Hi. It’s Mr. Andersen. And this AP Physics essentials 97. It is on rotational inertia.
When an object is rotating, like the fly wheel on this tractor, it has a certain amount of
angular momentum. And that is made up of the angular velocity, how fast it is spinning
and also this rotational inertia of the object. But to calculate the rotational inertia it
is going to depend on what that object looks like. And this fly wheel has a large amount
of mass that is distributed far away from that point of rotation. And so it is going
to have a large rotational inertia. And it is a great place where we can store energy
as the engine is idling or we could even start the engine using that fly wheel. And industry
is starting to use industrial fly wheels as a place in a factory where we can store a
certain amount of energy. And so angular momentum of a system is based on two things. It is
the position of all of the objects, and that is going to be its rotational inertia. And
then the velocity of all of those objects. And so due to the conservation of angular
momentum, if we change the position of where those objects are we are not going to get
new momentum. And so what we are going to do is see changes in the velocity of those
objects. And so remember momentum of a particle that is moving, p equals m v, where m is mass
and v is the velocity of the object. But what if the object is not moving through space.
Let’s say it is just rotating. It still has momentum. We call that angular momentum,
L. And that equals I times omega. Where omega represents the angular velocity, how fast
it is rotating. And then I is very similar to the mass. It is this rotational inertia.
And in fact mass is a large part of that. And so let’s say we have a record that is
spinning. So it has a certain amount of angular momentum. And I simply just want to stop the
record. Well what makes it harder for me to stop the record? It is rotational inertia
and the velocity at which it is spinning. Now that rotational inertia is made up of
two things. The mass of the record or the turntable in this case. And also the radius,
how far it is from the center point. And if you needed to calculate it, this is the formula
for rotational inertia. And I equals 1/2 m r squared. And so if we were to just try this
with a record, the mass of a record is 1/10th of a kilogram. So 0.10 kilograms. And then
the radius from the center point is going to be 15 centimeters or 0.15 meters. And so
using that formula I could calculate my I or rotational inertia to be 1.1 times 10 to
the negative third kilogram meters squared. And in AP they will have to give you the formula
for various objects if they want you to calculate rotational inertia. Let’s say however we
were to change that record. And we were to have all of the mass on the outside of the
record. Well we would have to use a different formula. Instead of it being 1/2 m r squared,
it is actually going to be m r squared. And so we are not taking it times 1/2. And so
by positioning that mass on the outside, what have we done to our rotational inertia? We
have actually doubled it. And so if we get it going at the same speed it is going to
have twice the angular momentum. And now you can start to see why fly wheels have there
mass distributed to the outside. And so the inverter di Vinci came up with a flywheel
that is variable. And so as it would rotate, these spheres would be lifted up and they
would start to move to the outside. And so if you think about it, what is going to happen
to our rotational inertia as the mass is all moving farther out from that center point?
Well I is going to increase as we start to crank this flywheel. Now what happens when
you let it go? Well friction is going to slow it down. And so our angular velocity is going
to start to drop. But what is really interesting about this is that as it starts to slow down
these spheres start to drop down and so what happens to our rotational inertia? It actually
becomes less and less. And so he designed this as a way that you could have a crank
that keeps moving at a constant velocity. It is pretty smart. We would also see rotational
inertia if we are looking at objects in orbit. And so if we look at a satellite as it orbits
around the earth you can think of it as two objects that are connected. And so if we look
at how long it takes to go around the earth, it is around 90 minutes. But let’s say we
change what is orbiting. Let’s say it is now the moon going around the earth. It has
a large angular momentum. The distances are so large that it takes around 28 days for
it to go around the earth. Let’s say we were to look at the earth as it moves around
the sun. Again it takes a year for it to go all the way around. But what would happen
if we could magically grab the earth and position it in a different spot and we conserve the
angular momentum. Well now, since that rotational inertia is so much closer to the middle, we
are actually increasing the velocity and so we are going to see a year that is going to
be about half the distance of what it is. And we would see the same thing if we look
at planets that are closer to the sun. So did you learn to calculate the angular momentum
of a rotating object? Remember it is dependent upon the speed but also this rotational inertia,
which itself is determined by the mass, the radius and what the shape looks like.
And I hope that was helpful.