Welcome back.
In the last couple of videos, actually the very first video
on centripetal acceleration, I told you that the necessary
centripetal acceleration, and we're just talking about the
magnitude, and actually the c tells you the direction, it's
centripetal so its inward acceleration, it equals the
velocity squared over the radius.
I told you that the real rigorous proof has to be done
with calculus.
But I looked on Wikipedia and there's actually a pretty neat
proof, although when you read Wikipedia, it's not so obvious
on what they're saying.
So I thought I would do a video on it, because this is
cool and you don't need calculus to understand it.
So let's just do something.
Let's just plot the distance vector, the velocity vectors
and the acceleration vectors as something
goes around a circle.
Let me draw two circles.
So let's say this is my first circle.
Let me draw another circle, another color just for fun.
OK.
This is my other circle.
This is the center of the circle.
And this is the center of the circle.
And so what's the position vector in any point in time?
Well, the position vector, you could just
draw it as a radius.
So the position vector at any given time looks
something like this.
So let's say initially this is the position vector.
That's the initial position vector.
Its magnitude is the radius of the circle.
And this direction is right here in the positive
x-direction.
And at that point, what is the velocity vector?
Well, the velocity-- let's assume we're going
counterclockwise.
I don't know why I assumed that.
It could go the other way.
Let's say that this is the velocity vector at that point.
The velocity vector is going to look something like that.
That is the velocity vector.
That's v.
It's going tangent to the circle.
Let's plot the velocity vector as a function of
time on this circle.
So if that's the velocity vector of that time, I can
draw the velocity vector here.
This the the same vector.
Remember, I'm just saying, at a particular time, what does
the distance vector look like and what does the velocity
vector look like?
So at that time, the velocity vector looks like this.
I'm trying to make it the same size to show you
it's the same vector.
And I'm doing it in the same color to show you it's the
same vector.
This is the exact same vector.
This is the actual circle.
Like if you were to draw it, this is the path of the dot or
whatever is moving around the circle.
And this, you can just view it as I'm plotting the velocity
over time, the velocity vector over time.
So let's say a few seconds later, or a few moments later,
what does the radius vector look like?
Well, then the radius vector looks like this.
I'm trying to write as neatly as possible.
And what does the velocity vector look like?
Go back to the purple.
The velocity vector once again is tangent to the circle.
And it'll look something like that.
It's going to have the same magnitude,
just different direction.
Actually, let me do it in a different color, just to show
you what I'm doing.
So let me do it in brown.
So the velocity vector is going to look
something like this.
It's brown, supposed to be a brown.
That's the velocity vector.
So after a few seconds, or a few moments, where is the
velocity vector here?
Well, it's going to look like this.
Remember, this vector I'm just plotting
here after a few seconds.
Let me use that line tool.
So it's the same magnitude and now the velocity is just at a
different angle.
It should be the exact same angle as what I just drew in
the other circle.
So that's this velocity vector.
So when we start here, the velocity is going to change at
the circle.
After a few moments, the object's rotating around the
circle, so now the velocity is the same magnitude.
It's just switched directions.
And so what's happening here?
When we plot the velocity vector over time, it has the
same magnitude, so it'll draw out a circle, but its
direction changes.
Just as in this case, what takes us from this point to
this point?
This was the velocity vector and the velocity vector's
always changing.
But, in general, this is the change in position.
And what causes the change in position?
Well, the velocity, or at least the speed, because the
vector's always changing.
So in this case, what's changing the velocity?
Well, just like velocity changes radius, acceleration
changes velocity.
Let me draw an acceleration vector, and acceleration is
going to be in this direction.
Because if the velocity is changing from here to here,
the acceleration is going along the direction of the
change in velocity.
So the acceleration vector might look
something like that.
It's going to be tangent to this velocity path.
And that's interesting, too, because if this is the
acceleration vector when the velocity is here, so when the
velocity is here, the acceleration vector's going
directly to the left, so that means that acceleration
vector's going directly to left, and so this is also the
acceleration vector, which coincides with what we learned
about centripetal acceleration.
The acceleration has to be going inwards.
And we see that when we actually plot
the velocity vector.
And if we plot acceleration vector here, once again it's
going to be going tangent to this plot of
the velocity vector.
Let me do it in a different color.
I've already used that color.
I'll do it in yellow.
So then the acceleration vector is going to look
something like this.
Remember, acceleration is nothing
but change in velocity.
So at this point, when this is the velocity vector, the
acceleration vector will just be this.
Once again, it's just the opposite direction of the
position vector.
So why am I doing all this?
Well, I'm doing all of this to set up the
analogy to show you.
As this object completes one entire path, what's happening
on this circle?
Well, this circle, the velocity is completing one
entire path, right?
However long it takes to go around this circle is the same
amount of time it takes to go from this velocity back to
this velocity.
The magnitude is the same the whole time, but the direction
is changing.
So if this takes 10 seconds to go around the circle in real
position, then it takes 10 seconds for the acceleration
to change the direction of this velocity enough that it
goes back to the original velocity direction.
So why am I doing all of that?
Well, how long does it take for the object to do one
rotation around this path?
Well, it's just the distance divided by the velocity.
So the time to do one revolution around this path is
the distance-- well, that's just the
circumference of the circle.
Well, that's 2 pi r divided by your speed, which is v.
And how much time does it take to go around this path?
Well, we know it's going be the same time, and now we're
going to say the time's going to be the change
in velocity, right?
In this circle, this is the change in distance, but it's
the change in velocity.
So here we had this much change in velocity.
I know this might be a little non-intuitive.
Then we have a little bit more change in velocity, a little
bit more change in velocity, a little bit
more change in velocity.
So the total change in velocity is just going to be
the circumference of this circle.
And what's the circumference of this circle?
The radius of the circle is the magnitude of the velocity.
So it's 2 pi times the velocity of the circle.
And so, what is the amount of time it takes a do one
revolution?
Well, it's the total change in velocity.
And that's 2 pi times the magnitude of the velocity
divided by acceleration.
And if this doesn't make complete sense, just remember,
acceleration is change in velocity over change in time.
I'm glossing over it a little bit, because you might say,
oh, well, the net change in velocity is zero.
But we're actually more concerned about what's the
total change in the velocity?
So it changes a lot, then it goes back to its original, so
that the net is still zero.
But this should hopefully give you a little intuition.
So think about what I said.
But the bottom line is we know that this expression and this
expression have to be equal, because in the same amount of
time it takes the object to do one complete revolution on
this circle, its velocity, direction, has also done one
complete revolution.
So we can set these two equal to each other.
So we could say 2 pi r over v is equal to 2 pi v over a.
Let me switch colors.
We can cross the 2 pi out on both sides.
And then we could multiply both sides times v.
You get r is equal to v squared over a.
Multiply both sides times a, I'm doing this little bit
circuitously, and you get ar is equal to v squared.
Divide both sides by r, and you get a is equal to v
squared over r.
So hopefully, that gives you a little bit of intuition about
why this works.
And I got this from Wikipedia, so I want to
give them proper credit.
And hopefully, my explanation helps clarify what the people
on Wikipedia are talking about a little bit.
Anyway, I'll also do the calculus proof, because that's
more just straight up math.
And I'll see you in the next video.