Hi. It’s Mr. Andersen and this AP Physics essentials video 67. It is on the vector properties
of angular quantities, which maybe the longest title in this whole series. But it gets at
rotating systems, like a gyroscope. And to really understand how a gyroscope does its
magic, you have to understand what are the properties of this rotating system and how
do they act as vectors in different directions. And so it is confusing when you look at rotating
systems. And so a good analogy that I came up with was to think about the North star.
So if you go out at night, you can always find Polaris, and Polaris is always going
to be pointed to the north. Now why do we use the North star? Because all of the other
stars are going to move in the night sky. If you were to shoot them with time-lapse
it would look like this, with the North star right in the middle. Now those stars are not
moving and it is the earth that is spinning, but since the North star is pointed right
where the earth’s axis is, then it is not going to appear to rotate. And in a rotating
system, when we are looking at angular quantities, all the parts of that rotating system are
going to be moving. The only thing that is not going to move is going to be that axis
in the center. And so we can use that axis in the center to figure out the vector forces
or accelerations of all of these objects using right hand rule. And so if you have a rotating
system like this, this one is rotating counter-clockwise, there are going to be a number of different
properties. And so one could be a torque. Maybe a torque was applied or a force was
applied to that rotating system. Maybe that increased its angular velocity. And as we
get an increase in angular velocity we have angular acceleration. And now since this thing
is moving it has motion. It also has rotational inertia. And so it is going to have angular
momentum. All four of these are going to be vectors. And it is very simple to figure out
in which direction they are pointing. And that is using the right hand rule. And so
if we have a rotating system that is rotating like that all you do is use your right hand
and I am going to move my fingers in the directions of the rotation and my thumb is going to point
right out at me. And so all of these are going to be vectors that point in the direction
of right out the screen. Let’s say it was rotating in the other direction, then you
use your right hand rule again and you move it upside down and now it is going to be pointing
away from you. And so if there is no rotation in a gyroscope, what is going to happen to
it when I let it go? Well it is simply going to fall over. Now is that a rotating system?
It is. The bottom is stationary and so we have a torque or a force being applied to
it. And that is the force due to gravity. But it is not going to stand up. And so if
we get a gyroscope spinning, so I applied a force to it, I applied a torque to it, and
we have rotation in that direction, we can start to figure out where was that torque
exerted, down. Where is the increase in angular momentum? It is going to be down as well.
And so now it has momentum, angular momentum and that is going to resist this torque due
to gravity. And so if I just let it spin it can even balance on top of this cup, because
we have momentum in that vertical direction. It is resisting changes in either direction.
Now what happens as it slows down? Eventually it is going to slow down and that force of
gravity is going to pull it off the cup. And so let’s look at those properties again.
In review, what is a torque? A torque is applying a force that is perpendicular to the radius.
In other words a point from the center of that axis. And so if we figure out where that
torque is, so it has a force that is perpendicular to this spinning disc, all I do is use my
right hand rule again. I am going to put my fingers in the directions of the force, where
is the torque going to be acting? It is going to be acting straight up. Or in other words
there is going to be a force in that direction. Let’s look at angular velocity. That is
going to be the rate of spin of this disc. Let’s say it is fixed at this point. And
so let’s say it is spinning in this direction, where is going to be the angular velocity?
What is the vector for that? Again, all the points of the disc are moving everywhere,
but where is going to be the direction of the angular velocity? Again using your right
hand rule it is going to be pointed straight up. Let’s say we start to increase that
angular velocity. Now we are getting angular acceleration. Which direction is that going
to be? It is going to be pointed up as well. And now let’s look at angular momentum.
And this is kind of a thought experiment. Imagine if we had a big disc that was kind
of frictionless and you were to start to walk on it. So you are going to start to move in
that direction. Well you are a point object, you are moving in that direction and you are
going to start to have angular momentum. Where is the angular momentum going to be? Again,
using your right hand rule it is going to be pointed up. Now using Newton’s third
law you are going to apply a force into that disc. It is going to apply a force back on
you. And so that disc is going to move in the other direction. And so if I move my hand
in that direction where is angular momentum of the disc itself going to be? It is going
to be in the opposite direction. And since it started with no angular momentum the sum
of those angular momentums is going to be equal to each other. I am going to move in
one direction the disc is going to move in the opposite direction. And so this helps
us explain how a gyroscope works. And so if we start a gyroscope spinning, let’s say
it is spinning in this direction, where is the torque? It is going to be towards the
center or where that string is. Where is the angular momentum going to be? It is going
to be in that direction as well. And so it is going to have a certain amount of angular
momentum in that direction. So let me start that gyroscope spinning. So we can see it
is resisting changes due to gravity. But why is it starting to precess? Why is it starting
to turn like that? Well there is another rotating system here as well. There is going to be
the force of gravity down and that is going to be perpendicular to that radius coming
from the center of axis. And so now I am going to use my right hand there. I am going to
point my fingers down where that force due to gravity is. And now where is the torque
now? It is pointed out towards me. And so where is going to be the increase in the angular
momentum going to be? It is going to be towards me as well. And so we can add those two vectors
and we can figure out what is the sum vector of the gyroscope itself. And that is going
to be pointed towards us and that is why this is going to precess like that. It is just
going to keep moving like that. Now what is going to happen as it keeps spinning and spinning
and spinning? It is going to start to lose some of that angular momentum and as it does
that the force of gravity is going to apply a greater torque and it will eventually fall
down. And so in the physics classroom you can do inquiry with this. You could start
with a gyroscope or a better way to do this is use something larger that has a lot of
mass or rotational inertia, like a bicycle wheel. And you can spin it and then you can
hold your finger on one side and you can start to feel that force or that turning force.
Or you can just get a gyroscope of your own. You just apply a torque to it. It is going
to spin. These things work really good unless it gets out of control and it falls off of
the table like that. And so again, could you a representation to study all the forces acting
on a rotating system? And then finally, could you collect data on the torque, angular velocity,
angular acceleration and momentum? I hope so. And I hope this was helpful.