Bo: Hey, guys.
Billy: Hey, Bo!
♫ (lyrics) Flipping physics ♫
Mr. P: Next we're going to talk about circular motion.
Now we'll talk about circular motion
where we have the angular velocity,
which is omega (ω) is equal to the derivative of
the angle in position as a function of time.
You can also have change in theta (θ) over change in time,
which would be the average angular velocity
as opposed to the derivative, which is of course
the instantaneous angular velocity.
We have the angular acceleration
which is alpha (α), or fishy thing,
which is equal to the derivative of
the angular velocity as a function of time,
which would be the instantaneous angular acceleration.
Where you could also,
if we're talking about circular motion,
we could also talk about the centripetal acceleration,
the acceleration that is responsible for
the circular motion,
the acceleration in the in direction
because centripetal means center-seeking.
That's equal to the tangential velocity squared
divided by the radius.
it's also equal to the radius times the angular velocity squared.
We need to talk about tangential velocity.
You'll notice I put tangential velocity
equals the radius times the angular velocity
in the middle.
This is actually the equation on your equation sheet.
You are not given either one of these two,
but the reason I put these three up together
is it turns out that if you take the integral
of the tangential velocity equals the radius
times the angular velocity
with respect to time,
you'll get the arc length,
which is equal to the radius times the angular position.
Just so you know, that's a lowercase cursive s.
It stands for arc length.
That's equal to the radius times the angular position.
If you take the derivative of the tangential velocity
is equal to the radius times the angular velocity,
you're going to get the tangential acceleration
equals the radius times the fishy thing,
or the angular acceleration.
They only give you the one in the middle,
but by giving you the tangential velocity
equals the radius times the angular velocity,
you actually have all three.
All you need to do is either take the integral
to get the arc length,
or the derivative to get the tangential acceleration.
Again, whenever you use these equations
or centripetal acceleration,
you need to make sure that you use radians
because the radians are dimensionless.
Torque, or lowercase Tau (τ), is equal to r cross F,
the lever arm cross-product with the force.
Please remember with the cross product.
It is a vector, and the order does matter,
so it is r cross F.
You need to remember how to use the unit vectors for it
using the matrices to go through and solve that,
or if you're not going to use unit vectors,
it's just rF sine theta.
We also have a couple of equations for torque,
the net torque.
One of the equations for net torque is it's equal to
the moment of inertia times the angular acceleration.
We haven't yet defined the moment of inertia.
I will in just a moment.
The net torque, whenever you sum the torques,
just like when you sum the forces,
there are things that you need to identify.
When you sum the torques,
there are also things you need to identify.
You need to define the axis of rotation
where you're summing the torques around, what point?
You need to identify the object
or objects you're summing the torques on.
You need to identify the direction
that you're actually summing the torques in,
and you need to identify what the positive direction
for torque is.
Moment of inertia.
Always called ro-
phlellel ...
Moment of inertia, also called rotational mass.
Again, just like the center of mass,
we have two different equations for moment of inertia,
or rotational mass.
One equation is for a rigid object with shape,
and just like for center of mass,
it is one that involves the integral,
and be careful.
A lot of times, people confuse these two, so ...
The original object with shape is 1 over the total mass
equal to the integral of the position with respect to mass.
For the moment of inertia,
there is no 1 over the mass,
and it's just the integral of r²dm,
and this is the distance from the axis of rotation squared,
and you're integrating with respect to mass.
When you do that, generally you need to deal with
one of your various densities.
Volumetric mass density,
surface mass density,
and linear mass density.
rho (ρ), sigma (σ), lambda (λ).
We don't really work with sigma,
the surface mass density much in mechanics.
I include that because you do end up
working with surface charge density
in electricity and magnetism quite often,
so just for completeness sake,
but you end up working with the volumetric mass density
or the linear mass density often
with the moment of inertia.
For a system of particles, the moment of inertia,
or rotational mass is equal to the sigma,
the sum of the mr², so you just have a system of particles,
and it's just going to be the mass of each particle
times r, the distance from the axis of rotation, quantity squared.
You also need to remember the parallel axis theorem.
It'll come in handy sometimes.
The parallel axis theorem,
what you're doing is you're taking the moment of inertia
about the center of mass,
and figure out the moment of inertia
of that rigid object with shape
about some other point.
That's going to be equal to the moment of inertia
of the center of mass
plus the mass of the whole thing times d squared,
where d is the distance from the center of mass
to the new axis of rotation.
One common mistake with this
is remember that you can only use this
when you're at a constant density of the object.
Only works, the parallel axis theorem
only works with a constant density.
Two types of equilibrium.
We have rotational and translational equilibrium.
Translational equilibrium
means that the net force is equal to zero,
which means the acceleration of the object
is equal to zero,
which means the object is either
moving at a constant velocity or at rest.
A lot of times people think that an object
in translation equilibrium has to be at rest.
That is not correct.
It has to not be accelerating,
which means it's either at rest
or moving at a constant velocity.
Rotational equilibrium,
the net torque is equal to 0,
which means the moment of inertia
times the angle acceleration is equal to 0,
which means that the angular acceleration is equal to 0.
Again, it's either at rest or moving at
a constant angular velocity.
Please, remember, whenever you're dealing with,
translational equilibrium or rotational equilibrium,
you need to draw a free-body diagram,
sum the forces, sum the torques,
identify the objects, the direction, the axis of rotation,
what the positive directions are.
Please remember those basic concepts.
Now that we have circular motion,
we can talk about rotational kinetic energy.
Whenever anything is rotating,
it's going to have rotational kinetic energy,
one-half the moment of inertia
times the angular velocity squared.
Rolling without slipping simply means
that it's rolling without slipping.
It means it's rolling without slipping.
To solve a problem that includes rolling without slipping,
you need to use conservation of mechanical energy.
Remember to identify your initial point, your final point,
your zero line, and don't forget you have both
rotational kinetic energy and translational kinetic energy,
and ...
the velocity of the center of mass of the object
equals the radius of the object
times the angular velocity of the object.
That's what's going to connect the rotational kinetic energy
and the linear kinetic energy, please.
Angular momentum for a particle, r cross p,
again the cross product, again, the order matters.
If you're not dealing with unit vectors,
you can just do r times momentum
times the sine of theta, the angle between
the position and the velocity,
so rmv sine theta, where r is the distance
from the axis of rotation to wherever the object is.
Again, angular momentum.
People often forget that angular momentum is
about a specific point.
Have to identify it.
Something comes up on almost every AP test
having to do with this, and I'm going to walk through it.
It's a very basic concept right now.
Let's say we have a piece of clay
moving toward a circle, a cylinder of some sort
that is rotating around
its center of mass here, at point A.
Now, this piece of clay has angular momentum
as its moving toward this cylinder, if you will,
and we can figure out what it is.
It's equal to r, the distance from the object
to the axis of rotation, which is at point a,
times the mass of the object,
times the velocity of the object,
times the sine of theta.
Now, the issue here is that both r and theta
are changing as a function of time,
as a function of position.
The way you deal with that is this.
Sine of your angle theta is equal to
opposite over hypotenuse, where opposite that theta is d,
and the hypotenuse is r.
In other words, d is equal to r sine theta.
Now, r may be changing as a function of position,
theta may be changing as a function of position,
however, d is constant.
In other words, the angular momentum
of the object as a function of time
about point a is actually equal to
d times the mass times the velocity.
Comes up on just about every AP test at some point,
probably on a multiple choice.
That's angular momentum for particles.
Now we need to talk about rigid objects with shape.
Rigid objects with shape.
The angular momentum is equal to the moment of inertia
times the angular velocity.
Ooh, I had something else.
Oh! I just remembered!
Please, going back to rolling without slipping.
This is the condition for rolling without slipping.
People get really confused.
Rolling with slipping,
well, when it starts rolling with slipping,
this is no longer true,
so it asks sometimes,
"Okay, when does it stop?
"When does it start rolling without slipping?"
That's when this condition becomes true,
when the velocity of the center of mass
equals the radius of the object times the angular velocity.
Coming back to rigid objects with shape
and the angular momentum is equal to
moment of inertia times the angular velocity.
Again, the net torque,
the rotational form of Newton's 2nd Law.
It is equal to the moment of inertia
times the angular acceleration,
and is also equal to ...
The net torque is equal to the moment of inertia
times the angular acceleration.
It is also equal to the derivative of the angular momentum
as a function of time,
just like the net force is equal to mass times acceleration,
but more generally, it is equal to
the derivative of momentum as a function of time.
The net torque is equal to the derivative of the
angular momentum as a function of time.
Now, if all the torques are internal,
and again, you have to identify your axis of rotation
whenever you're summing the torques.
That means if the net torque is equal to zero,
in other words, there are no external torques,
this means that the momentum,
angular momentum is not changing.
Therefore, some of the initial momentums equals
the sum of the final momentums.
Conservation of angular momentum,
and again, you have to identify your axis of rotation.
Please.
Newton's universal law of gravitation.
The force of gravity between any two objects
is equal to negative big G, mass-1, mass-2,
divided by r-squared, times the unit vector r.
Big G is universal gravitational constant,
6.67 times 10 to the negative 11, what is it?
Newtons, times meters squared over kilograms squared.
The force of gravity, this is the force of gravity
that exists between any two objects.
They're always being pulled toward one another,
whenever you have two objects.
So, one thing that people forget about this,
This is a unit vector,
so it just has to do with the direction,
but the main thing people forget
is that r is not the radius.
r is defined as the distance between
the center of masses of the two objects,
and it's especially confusing because sometimes r
actually does work out to be the radius,
but by definition, it is not.
It is the distance between the center of masses
of the two objects.
I want to talk about Kepler's 3rd Law.
Please, do not memorize Kepler's 3rd Law.
Instead, remember how it's derived.
In Kepler's 3rd law,
we start with a net force in the in direction,
which is equal to the force of gravity,
which equals mass times the centripetal acceleration.
Centripetal acceleration because it's the acceleration
in the in direction.
the force of gravity is Newton's Universal Law of gravitation
because we're talking about two objects.
You're usually talking about
the thing that's in an orbit around the giant thing.
We'll call the thing that's in orbit mass-1,
so we have big G, Newton's universal gravitational constant
times mass-1, times mass-2, divided by r-squared,
where r is the distance between the center of masses
of the two objects.
Now, on the other side, you have mass 1,
which is the satellite times the centripetal acceleration.
We can use the radius times the angular velocity squared.
Now, notice in this particular case,
even though the r is defined as
the distance between the center of masses
of the two objects,
it is also the radius.
In other words, these two r's are the same,
and we can end up with
Big G times the mass of the planet
is equal to the radius cubed,
times the angular velocity squared.
This is, of course, the radius of the orbit of the object.
Now, with Kepler's 3rd Law,
we generally need to look at the angular velocity.
Angular velocity is the change to angular position
over change in time.
That would be the average angular velocity.
If we're talking about one full revolution,
the time would be the period,
that's the definition of the period,
and for one full revolution,
the change in theta would be 2 pi radians.
In other words, the angular velocity is equal to
2 pi divided by the period.
We could substitute that back into our equation,
and this is Kepler's 3rd Law.
If you rearrange it for the period squared,
you'll get Kepler's 3rd Law,
but that's the basic idea,
the fact that you end up using this,
and who knows exactly what they're going to
ask you to solve for.
Please do not memorize Kepler's 3rd Law.
It's a bad idea.
Please just remember how it is derived.
Another basic thing that I want to make sure we get here
is that frequency is equal to 1 over the period,
which means ...
that the angular velocity is equal to 2 pi times the frequency.
When we get to simple harmonic motion,
this is going to be called the angular frequency,
and often people think that the angular frequency
and the frequency are the same thing,
angular frequency and the frequency
are clearly not the same thing.
Universal gravitational potential energy.
Negative big G m1 m2 over r.
Notice the similarity between the two,
often confusing, leaving or adding the square,
so be careful of that.
Universal gravitational potential energy,
big G, m1, m2, over r.
Notice that the zero line is already preset
for universal gravitational potential energy.
The zero line is at r is equal to infinity.
1 over infinity is 0, therefore the zero line
is infinitely far away.
Therefore, notice the gravitational potential energy
is universal gravitational potential energy
is always negative, and you have to have two objects.
You cannot have gravitational potential energy,
universal gravitational potential energy without 2 objects.
Please.
Simple harmonic motion, SHM.
The condition that must be met
in order to have simple harmonic motion
is that the 2nd derivative position as a function of time
is equal to negative omega squared
times the position where omega is the angular frequency.
That is, if you could show that
the motion of an object follows that pattern,
whatever you have for angular frequency,
you could figure out the period.
We'll go through that in just a moment.
One equation that fits for that
is that the position as a function of time
is equal to the amplitude
times the cosine of the angular frequency
times time plus the phase constant, the phase shift.
That's all it is.
It's just how much you're shifting from cosine or sine,
or which ever one you want to have.
You can take the derivative of this to get the velocity.
You could take the second derivative
to get the acceleration, so on and so forth.
Let's go through a specific example.
We're just going to talk about a mass spring system.
Let's talk about a mass spring system.
In a mass spring system,
the net force in the x direction's going to be
equal to the negative of the force of the spring,
assuming the force this spring is going to be left.
The force of the spring is k.
The spring concept times the position,
so we have negative kx equals the mass
times the acceleration in the x direction.
In other words, the acceleration with respect to x
is equal to negative k over m times x
or acceleration, which is the 2nd derivative position
as a function of time in the x direction
is going to be equal to the negative, the k,
the spring constant divided by mass times the position.
In other words, omega, the angular frequency
for a mass spring system,
is equal to the square root of the spring constant
divided by the mass,
The angular frequency, then, is equal to 2 pi
divided by the period,
we already figured that out before,
which means that the period of a mass spring system
is equal to 2 pi times the square root of mass
times the spring,
or the square root of mass divided by the spring constant.
Just deriving at the period for a mass spring system.
Anything that's in simple harmonic motion,
you can figure out the period using that basic concept.
We also have the period for a pendulum,
For a pendulum, the period is going to be equal to 2 pi
times the square root of length of the pendulum,
the distance from the center of suspension
to the center of mass of the object,
or the pendulum divided by the acceleration
to gravity on whatever planet you're talking about.
In simple harmonic motion,
generally we deal with non-damped simple harmonic motion,
which means energy is going to be conserved,
which means that the total energy
is going to be equal to, if you're talking about
all the way at the amplitude,
the total energy would be one-half k
times the amplitude squared,
which would equal to, if we're talking about the total energy,
we could be talking about the equilibrium position.
This would be the point at which it has
the maximum velocity.
It has zero elastic potential energy,
therefore the energy would be entirely kinetic energy,
one-half mass times velocity maximum squared.
That is my review of the entire mechanics,
AP Physics C curriculum.
A brief one at that,
but I hope you enjoyed learning with me today.
I enjoyed learning with you.
Voiceover: Lecture notes
are available at FlippingPhysics.com.
Please enjoy lecture notes responsibly.