Mr.p: Good morning. Let's learn about angular momentum and do a simple example problem.
"Flipping Physics" Intro
Bobby, what is the equation for linear momentum?
Bobby: Linear momentum equals mass times velocity.
Billy: Don't don't forget momentum and velocity are vectors.
Bobby: Right.
Mr.p: In addition to linear momentum, objects can also have an angular momentum.
The symbol for angular momentum is a capital L.
Bo, what do you think an equation for angular momentum is?
Bo: Well, linear momentum equals mass times velocity.
So, instead of mass, it is rotational inertia, and instead of linear velocity, it is angular velocity.
So, angular momentum equals rotational inertia times angular velocity.
Billy: Don't forget angular velocity is a vector, so I bet angular momentum is a vector too.
Bo: Right.
Mr.p: Billy, you are correct that angular momentum is a vector
and Bo, you are correct that an equation for angular momentum is
angular momentum equals rotational inertia times angular velocity.
Notice that, because angular momentum and angular velocity are both vectors,
the direction of the angular momentum of an object
will be the same as the direction of the angular velocity of the object.
And, just like rotational inertia has an axis of rotation,
angular momentum, an equation for which is rotational inertia times angular velocity
also has an axis of rotation
Please make sure you identify the axis of rotation whenever working with angular momentum.
Angular momentum equals rotational inertia times angular velocity
is the equation for the angular momentum of Rigid Objects with Shape.
In other words, objects which do not change shape easily and are larger than point particles.
For example, discs, cylinders, spheres planets, hammers, records...
Which brings us to the example problem. Billy, please read, and Bobby, please translate.
Billy: Determine the angular momentum of a 141 g, 31.4 cm diameter record rotating clockwise at 45 rpm.
Bobby: Mass equals 141 g, diameter equals 31.4 cm.
I bet we will need the radius, so radius equals 31.4 divided by 2 or 15 .7 centimeters.
Angular velocity equals 45 revolutions per minute and
angular momentum, capital L, equals question mark.
Billy: The rotational inertia of a disc about its center of mass equals
one half times its mass times its radius squared.
Mr.p: Notice that the center of mass of the disc
is then the axis of rotation for the angular momentum of the disc.
Bobby: Right, so the rotational inertia of a disc about its center of mass equals
one half times its mass times its radius squared.
Billy: I just said that. Bobby: Right, but I was asking mr.p to write it down.
Billy: Did I not do that? Bobby: No, you read, I translated.
Mr.p: And Bo, please solve the problem.
Bo: Angular momentum equals rotational inertia times angular velocity.
Substitute in the equation for rotational inertia and then substitute in numbers.
One half times 141 times 15.7 squared times 45 or 781,989.5 or 782,000 with three sig figs
and the units are grams times centimeters squared times revolutions per minute?
That does not make sense.
Billy: We need to convert to base SI units and radians
because radians are a placeholder and should cancel out.
Bo: Right. So, multiply grams by one kilogram over a thousand grams.
Multiply centimeters by one meter over 100 centimeters.
Multiply 45 revolutions per minute by 1 minute over 60 seconds and 2 pi radians over one revolution to get
4.7124 radians per second for the angular velocity.
That means the angular momentum of the record equals
one half times 0.141 times 0.157 squared times 4.7124 which equals 0.00818899 or 0.0082 with 2 sig figs
and the units are kilograms times meters squared times radians per second.
Billy: But the radians have no units and are not needed.
So, the units are just kilograms times meters squared over seconds
Bobby: Those are just the base SI units for newtons, right?
Bo: Actually, no. A newton is kilograms times meters per second squared.
Billy: But kilograms times meters squared per second is equal to the units for linear momentum times meters
because it's the same as kilograms times meters per second all times meters.
Bo: Sure. Bobby: Okay.
Billy: I don't know. It seems useful.
Mr.p: I think y'all forgot something.
Bobby: Angular momentum is a vector. Bo: We forgot direction.
Billy: Right. So, it's rotating clockwise.
Okay, curl, the fingers of my right hand clockwise and stick up my thumb
and the angular momentum of the record is away from you, which is into the board and negative.
Bo: Right. Because if we were looking at the record from above
the direction of the angular momentum would be down, which is negative.
So, the angular momentum of the record is negative 0.0082 kilograms times meters squared per second.
Mr.p: Very nice.
But remember the angular momentum and angular velocity of the object are in the same direction.
Therefore the angular velocity is also negative.
Thank you very much for learning with me today. I enjoyed learning with you.