JULIE HOOD: I'm Dr. Julie Hood, an AP Physics teacher for MAST
Academy in Miami, Florida.
In today's video, we're going to work through a free response
problem that is primarily numerical.
And it focuses on potential energy.
If you want to work through this problem with me,
click on the link over the video now.
And then you can download the PDF of the question featured
in this session.
So in this problem, we're going to calculate a lot of things.
Mechanical energy, both potential and kinetic,
momentum, acceleration.
We're going to sketch some graphs.
And one of my favorite topics, we'll
talk a little bit about simple harmonic motion.
So here we go.
We've got a 3 kilogram object moving along the x-axis
in a region, where the potential energy
as a function of position is given as 4x squared.
U is in joules and X is in meters.
When the object passes the point at negative 1/2 a meter,
its velocity is 2 meters per second.
And all the forces that act on the object today
are conservative.
Now, the first part of this asks us
to find the total mechanical energy of the object.
And then we're going to calculate
the x-coordinate of any points at which the object has
zero kinetic energy.
And we'll find the momentum of the object
when the object is over at 0.6 meters and the acceleration
when it's there as well.
And we'll even sketch some graphs of x versus t
in kinetic energy versus time.
So what you're expected to understand and do
in this problem is find the total mechanical energy
of a system and how to use it to find positions
for the kinetic energy is 0.
And we'll also find momentum using
the conservation of energy.
We'll find the acceleration at a given point.
And we'll even graph the object's position
and kinetic energy as functions of time.
So again, this is the question stem.
And the first thing we're going to do
is calculate the total mechanical energy
of the object.
Now, all the forces that act on the object are conservative.
And that's a beautiful thing because that
means energy is conserved.
All right, we'd want to find the total mechanical energy.
So that's going to be potential plus kinetic.
Our potential energy function is 4x squared.
And of course, kinetic is 1/2 mass,
velocity at a particular position squared.
Now, we're told-- well, we could actually
calculate the potential energy when the object
is at negative 1/2 a meter.
We'll just use that function above
and substitute in negative 1/2 a meter squared.
And we have 1 joule of energy tied up in potential energy
when the object is there.
And the kinetic energy there, well, we've
got 3 kilograms of mass.
And when the object is at negative 5.5 meters,
it's going 2 meters per second.
So there's 6 joules of kinetic.
So the total energy in the system
is summing the two energies together.
And we have 7 joules.
That's how much energy we have in the system, wherever it is.
In the second part of this question,
we need to find the x-coordinates when the object
has zero kinetic energy, OK.
Well, remember if we have zero kinetic energy,
then we have 7 joules of potential energy
because the total energy was 7 joules.
And the function U is 4x squared.
So we could just solve for x.
And we find there are two routes.
Don't forget that second route.
When you take the square root of something
both plus and minus 1.3 meters for position will work.
So that doesn't-- it shouldn't come as a surprise.
I mean, look at the function, it's a parabola.
It crosses the x-axis in two places.
So both plus and minus.
And that's your answer.
The next part of this problem, they
want you to calculate the magnitude
of the momentum of the object at position 0.6 meters.
So momentum is mass times velocity.
And we have 3 kilograms of mass.
And this is our potential function.
So when it's at 0.6, that means 1.44 joules of the total 7
joules are tied up in potential.
And because energy is conserved, that
leaves us with 7 minus 1.44 or 5.56 joules of kinetic energy.
And kinetic energy, it's 1/2 mass times velocity squared.
So we can find the velocity.
It's going to be 1.92 meters per second.
We know the mass.
Now, we know the velocity.
We can determine the momentum.
There it is.
Nice.
It was really not that bad.
You'd have to practice.
Now, the next part of this question
asks you to calculate the magnitude of the acceleration
as it passes that particular point.
When I think of acceleration, I often think of forces.
But I have to find the force in order to find the acceleration.
I've got a potential energy function.
And I want to remind you, I hope you've
seen this equation before.
It's not on your equation sheet.
It breaks my heart that it's not there.
Memorize it.
Use it.
Practice with it.
You'll see how powerful it is.
So all we have to do is take the derivative
of the potential energy function.
Don't forget the negative sign.
And you'll get the force function.
So the acceleration at 0.6 meters
is going to be negative 8x over the mass.
And now we know what the x has to be.
We know what the mass is.
So we're looking for magnitude.
We can ignore the negative sign.
And there's our acceleration, 1.6 meters per second squared.
So the answer to this question.
Now, the last part of this question,
they want you to sketch graphs of the object's
position versus time, as well as kinetic energy versus time.
And these are the graphs they give you.
They'll line them up for you.
That should be your first clue that you want.
This is just a sketch problem so you
don't have to get real detailed, but you need to match trends.
Draw general trends.
And match key maxes and mins, for example.
Let's consider the displacement function.
Remember, this was our acceleration.
The acceleration is proportional to the negative displacement.
That should ring a bell.
Simple harmonic motion comes to mind.
Whenever you have acceleration proportional
to negative displacement, you've got an oscillating system.
And they told us to assume that at the origin,
the object was x equals 0 at time equals 0.
So that's where we're going to start our harmonic motion
cosine function.
Now, although they don't ask for it, it turns out--
let's review a little simple harmonic motion,
acceleration being proportional to displacement
by negative omega squared.
And omega is just 2 pi over the period.
And we had 3 kilograms of mass.
So we can figure out exactly what the period is.
It's 0.883 seconds.
They don't ask for it, but this is a good thing for you
to practice.
Practice, practice, practice.
So we've got this displacement function.
And now, we need to figure out what
our kinetic energy is at each of these places at time.
Position versus time.
Kinetic energy versus time.
I went ahead and I lined up all the places,
where the cosine crossed the x, the time axis, and the mins
and maxes.
And I brought them down to the kinetic energy graph.
Because notice when the potential energy is
at the origin, you've got 0.
In fact, everywhere you cross the x-axis,
you've got a maximum kinetic energy
because you have no potential.
So let's just go ahead and put all those maximum points there.
And at a maximum x squared, you're
going to have no potential energy.
Because you'll have all--
I'm sorry, you'll have no kinetic energy
because you'll have all potential energy.
So let's go ahead and put those zero points in.
And don't just connect the dots.
Make nice little curves.
But there you have it.
That's it.
As long as you draw at least one cycle of motion
and match up the minimums and maximums, you're good to go.
Full credit.
So for this problem, you needed to know
what's expected when you're asked to calculate, derive,
sketch.
These are specific terms that have
specific expectations of you.
So here's the deal, though, with this problem.
There were some very common errors and misconceptions.
For one thing, students did very poorly with basic algebra.
Please, excuse my dear aunt Sally
is always a good rule of thumb.
Practice, practice.
And here's another thing that came up.
An object, if it doesn't have constant acceleration,
be careful and don't apply these kinematics equations.
They don't work.
When an object is oscillating, for example,
simple harmonic motion, you can't use these equations.
You've got to resort to calculus and such.
So this is a good problem to review.
Potential energy comes up a lot.
And I really think you've got this.
You're what?
Two weeks away from game day.
Go for the five.