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# Work/energy problem with friction

A conservation of energy problem where all of the energy is not conserved. Created by Sal Khan.

Video transcript

Welcome back. I'll now do another conservation
of energy problem, and this time I'll
add another twist. So far, everything we've been doing,
energy was conserved by the law of conservation. But that's because all of the
forces that were acting in these systems were conservative
forces. And now I'll introduce you to
a problem that has a little bit of friction, and we'll see
that some of that energy gets lost to friction. And we can think about
it a little bit. Well where does that
energy go? And I'm getting this problem
from the University of Oregon's zebu.uoregon.edu. And they seem to have some nice
physics problems, so I'll use theirs. And I just want to make
sure they get credit. So let's see. They say a 90 kilogram
bike and rider. So the bike and rider combined
are 90 kilograms. So let's just say the mass is 90
kilograms. Start at rest from the top of a 500 meter
long hill. OK, so I think they mean
that the hill is something like this. So if this is the hill, that
the hypotenuse here is 500 hundred meters long. So the length of that,
this is 500 meters. A 500 meter long hill with
a 5 degree incline. So this is 5 degrees. And we can kind of just view
it like a wedge, like we've done in other problems.
There you go. That's pretty straight. OK. Assuming an average friction
force of 60 newtons. OK, so they're not telling us
the coefficient of friction and then we have to figure
out the normal force and all of that. They're just telling us, what
is the drag of friction? Or how much is actually friction
acting against this rider's motion? We could think a little bit
about where that friction is coming from. So the force of friction is
equal to 60 newtons And of course, this is going to be
going against his motion or her motion. And the question asks us, find
the speed of the biker at the bottom of the hill. So the biker starts up
here, stationary. That's the biker. My very artful rendition
of the biker. And we need to figure out the
velocity at the bottom. This to some degree is a
potential energy problem. It's definitely a conservation
of mechanical energy problem. So let's figure out what the
energy of the system is when the rider starts off. So the rider starts off at
the top of this hill. So definitely some
potential energy. And is stationary, so there's
no kinetic energy. So all of the energy is
potential, and what is the potential energy? Well potential energy is equal
to mass times the acceleration of gravity times
height, right? Well that's equal to, if the
mass is 90, the acceleration of gravity is 9.8 meters
per second squared. And then what's the height? Well here we're going to have
to break out a little trigonometry. We need to figure out this side
of this triangle, if you consider this whole
thing a triangle. Let's see. We want to figure out
the opposite. We know the hypotenuse and
we know this angle here. So the sine of this angle is
equal to opposite over hypotenuse. So, SOH. Sine is opposite over
hypotenuse. So we know that the height--
so let me do a little work here-- we know that sine of
5 degrees is equal to the height over 500. Or that the height is equal
to 500 sine of 5 degrees. And I calculated the sine of
5 degrees ahead of time. Let me make sure I
still have it. That's cause I didn't have my
calculator with me today. But you could do this
on your own. So this is equal to 500,
and the sine of 5 degrees is 0.087. So when you multiply these
out, what do I get? I'm using the calculator
on Google actually. 500 times sine. You get 43.6. So this is equal to 43.6. So the height of the hill
is 43.6 meters. So going back to the potential
energy, we have the mass times the acceleration of gravity
times the height. Times 43.6. And this is equal to, and then
I can use just my regular calculator since I don't
have to figure out trig functions anymore. So 90-- so you can see the whole
thing-- times 9.8 times 43.6 is equal to, let's
see, roughly 38,455. So this is equal to 38,455
joules or newton meters. And that's a lot of
potential energy. So what happens? At the bottom of the hill--
sorry, I have to readjust my chair-- at the bottom of the
hill, all of this gets converted to, or maybe I should pose that as a question. Does all of it get converted
to kinetic energy? Almost. We have a force
of friction here. And friction, you can kind of
view friction as something that eats up mechanical
energy. These are also called
nonconservative forces because when you have these forces
at play, all of the force is not conserved. So a way to think about it is,
is that the energy, let's just call it total energy. So let's say total energy
initial, well let me just write initial energy is equal
to the energy wasted in friction-- I should have written
just letters-- from friction plus final energy. So we know what the initial
energy is in this system. That's the potential energy
of this bicyclist and this roughly 38 and 1/2 kilojoules
or 38,500 joules, roughly. And now let's figure out the
energy wasted from friction, and the energy wasted from
friction is the negative work that friction does. And what does negative
work mean? Well the bicyclist is moving 500
meters in this direction. So distance is 500 meters. But friction isn't acting
along the same direction as distance. The whole time, friction is
acting against the distance. So when the force is going in
the opposite direction as the distance, your work
is negative. So another way of thinking of
this problem is energy initial is equal to, or you could say
the energy initial plus the negative work of friction,
right? If we say that this is a
negative quantity, then this is equal to the final energy. And here, I took the friction
and put it on the other side because I said this is going to
be a negative quantity in the system. And so you should always just
make sure that if you have friction in the system, just as
a reality check, that your final energy is less than
your initial energy. Our initial energy is, let's
just say 38.5 kilojoules. What is the negative work
that friction is doing? Well it's 500 meters. And the entire 500 meters, it's
always pushing back on the rider with a force
of 60 newtons. So force times distant. So it's minus 60 newtons,
cause it's going in the opposite direction of the
motion, times 500. And this is going to equal
the ending, oh, no. This is going to equal the
final energy, right? And what is this? 60 times 500, that's 3,000. No, 30,000, right. So let's subtract 30,000
from 38,500. So let's see. Minus 30. I didn't have to do that. I could have done
that in my head. So that gives us 8,455 joules is
equal to the final energy. And what is all the
final energy? Well by this time, the rider's
gotten back to, I guess we could call the sea level. So all of the energy
is now going to be kinetic energy, right? What's the formula for
kinetic energy? It's 1/2 mv squared. And we know what m is. The mass of the rider is 90. So we have this is 90. So if we divide both sides. So the 1/2 times 90. That's 45. So 8,455 divided by 45. So we get v squared
is equal to 187.9. And let's take the square root
of that and we get the velocity, 13.7. So if we take the square root of
both sides of this, so the final velocity is 13.7. I know you can't read that. 13.7 meters per second. And this was a slightly more
interesting problem because here we had the energy wasn't
completely conserved. Some of the energy, you could
say, was eaten by friction. And actually that energy
just didn't disappear into a vacuum. It was actually generated
into heat. And it makes sense. If you slid down a slide of
sandpaper, your pants would feel very warm by the time you
got to the bottom of that. But the friction of this, they
weren't specific on where the friction came from, but it
could have come from the gearing within the bike. It could have come
from the wind. Maybe the bike actually
skidded a little bit on the way down. I don't know. But hopefully you found that
a little bit interesting. And now you can not only work
with conservation of mechanical energy, but you can
work problems where there's a little bit of friction
involved as well. Anyway, I'll see you
in the next video.