For a particle that's moving in a circle,
we found that when it's moving at a constant rate of d theta
dt--- and let's recall what we meant by theta of t--
and here's our particle, and we introduced our polar
coordinates r hat and theta hat, then we found that the velocity
was r d theta dt theta hat.
And so let's assume that this quantity is positive,
in which case the velocity is pointing in the positive theta
direction.
And that means that everywhere in the circle,
the velocity is tangential to the circle,
and the magnitude is a constant.
So for this case of uniform circular motion,
we calculated that the acceleration
was equal to minus r d theta dt quantity squared r hat, which
means that at every point, the acceleration vector is pointing
towards the center.
Now we can write that acceleration vector
as a component a of r-- r hat-- where this component is given
by r times d theta dt squared.
It it's always negative, because when you square this quantity,
it's always a positive quantity.
The minus sign, just to remember-- that
means that the acceleration is pointing inward.
Now how can we think about that?
Well, if we look at the velocity vector, what's happening here
is the velocity is not changing magnitude
but changing direction.
And if you compare two points-- and let's
just pick two arbitrary points.
So let's remove this acceleration for a moment
and consider two arbitrary points-- say, a time t1 and t2.
So our velocity vectors are tangent.
The length of these vectors are the same.
And if we move them tail to tail-- [? Vt2-- ?]
and take the difference, delta v, where
delta v is equal to v of t2 minus V of t1,
then we can get an understanding why the acceleration
is pointing inward, because recall
that acceleration by definition is
a limit as delta t goes to 0.
That means as this point approaches
that point of the change in velocity over time.
And so when we look at this limit
as we shrink down our time interval between t2 and t1,
then this vector will point towards the center
of the circle.
And that's why the direction of a
is in the minus r hat direction.
Again, let's just recall that this is the case
for we called uniform circular motion, which
is defined by the condition that d theta dt is a constant.