- Good morning, today we're going to review rotational
kinematics as a part of the AP Physics
C mechanics curriculum.
♫ Flipping Physics ♫
Billy what are the equations for average
and instantaneous angular velocity?
- The symbol for angular velocity is the lower case
Greek letter omega, and it looks like a curly w.
Average angular velocity equals the change in
angular position over change in time and instantaneous
angular velocity velocity equals the derivative
of the angular position with respect to time.
And, the units for angular velocity are usually
radians per second when we use them in physics.
However, angular velocity is often in revolutions
per minute in the "real world".
- Bo, what about angular acceleration?
- The symbol for angular acceleration is alpha--
- Fishy thing. - Fishy thing.
- Some of us call alpha fishy thing.
Average angular acceleration is change in
angular velocity over change in time.
Instantaneous angular acceleration is
the derivative of angular velocity with respect to time.
And, the units for angular acceleration are
radians per second squared.
- Now, let's talk about uniformly
angularly accelerated motion.
- [Students] U fishy m.
- That's right, u fishy m.
These equation work just like the uniformly
accelerated motion equations and you can use them
whenever the angular acceleration is constant.
Class, there are how many u fishy m variables?
- [Students] Five.
- And, how many u fishy m equations?
- [Students] Four.
- If you know how many of the u fishy m variables?
- [Students] Three.
- You can find the other?
- [Students] Two.
- Which leaves you with one?
- [Students] Happy physics student.
- I don't know why only the first two of the
u fishy m equations appear on the AP equation sheet.
And, I know that those equations on the AP equation sheet
look a little bit different than the ones
that I have listed here, but we already discussed
those differences during the kinematics review,
so I'm not gonna do it again.
Bobby, what is the equation for arc length,
and what is arc length?
- The equation for arc length is s equals r delta theta
where s is the arc length, r is the radius of
the object moving along an arc, and delta theta
is the angular displacement of the object.
- But, he uses a weird s for arc length.
- Mr. P. uses a lowercase cursive s
because his esses and fives look identical.
- Yeah, so s for arc length, and arc length is
the linear distance traveled by an object
when it is moving along part of a circle or arc.
- Remember, you must use radians for the angular
displacement when using the equation for arc length.
And, speaking of radians,
one revolution equals 360 degrees which equals 2 pi radians.
And because arc length is a linear measurement
it has linear units like meters, centimeters,
millimeters, et cetera.
Bobby, please take the derivative of the entire
arc length equation with respect to time.
- Okay, arc length equals radius times angular
displacement, take the derivative of that equation
with respect to time.
That means we have the derivative of arc length
with respect to time equals, well let's assume
r is constant so r doesn't change with respect to time.
So, it is r times the derivative of change in
angular position, actually we need to take
the derivative of just the angular position
with respect to time.
The derivative of arc length with respect to time
is a linear velocity when moving along a circle.
- That's tangential velocity.
- Right, and the derivative of theta, or angular position
with respect to time is angular velocity.
So, we now have tangential velocity equals radius
times angular velocity.
- Tangential velocity is the linear velocity
of an object moving in a circle.
And, it has linear units, so meters per second
in base SI units.
Bo, please take the derivative of the entire
tangential velocity equation with respect to time.
- The derivative with respect to time of the equation
tangential velocity equals radius times angular velocity
is, well the derivative of tangential velocity
with respect to time is tangential acceleration.
And, the derivative of angular velocity
with respect to time is angular acceleration.
So, assuming the radius stays constant tangential
acceleration equals radius times angular acceleration.
- Tangential acceleration is the linear acceleration
of an object when it's moving in a circle.
So, it has linear units, meters per second squared
in base SI units.
And, both of these tangential quantities,
tangential velocity and tangential acceleration
are tangent to the circle the object is describing.
This means they're also perpendicular to the radius
of the circle the path of the object is describing.
Now, I don't know why however,
the equation of tangential velocity is
the only one of these three equations
which does appear on the AP equation sheet.
However, it's relatively easy to remember
the other two if you can remember how they are derived.
And, just like when using the equation for arc length
when using the equations for tangential velocity
and tangential acceleration you must use radians
for the angular quantities.
Now, let's take a look at the motion of
this sphere on a string.
We're going to assume this sphere on a string
is moving with zero angular acceleration.
When the angular acceleration of an object equals zero
the motion of the object is called Uniform Circular Motion.
Considering the angular acceleration of
this sphere is zero class is the angular velocity
of the sphere constant? - Yes.
- Yes, an angular acceleration of zero means
the sphere has a constant angular velocity.
Class is the linear velocity, or the tangential
velocity of the sphere on the string constant?
- No. - Zero.
- Yes?
- Okay, let's try something slightly different.
Class, is the tangential speed of the sphere constant?
- [Students] Yes.
- Correct.
Is the magnitude of the tangential velocity
of the sphere constant?
- [Students] Yes.
- Correct.
Is the tangential velocity of the sphere constant?
- Yes. - No.
- No, the direction of the tangential velocity
is changing the whole time, and tangential velocity
is a vector.
So, if the direction of the tangential velocity
is changing, then the tangential velocity is changing.
- Correct.
Because the direction of the tangential velocity
is not constant the tangential velocity is changing.
And, because the tangential velocity is changing
there must be a linear acceleration causing
that change in velocity.
The linear acceleration which causes circular motion
is called centripetal acceleration
and centripetal means center seeking.
So, the centripetal acceleration is the acceleration
which is in toward the center of the circle,
which the path of the object is describing.
The equation for centripetal acceleration is
tangential velocity squared divided by the radius
of the path the object is describing.
It also equals that radius times
the angular velocity squared.
Now, according to Newton's Second Law where there
is a linear acceleration there must be a net force.
Therefore, if an object is moving in a circle
there must be a centripetal acceleration
and there must be a centripetal force.
The centripetal force is the net force in the in direction.
I'll say that again.
When an object is moving along a circular path
there must be a net force acting inward on
the object and that net force in the in direction
is called the centripetal force.
It is equal to, according to Newton's Second Law,
mass times the acceleration in the in direction.
And, the acceleration in the in direction acting
on the object is called the centripetal acceleration.
Class, what three things did I ask you to remember
about the net force in the in direction,
the centripetal force?
- It's not a new force.
- So, it's never in a Free Body Diagram.
- The inward direction is positive, and therefore
the outward direction is negative.
- I would do the conical pendulum example
right now, however we already did that
during our AP Physics 1 review,
so let's not do it again.
But, you can watch the link if you want.
Let's now talk about non-uniform circular motion.
Billy, what needs to be true to have non-uniform
circular motion.
- [Billy] The angular acceleration of the object
needs to not be equal to zero.
- [mr.p] When the angular acceleration is non-zero
then the object has not only a centripetal acceleration
which is directed in towards the center of the circle
but also a tangential acceleration which is tangent
to the path the object is describing.
This means the acceleration of an object
in non-uniform circular motion is the sum
of both the centripetal and tangential accelerations.
The object in non-uniform circular motion
will have a total acceleration which has a direction
which is between the direction tangent to the circle
and the inward direction.
- [Billy] Okay, when an object has an angular
acceleration it will then have both a centripetal
acceleration and a tangential acceleration
and therefore it will have a net acceleration
which is the sum of those two accelerations.
Yeah, that makes sense.
- [mr.p] Let's return back to the equation
for angular velocity.
Bobby, what was the equation for
angular velocity, again please?
- Average angular velocity or omega equals change
in angular position or change in theta
or angular displacement divided by change in time.
- And Bo, if an object has traveled through
one full circle how many radians has it traveled through?
- One revolution is 360 degrees which is two radians.
- Two pi radians. - Right, two pi radians.
Thanks. - It's okay.
- And, the time for one full revolution is called
the period of the object and it has a symbol
of capital T.
Therefore the angular velocity of an object
equals two pi divided by the period of the object
which we could rearrange to get
period equals two pi divided by the angular velocity
of the object which does appear on the AP equation sheet.
That completes my review of rotational kinematics.
Next, please enjoy my review of rotational dynamics.
Or, you could visit my AP Physics C review webpage.
Thank you very much for learning with me today,
I enjoyed learning with you.