# 9.2 Uniform Circular Motion: Direction of the Acceleration

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.