Good morning. In our previous lesson introducing conservative and nonconservative forces
we showed that the work done by a conservative force equals the negative change in potential
energy associated with that force. What I am going to do now is slide the integral definition of work
into the equation. What we get is the work done by a conservative force equals the integral with
respect to x-position from position initial to position final of force in the x-direction
which equals the negative of the change in potential energy associated with that force.
Billy, please rearrange that equation to make it into a derivative instead of an integral.
Absolutely. ♪ Flipping Physics ♪
Okay, remember that the integral means the sum of the areas of an infinite number of
infinitesimally small rectangles, in this case with sides F sub x and dx,
we can instead look at the area of only one of those infinitesimally small rectangles which is
just height F sub x times width dx. Then we would also be looking at an infinitesimally
small change in potential energy dU. We can then rearrange that equation to show that the component
of a conservative force in the x-direction equals the negative of the derivative,
with respect to x-position, of the potential energy associated with that force.
Very nice Billy. Bo, please show that equation is valid using elastic potential energy.
Sure. That equation then is the spring force equals the negative of the derivative,
with respect to x-position, of elastic potential energy. The equation for elastic
potential energy is one half spring constant times displacement from equilibrium position squared.
The derivative of that with respect to x-position is well, 2 times 1 half equals 1,
so negative spring constant times displacement from equilibrium position.
And that is Hooke’s Law, or the equation for the spring force. So, the equation is
valid because we get Hooke’s Law out of it. (And why is the negative in the equation?)
Because the spring force and the displacement from equilibrium position are opposite in direction.
Correct. Thanks Bo. Bobby, please do the same thing with gravitational potential energy.
Uh Well The force of gravity equals the negative of the derivative, with respect to x-position,
of gravitational potential energy. Actually, that should be with respect to y-position because
the force of gravity is in the y-direction. And the equation for gravitational potential energy
is mass times acceleration due to gravity times height, h, above the zero line, however, let’s use
a y instead, because that will make it more clear that we are taking the derivative with respect
to that variable. The derivative of that equals negative mass times acceleration due to gravity.
That means the equation relating conservative force to change in potential energy
is valid because we get the equation for the force of gravity out of it.
And the force of gravity is negative because the force of gravity is down.
Great. Thanks Bobby. Now let's work through an example. Let’s say we have a conservative force
of negative 2.0 x plus 2.0 x squared all times unit vector i newtons that acts on an object
as it moves from an x-position of 1.0 meters to 2.0 meters. What is the change in potential
energy of the object? You may assume this force is the only force acting on the object.
Before we begin answering the question, realize when you see the term “conservative force”,
you need to immediately think of the equation conservative force in the x-direction equals the
negative derivative with respect to x-position of the potential energy associated with that force.
And realize this equation is not on the AP Physics C equation sheet. So, you need to memorize it.
Okay, Billy, please solve this problem. Certainly. We start with the equation
conservative force in the x-direction equals the negative derivative with respect to x-position of
the potential energy associated with that force. Then we can multiply both sides by dx and then
take the definite integral of both sides. The left-hand side is the integral with respect to x,
so it is from position initial to position final. The right-hand side is the integral with respect
to potential energy, so that is from potential energy initial to potential energy final.
And that equals the negative of the quantity potential energy final minus potential energy
initial or the negative of the change in potential energy. Oh, and that’s funny,
the left-hand side of that equation is the work done by the force, and the right-hand side is the
negative of the change in potential energy, which is the equation you started this lesson with.
Anyway, multiplying through by negative one gives us that the change in potential energy then
equals the negative of the integral of the force with respect to position or the negative of the
integral from 1 to 2 of the quantity negative 2 x plus 2 x squared all with respect to x.
That equals the negative of the quantity negative 2 x squared over 2 plus 2 x
cubed over 3 with the limits from 1 to 2. Before we substitute in the limits, let’s simplify to get
x squared minus 2 x cubed over 3. And substituting in limits gives us 2 squared minus 2 times 2 cubed
both over 3 and then minus the quantity 1 squared minus 2 times 1 cubed both over 3. That equals
negative four thirds minus
one third or negative five thirds which equals negative 1.7 joules with 2 sig figs.
Why is the change in potential energy of the object negative?
Thank you Billy and yes, the change in potential energy associated with this conservative force
as the object goes from an x-position of 1 to an x-position of 2 meters is negative.
While we do not know much about what is physically going on in this example,
because we do not know much about this conservative force which I completely made up,
we do know that the potential energy of the object has decreased in the direction of the
motion of the object. What we can do is make a comparison to gravitational potential energy
and the force of gravity. In that gravitational analogy, the object has moved down. The force
of gravity and displacement of the object are in the same direction,
so the work done on the object by the force of gravity is positive, that means the change in
gravitational potential energy of the object is negative, so the gravitational potential energy
of the object has decreased. If you look at the graph of the force in our example, you can see
the conservative force in our example is always in the positive x-direction between 1 and 2 meters.
Because the displacement of the object is also in the positive x-direction, the work done by
the force, or the area “under” the function, is positive. We know the change in potential energy
associated with this conservative force equals the negative of that work, which is why the change in
potential energy of the object is negative; the potential energy of the object has decreased.
Thank you very much for learning with me today, I enjoyed learning with you.