seconds to move the box,
then your average power output was 125 Watts. Youre basically a lightbulb! Now, we can also describe power in another
way, by putting two different facts together: One, that work is equal to force times distance.
And two, that average velocity is equal to
distance over time Knowing this, we can say that power is the net force applied to something with a particular average velocity. If you moved the box 5 meters in 2 seconds,
then its average velocity was 2.5 Meters per second. And we already said that you were pulling
the box along with a force of 50 Newtons. So, the force you were using to pull the box,
times the boxs average velocity, would also give you an average power output of 125
Watts.
The two equations for average power are
really describing the same relationship; theyre just using different qualities to do it. Were going to be talking about power a
lot when we discuss electricity in later episodes. Its the best way to calculate how
energy moves around in a circuit. But thats a story for another day.
For
now, our work is done. Today, you learned the two equations we can use to describe work, and that energy is the ability to do work. We also talked about kinetic and potential
energy, and how they play into non-conservative and
conservative systems. Finally, we found two different equations
for power.
Crash Course Physics is produced in association
with PBS Digital Studios. You can head over to their channel to check
out amazing shows like The Art Assignment, PBS Idea Channel, and PBS Game Show. This episode of Crash Course was filmed in
the Doctor Cheryl C. Kinney Crash Course Studio with the help of these amazing people and
our equally amazing graphics team is Thought Cafe..
No comments:
Post a Comment