Now that we've discussed the equations of motion,
how things just move on their own, as well as forces,
how things operate under the influence of forces
especially using Newton's second law.
We're now ready to move to a few last topics in the topic of forces
which includes circular motion and how to find the center of mass.
So first let's go to circular motion.
We'll start with that and go to center mass at the end where with circular motion,
we're going to go right back to where we first said about Newton's idea of things falling and orbiting.
When Newton had this idea that the same force that kept an object on orbit
as well as brings an apple to fall to the ground. His idea was what if the apple were moving,
or the object, in this case the moon around the earth, or something
was moving to the side as it was falling.
Then in this picture here you can sort of imagine to yourself that
if the object is moving to the side, but then also fell towards the center of the earth a little bit,
it would stay on a circular path by both moving and falling a little bit.
So that an object that's actually moving in a circle, for example the moon orbiting the earth,
is in fact always falling towards the earth but the fact that it's also moving sideways as it falls.
So it moves and falls, it moves and falls in its continual orbit and this was Newton's idea.
There's one tricky thing which might have come to your mind, as I said this which is,
it seems like a very particular balance to strike and what if it's moving a little too fast,
for the following motion and what if it's not moving fast enough, for the following motion,
in which case it might go too far or pass its orbit, or start falling in to earth's orbit.
What does it need to stay on an exactly perfect circular path?
That's a question we can ask ourselves is how fast must the object be moving
in order to maintain a perfect circular motion and this is uniform circular motion which is the question right now.
It turns out and we won't derive this right now but the acceleration that an object needs
towards the center of the earth or the center of any circular orbit,
has to be related to the velocity of the object in a very particular way and that is this,
A, towards the center, this a-sub-c has to be equal to the velocity of the object squared
divided by the radius of whatever circle, whatever path it's going in.
This means, since we know the acceleration of the object, we can use Newton's second law
to also find the force that an object would need to have pulling it towards the center of whatever path it's going in.
For example, the force of the earth on the moon,
keeping it in its orbit, the force that the object needs in order to continue going in a circle,
and this will be f equals ma just using Newton's second law again.
So the force is the mass times the acceleration,
where we've solved for the acceleration as v squared over r.
So the force an object need to stay in a circle has to be equal to v squared over r,
if it is to maintain a uniform circular motion.
There's something tricky though about this force that we've just introduced.
This uniform circular motion force, and that is that it's different than the other forces we've introduced,
because it's not an actual force acting on objects, this mv squared over r.
What we've done in the derivation that I just talked about
is actually discuss what we need the result of many forces to be,
so looking at Newton's second law, here's what I mean, on the left hand side of the second law,
we always write all of our physical forces. That's what the f means in this equation.
So you the catalog, you look at a force diagram, as we've done a few times.
You catalog all the different forces, and you write them down.
These are actual physical forces in a problem which can include gravity or the normal force or many others.
On the right hand side of the equation, what we have is the resulting acceleration.
In other words what happens because of all these forces that we've introduced.
So this new notion of this circular motion and the acceleration you need to go in circular motion,
is a derivation telling us what the left hand side if we've written it all out and compactified it and simplified it.
What it needs to end up being equal to, at the end of the day
if we want our object to go in a perfect circle.
In other words, if I have many forces acting on an object,
all sorts of things whether its gravity or friction or anything like that,
pulling an object in some given direction, what I need is that when I've added all those up together,
I end up with an acceleration which is equal to v squared over r if I want my object to go on a circle.
And for that reason, instead of the force I've just introduced here
mv squared over r being something you put on the left hand side in Newton's second law
as we've been doing with the other forces, this is actually a force you put on the right hand side
because again it's not a physical force.
It's actually just telling us what we need our physical forces to do for us if we want to go in perfect circular motion.