Hi. It’s Mr. Andersen and this is AP Physics essentials video 95. It is on the center of
mass. I want you to take a look at this tire for a second. And I am going to apply a force
to it. You can see the motion is fairly regular. And the reason why is that the center of mass
is right in the middle of that tire. And it is rotating around that center of mass. Now
let me apply a force to a hammer. And you will see that the motion is not as uniform.
It is tricky to our eyes. And the reason why is that the center of mass is off-center.
What I am going to do though is keep the center of mass where it is and then I am going to
rotate that hammer. And you are going to see that the center of mass stays where it is.
It is just the motion of that object above and below the center of mass that fools your
brain. Now let me show you a system. So this is the Saturn 5 rocket. It is made up of a
number of different stages. And so where would you say the center of mass is? The average
mass? Well it is maybe somewhere like that. What do you think would happen to the center
of mass if I were to break this rocket apart into its two stages at this point? Well it
will stay in the same location. And so if we do not have an external force acting on
the system, the center of mass stays where it is. Even though that might be rotational
motion, as we break those stages apart, the center of mass remains where it is. And so
any system is going to have a center of mass. And that center of mass is made up of all
of the objects within the system. And what is neat about physics, according to Newton’s
second law, is if we do not apply a net external force then there is going to be constant velocity.
In other words we are not going to have acceleration in that object. It will either remain still,
the center of mass, or it is going to move with a constant velocity. And so how do you
figure out where the center of mass is? Well a good place to get started is with a simulation.
And this is center of mass builder from simbucket. And what you can do is you can click on the
screen and it starts to make mass. It will then show you where the center of mass is.
So you can see if I make a uniform mass, center of mass is right in the middle. What happens
if I add another uniform mass in the x direction? It is right between the two. What if I move
in the y direction? You can see that it is going to be the average of that as well. So
the center of mass does not have to be within a specific object. What is neat is you can
use the hang test in physics. So if we simply hang this object, if we let it settle out
the vertical line through that hanging point is going to go right through the center of
mass. So you can calculate the center of mass experimentally, and I will show you how to
use an equation to do it a little bit later in the video. And so if we apply a force to
a object we can get what looks like irregular motion in this hammer. But since we have applied
a force, we know that we are going to have constant velocity. And so if I mark where
that center of mass is, slow it down, watch what happens. We are going to have constant
velocity. And that movement above and below that center of mass is confusing to our brain.
And so let me show you how this works with a little video. I have two pieces of ice that
are connected together with a spring. I am going to apply a force to one and they are
going to slide across the ice. And so we get this caterpillar like motion. So to our brain
it looks like we are having acceleration and then deceleration. But what I want you to
do is get involved. So I want you to hold your finger on the screen where the center
of mass is as I play this video. Where would that center of mass be? It is going to be
on the spring right between those two blocks of ice. And so what you are going to try to
do is as I play the video keep your finger between the objects on the spring. And so
let’s do that. So you should be tracing it out. Keep your finger between the blocks.
And you should find that as you trace it you are showing movement from left to right with
constant velocity. It is slowing down at the end a little bit. Now let’s try a little
harder one. We have a force on both of these bricks, on either side. So again what you
are trying to do is hold your finger in between those two bricks on the spring itself. So
let me play this video. And I am doing this as well. So what I am finding is that it is
not moving as quickly but it is still moving with constant velocity from left to right.
And we could do even a complex system, like this. We have four objects that are connected
together with springs. So try to hold your finger where the center of mass is. That is
going to be between all four of these objects. And if you do that, it is hard to kind of
keep it in-between all four. But your brain is pretty good at this. Again, what am I finding?
There is crazy motion on the screen, but my finger is moving across with constant velocity.
And that is because there is no external force. And so we could look at a rocket, as a rocket
flies through the sky, it takes a parabolic path, like that. So we could trace it out
and it would look like that. But let’s say we were to take that same rocket and it were
to fly through the sky but half way through it were to break apart into its two different
parts. So it looks odd. In other words that motion looks odd as it breaks apart. But if
we were to plot the center of mass, it would follow the same parabolic path of that rocket
by itself. And that is because we have no external force. And so what is neat about
the center of mass is you can take systems that are incredibly complex, and if we just
figure out where the center of mass is, it makes solving that problem much easier. So
as you move into AP Physics 2 you will have to calculate where that center of mass is.
And so if we have this object right here, so we have two masses, they are connected
with a bar, but let’s say there is no mass in that bar, here is the formula that shows
you how to calculate where the center of mass is. And what you do is really find the point
where you can balance that object on that point. And so we set up an arbitrary distance
a distance away in the x from our object. And we will call that Xcm. That is going to
be the center of mass. We then set up our two distances, this is going to be x1. It
is the distance to the center of mass of mass 1. And then x2 is the distance to the center
of mass of mass 2. So our formula, all you do is simply multiply mass 1 times x1. So
it would be this mass times that x plus this mass times that x and then you divide the
whole thing by the sum of the two masses. In other words mass 1 and mass 2. So if I
give you a problem you could pause the video at this point and try to solve it. But I will
show you how to work this one out. And so what you would do is you would simply multiply
mass 1, which is 1 kilogram times the distance which is 2 meters. Plus mass 2 which is 2
kilograms times the distance of 5 meters. And then you divide it by 1 plus 2 kilograms
or 3 kilograms. It looks like that. And you should get a distance 4 meters. In other words
Xcm is going to be 4 meters from here. And so it would be 2 meters from mass 1 and it
would be 1 meter from mass 2. That is a pretty easy problem. But if you could throw in different
values then you are going to get a harder problem. What if I throw in another mass,
like this. How would you solve it? Now we have 3 masses and they are moving in the x-axis.
All we do is simply add this. Mass 3 times x3. Where x 3 is the distance to the center
of mass of mass 3. And then we are going to divide by the sum of all 3 masses. And so
solving it, you could have, you know 12 masses. It is pretty easy to figure out where that
center of mass is. What if I were to turn it like this. Instead of going in the x, it
were also to go in the y. Well how would you do this? First of all we would have to get
the x coordinate at the center of mass or, we are going to use this formula down here.
You would then chose an arbitrary distance in the y and then you are going to use the
same equation. But this time instead of figuring out how far we are going in the x, you would
also figure out how far we are going in the y. Now I know my x coordinate, my y coordinate
and we could even move into the z to figure out where that center of mass is. Remember
if we do not apply a force to it it is going to have constant velocity. And then as you
move into AP Physics 2 you have to be able to calculate in one and two dimensions where
that center of mass is. So I hope that made sense. And I hope that was helpful.