Bo: Hey, guys.
Billy: Hey, Bo!
♫ (lyrics) Flipping Physics ♫
Mr. P: Ladies and gentlepeople, the bell has rung,
therefore class has begun.
Therefore you should be seated in your seat,
ready and excited to start class.
You should be aware that today we're going to be
reviewing everything we've ever learned
in mechanics in AP Physics C,
so let me hear you get excited.
Bobby: Oh, wow.
Billy: Oh, yeah!
Mr. P: Please understand that today,
I just want to move as quickly as possible.
We're going to review.
I'm not going to call on anybody.
I'm not going to ask any questions.
That's just the way it needs to be today, okay?
Billy: Oh, I don't know if I can do that!
Bo: No problem.
Mr. P: Here we go.
The basics.
Vector versus scalar.
It may seem very elementary, and it is.
A vector has both magnitude and direction
and a scalar has magnitude only.
However, it comes up all the time.
Just one basic example comes up
on multiple choice questions.
Very often, that momentum is a vector,
and they do some sort of conservation of momentum problem
and people forget that it's a vector,
and they just treat it as a scalar,
so please remember the basic concept of a vector
versus a scalar.
Next, the concept of uniformly-accelerated motion.
Uniformly-accelerated motion,
the basic idea that
when the acceleration is uniform,
in other words, the acceleration equals a number
that there are a basic set of equations that you can use.
There are only three equations given
on the equation sheet,
however there are actually four.
I'm going to write them out.
Again, there are four UAM equations.
I don't know why this last one,
they choose not to give you on the equation sheet.
You're more than welcome to use it.
You'll notice I have a slightly different form,
but this is the form I prefer them to be in.
Again, this is UAM, uniformly-accelerated motion.
You can use this whenever the acceleration
equals a number.
In other words, the acceleration does not change.
It is constant, UAM.
Now, there are 5 UAM variables.
There are 4 UAM equations.
If you know 3 of the variables,
you can figure out the other 2,
and this leaves you with 1 happy physics student.
Next, we're going to talk about acceleration.
You can see I've actually written
2 equations for acceleration.
We have acceleration equals the derivative
of velocity is a function of time,
and acceleration equals the change in velocity
over change in time.
The derivative would be the instantaneous acceleration.
the change of velocity over change in time
would be the acceleration over a time period,
which would be the average acceleration.
Please remember that with any derivative,
you can actually rearrange it to get an integral.
In other words, velocity is also equal
to the integral acceleration as a function of time.
We can do the same thing with velocity
where we have the equation for instantaneous velocity,
the derivative of position as a function of time,
and the average velocity, the change in position
over the change in time.
We can rearrange the derivative to get
that the position is equal to the integral
of the velocity with respect to time.
Also, you need to remember what the derivative
and the integral actually mean.
The derivative means literally the slope of the line,
and the integral means the area under the curve.
Now, I put under in quotes
because what that means is the area
between the curve and whatever is on the x-axis.
In fact, one of the things you also need to remember
is that area under or below the x-axis
is actually negative.
So under the curve specifically means
that the area between the line and the x-axis
where above the x-axis, you have positive area.
Below the x-axis, you have negative area.
Bo: You guys do realize he posts all of his lecture notes
at flippingphysics.com, right?
Billy: Yes, I know that.
However, I find that I learn it better,
and I remember it when I write it all down.
Thank you.
Bo: Suit yourself.
Mr. P: Next, let's talk about the basic concept
of projectile motion.
Projectile motion is the basic idea
when you have an object flying through the vacuum
that you can breathe in 2 dimensions,
which means that in the x direction,
we don't have any forces
so the net force is equal to 0 in the x direction.
Therefore, the acceleration is equal to 0.
Therefore, you have a constant velocity.
You can then use the equation for constant velocity
in the x direction.
In the y direction,
you know that the acceleration in the y direction
equals negative g, where g on planet Earth
is a positive 9.81 meters per second squared.
This is uniformly accelerated motion
in the y direction.
In fact, it's actually called "freefall" in the y direction.
Notice that the change in time,
is the only variable which is a scalar
in these equations,
is therefore independent of direction,
and usually what you're doing
is you're solving for the change in time
in one direction then applying it to the other direction.
The net force or sum of the forces
equals mass times acceleration
where both force and acceleration are vectors.
Remember, whenever you're summing the forces,
you have to draw a free-body diagram.
A free-body diagram, or FBD for free-body diagram.
Sometimes you also call it a force diagram.
Whenever you sum the forces, you have to identify
the object or objects you're summing the forces on,
and you need to identify the direction.
If the direction isn't clear what's positive,
you also need to identify what the positive direction is.
Now, an interesting thing about Newton's second law is
this is not quite Newton's second law.
Better approximation of Newton's second law
actually has to do with the derivative.
This is Newton's second law using the derivative,
where the net force is equal to the derivative
of momentum as a function of time
where, again, force and momentum are both vectors.
What this means is where momentum
is mass times velocity,
a more accurate representation of Newton's second law
is that the net force is equal to the derivative of mass
as a function of time times velocity,
plus the mass times the derivative of the velocity
as a function of time.
One of the things that we usually assume
is that the mass of the object is not changing,
and therefore, this whole piece will go to zero,
and you just end up with a net force
equals mass times the acceleration.
The next basic concept
is the concept of impulse.
The symbol for impulse is a capital J.
Impulse is the integral of force with respect to time,
and it's also equal to the change in momentum.
Again, let's just review the fact that
that the impulse, then, is the area underneath the curve
because it's the integral of the force
with respect to time
and it's not unusual to see some sort of
multiple choice question that has to do with,
it gives you a graph, you have to figure out the impulse,
or the change in velocity, or something of that sort,
and you just need to figure out the area under the curve.
Sometimes they also do it with work.
Returning back to our net force equation
equals the derivative of momentum as a function of time.
If the net force on the system is equal to 0,
that means the derivative of the momentum
as a function of time is equal to 0,
and that means that the momentum does not change
as a function of time.
In other words, you end up with conservation of momentum,
the sum of the initial momentums
equals the sum of the final momentums,
again, where momentum is a vector.
That is where all the forces are internal,
that it means that the sum of the force is equal to 0,
and therefore you have conservation of momentum.
Let's talk about specific forces.
We could talk about the force of friction, for example.
The force of friction by definition
is less than or equal to mu times the force-normal
where mu is called the coefficient of friction
and does not have any dimensions,
and is dependent on
the 2 different surfaces that are interacting.
We actually have 2 different types of friction
or 2 different types of coefficients of friction.
We have static, which is non-moving friction
and kinetic, which is moving friction.
It ends up, we have to have
a couple of the different equations here.
In other words, because we can have both kinetic
and static friction,
we have that the force of kinetic friction
is equal to mu K times the force normal.
When the force of static friction
is actually less than or equal to
the coefficient of static friction times the force-normal.
However, you usually end up using the max-known force
of static friction, which is equal to mu-S times force-normal,
but it is important to remember
that the force of static friction actually adjusts
up and down to prevent the object from moving.
Some basic ideas about the direction
of the force of friction.
One, it opposes motion,
meaning it tries to prevent the object
from moving, or if it is moving,
it is in the opposite direction of the motion.
That's not quite always true.
In AP Physics, you can have an example
for if you have a ...
For example, if you have a box on a truck,
and the truck is accelerating,
there's a force of static friction on the box
that's in the back of the truck,
and that force of static friction would actually
be pushing the box forward.
That's sort of an unusual case,
but for most of the time,
the force of kinetic friction will be
opposite the direction of motion,
and the force of static friction
would oppose what direction the object is trying to move.
The force of friction is always parallel to the surface,
and one of the things that I always add
is that the force of friction, the direction of it is,
it's independent of the direction of the force applied
because so often students simply think
the force of friction is opposite the direction
of the force applied, which is not true.
Work.
Work is equal to the integral of force dot product
with respect to the position.
If you have a constant force,
that means that the force is
just the dot product of the force,
and the position or the r-vector,
and you could just do F r cosine theta,
the force times the change in position
times the cosine of the angle between the force
and the change in position.
Next, we have three different kinds of mechanical energy.
Kinetic energy, one-half mass times velocity squared.
Gravitation potential energy, mass times
the acceleration due to gravity times h,
the vertical height above the 0 line.
Please make sure you always set the 0-line
before you do a problem.
Elastic potential energy, one-half kx^2,
where k is the spring constant,
and x is the displacement from equilibrium position.
3 different kinds of mechanical energy.
3 different equations that have to do with mechanical energy.
First equation is conservation of mechanical energy,
or mechanical energy initial equals
mechanical energy final.
With all three of these equations,
please always identify your initial and final points.
If you're going to use conservation of mechanical energy,
you have to have no force applied
and no force of friction,
or at least no work,
no energy converted to heat and sound, etcetera,
via the force of friction.
For the work due to friction equals change
in mechanical energy,
that you need to have no force applied
in order to use that equation,
and the net work equals the change in kinetic energy.
It's actually always true.
Just so you know, these 2 equations often get confused.
The work due to friction equals
the change in mechanical energy,
whereas the net work equals the change in kinetic energy.
Be careful to remember those correctly.
Power, which is the derivative of work with respect to time,
which is equal to the derivative of the quantity for work,
which is the dot product of force
and position with respect to time,
or the derivative of position as a function of time,
is velocity if you have a constant force,
it would just be the dot product of a force and velocity.
Power is the rate at which work is being done,
the joules per second.
Please remember that you can rearrange this equation again
where than work would be equal to the integral of power
with respect to time.
There is an equation that is always true
whenever you have a conservative force.
A non-conservative force
would be friction, the force of friction,
and a conservative force
is a force for which the path does not matter,
where the work done is independent of the path.
If you have a conservative force,
this equation, which is not on your equation sheet,
and often comes up, you need to have memorized,
again, for a conservative force.
For a conservative force,
that force is equal to the negative derivative
of the potential energy associated with that force,
with respect to position.
An example.
For the force of the spring,
which is equal to the negative derivative
of the potential energy associated with that spring
with respect to position.
That's going to be equal to the negative derivative
of one-half kx^2 with with respect to position.
The derivative of that ends up being equal to
the negative kx, which you know is the force of a spring,
just as an example.
Center of mass.
The equation for the position center of mass
for a system of particles would be
mass 1 times position 1, plus mass 2 times position 2,
plus however many you have
divided by the sum of the masses, mass 1 plus mass 2,
however many we have,
where x1 and x2 are the position
relative to some reference line
which you get to choose.
Again, that's for a system of particles,
and you could do it, the x center of mass,
you could do the y center of mass.
You could do the r position center of mass,
which would be in 3 dimensions.
You can even take the derivative of this
to get the velocity of the center of mass.
You can take the 2nd derivative of it
to get the acceleration of the center of mass,
just as examples.
Again, for a system of particles.
If instead you have an object, a rigid object with shape,
the equation is the integral.
For that, we have the r center of mass,
or position center of mass is equal to
one over the total mass times the integral of r dm,
so the position with respect to mass.
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Please enjoy lecture notes responsibly.