mr.p: Good morning.
Today we are going to review the Rotational Kinematics
topics covered on the AP Physics 1 exam.
Bo: Hi guys.
Billy: Hey Bo. Bobby: Hi Bo.
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mr.p: Billy, please tell me
everything you know about angular velocity.
Billy: The symbol for angular velocity is omega
which looks like a curly W.
The equation is angular velocity equals
Delta theta, or change in angular position,
over change in time.
Change in angular position is the angular displacement,
which would be the angular position final
minus the angular position initial.
The dimensions for angular velocity are
radians per second, although
we often give them in revolutions per minute.
In one revolution an object goes through 360 degrees,
which is also two pi radians,
which we need to know in order to do conversions.
mr.p: And Bobby, please tell me
everything you know about angular acceleration.
Bobby: The symbol for angular acceleration is alpha...
All: Fishy thing!
Bobby: which we lovingly call "fishy thing".
The equation is angular acceleration equals
change in angular velocity over change in time.
And, the dimensions for angular acceleration are
radians per second squared.
mr.p: Make sure you notice the relationship
between the linear and rotational equations
for velocity and acceleration.
Instead of linear displacement,
we substitute in angular displacement,
and instead of linear velocity
we substitute in angular velocity.
The equations really are quite similar.
Which leads us to the Uniformly,
Angularly Accelerated Motion equations.
All: U fishy M.
mr.p: Which I like to call the "U fishy M" equations.
These are your two U fishy M equations
on your equation sheet.
And, I'm about to write down their two
Uniformly Accelerated Motion or "UAM" equation equivalents.
But there are actually four Uniformly Accelerated Motion,
or UAM equations, which means there are
also four U fishy M equations.
Remember, the U fishy M equations are the equations
you can use when the angular acceleration is constant.
And they work just like the UAM equations.
Now, I'm going to take this stopper on a string
and I'm going to rotate it at a constant angular velocity.
Now, it actually can't quite be a constant angular velocity
because on the way down the force of gravity
will increase the angular velocity,
and on the way up the force of gravity
will decrease the angular velocity.
However, for the purposes of this exercise,
let's assume that the stopper is moving
at a constant angular velocity.
Bo, what can you tell me about
the angular acceleration of the stopper.
Bo: The angular acceleration for the stopper
which is moving at a constant angular velocity.
Well, if the angular velocity is constant,
then the change in angular velocity is zero,
and therefore, the angular acceleration is zero.
mr.p: When an object is rotating
with an angular acceleration equal to zero,
like this stopper on the string,
it is called Uniform Circular Motion.
And Billy, which one has a larger angular velocity,
the stopper on the string, or the piece of blue tape,
which is also on the string?
Billy: Okay, which one has the greater angular velocity,
the stopper or the blue tape in the middle?
Well, the stopper is clearly moving faster,
so it has a larger angular velocity.
Bobby: Actually, no.
Remember the stopper and the blue tape go
through the same the same number of revolutions per minute,
so they have the same angular velocity.
You're probably confusing
angular velocity and linear velocity.
Billy: Yeah, that's right.
They both have the same angular velocity,
however, the stopper has a larger radius,
so it goes through a larger arc length,
and therefore a larger linear velocity,
which is called the tangential velocity.
Thanks Bobby.
Bobby: You're welcome.
mr.p: Right, this question gets to the difference
between angular and tangential velocities.
The tangential velocity of a rotating object
is its linear velocity.
And, as the radius increases,
the arc length through which the object goes increases,
and therefore the tangential velocity
of the object also increases.
Bo: Mr. P?
mr.p: Yes Bo?
Bo: What are the dimensions for tangential velocity?
Billy: Tangential velocity is still a linear velocity,
so it has dimensions of meters per second,
'cause it's a linear velocity.
mr.p: Yes, tangential velocity
is a linear velocity in meters per second.
And remember when you use this equation
that you need to use radians per second
for the angular velocity.
And the tangential velocity is
the velocity tangent to the circle,
or, if you prefer, normal to the radius,
or perpendicular to the radius.
Now, just looking at the stopper,
Bobby, considering the stopper has
a constant angular velocity,
does the stopper also have a constant tangential velocity?
Bobby: I remember this one.
The answer is no.
The stopper has a constant tangential speed,
but not a constant tangential velocity.
Bo: Yeah, because velocity is a vector,
it has both magnitude and direction,
therefore, because the direction of the tangential velocity
is changing, the tangential velocity is not constant.
Bobby: And because the tangential velocity
is not constant, there must be a linear acceleration
and therefore a net force acting on the object.
And that linear acceleration is called
the centripetal acceleration.
Class, what does the word "centripetal" mean?
All: Center-seeking.
mr.p: Yes, "centripetal" is Latin for "center-seeking".
Because, the net force in the in-direction
is the centripetal force, the center-seeking force,
which causes the centripetal acceleration,
or the center-seeking acceleration,
which is in towards the center of the circle.
The equation for centripetal acceleration is that
it equals the tangential velocity squared
divided by the radius.
Centripetal acceleration also equals the radius
times the angular velocity squared,
however, that is not on your equation sheet.
Bo, please tell me the three things
I asked you to remember about the centripetal force.
Bo: The centripetal force is not a new force.
It is the net force in the in-direction.
So it could be a single force,
like the force normal, or the force of tension,
or it could be a combination of forces.
Because the centripetal force is not a new force,
it is never in a Free Body Diagram.
And we end up summing the forces in the in-direction,
therefore we need to remember that
the in-direction is positive,
and the out-direction is negative.
mr.p: Notice in this particular example,
which is called a "conical pendulum",
that the in-direction is not in the direction of the string.
If you look at it from a top view,
you can see that the circle described by the stopper
is actually in a horizontal plane,
and therefore the in-direction is actually horizontal.
Billy, how would you solve a problem like this?
Billy: First you draw the Free Body Diagram
of the forces acting on the stopper,
which are the force of gravity, which is straight down,
and the force of tension, which is along the string.
Before you can sum the forces in the in-direction,
you need to break the tension force
into its components in the y direction and the in-direction.
After we find the components,
we can redraw the Free Body Diagram.
The force of gravity is still down,
however now instead of the tension force,
we have the tension force in the y direction,
and the tension force in the in-direction.
We can sum the forces in the y direction,
the tension force in the y direction is up,
and the force of gravity is down,
and the acceleration in the y direction is equal to zero,
so we can get the tension in terms of the force of gravity.
And then we can sum the forces on the stopper
in the in-direction, which would give us
the net force in the in-direction equals
the tension force in the in-direction,
which equals the mass of the stopper
times its centripetal acceleration.
And its centripetal acceleration equals
the tangential velocity squared divided by the radius.
Depending on what we know,
we could find the tangential velocity
either by using the equation:
tangential velocity equals the radius
times the angular velocity,
or: we could use the fact that velocity equals
displacement over change in time,
which for one full revolution would be
the circumference divided by time, for one full revolution,
which is called the period,
the symbol for which is a capital T.
But, we don't really know any variables
or really what we're solving for,
so I don't really know what to do from there.
mr.p: Billy, that is plenty, thank you.
I just wanted to set it up.
I didn't want to actually solve the problem.
One important thing to notice is that
the radius for the circle would not be
the length of the string, but would be
this side of the triangle,
and you could use soh-cah-toa to figure that out.
So, thank you.
mr.p: You are welcome.
mr.p: Now that you've brought up period,
let's talk about both period and frequency.
Period and frequency are inversely related.
The period is the time for one full cycle,
or time for one revolution, and the dimensions
for period are either seconds or seconds per cycle.
Frequency is the number of cycles per second,
or revolutions per second, and the dimensions
for frequency are cycles per second, which we call "Hertz".
Now, going back to the equation for angular velocity,
Angular velocity equals the change in angular position
over the change in time.
If the object goes through one full circle,
then the angular displacement is 360 degrees,
or two pi radians, and the change in time
would be defined as the period.
Solving for the period gives us one of the equations
on your equation sheet.
The period equals two pi divided by the angular velocity,
which is then also equal to the inverse of the frequency.
Bobby: Mr. P?
mr.p: Yes Bobby?
Bobby: Where did the radians go?
mr.p: The radians dropped out
because radians are dimensionless,
and we no longer needed them as a place-holder.
We have reached the end of my review
of Rotational Kinematics for AP Physics 1.
My next video is a review of rotational dynamics,
which you are more than welcome to enjoy.
You can also visit my website, flippingphysics.com,
where all of my AP Physics 1 review videos
are organized with lecture notes.
Thank you very much for learning with me today.
I enjoyed learning with you.