# 9.1 Uniform Circular Motion

When we analyzed how the

position vector changed, we know that the velocity

for circular motion is given by the

radius times the rate that the angle is changing. And it points tangential

to the circle. So let’s draw a few

characteristic arrows to show that. At this point, we’ll draw

these pictures with d theta dt positive. So the velocity

points like that. It points like this. It points like that. And these are all the velocity

vectors at different times. Notice that if we make–

consider the special case in which d theta dt is a

constant, in that instance, the magnitude of

the velocity, v, is given by r magnitude

of d theta dt. And that is also a constant. But the velocity vector

is changing direction. And we know by definition

that the acceleration is the derivative of velocity. And so what we see

here is where we have a vector that’s constant

in magnitude but changing direction. And we now want to

calculate the derivative in this special case. We refer to this case as

uniform circular motion. So this special case is often

called uniform circular motion. OK. How do we calculate the

derivative of the velocity? Well, recall that the

velocity vector, r d theta dt– those are all constants–

because it’s in the theta hat direction, once again,

will decompose theta hat into its Cartesian components. You see it has a minus i hat

component and a plus j hat component. The i hat component

is opposite the angle. So we have minus sine theta

of t i hat plus cosine theta of t j hat. So when I differentiate

the velocity in time, this piece is constant, so I’m

only again applying the chain rule to these two functions. So I have r, d theta dt. And I differentiate sine. I get cosine with a minus sign. So I have minus cosine theta. I’ll keep the function of

t, just so that you can see that– d theta dt i hat. Over here, the

derivative of cosine is minus sine d theta dt. That’s the chain rule– sign

of theta dt, d theta dt, j hat. And now I have this

common d theta dt term, and I can pull it out. And I’ll square it. Now whether do you think that

dt is positive or negative, the square is always

positive, so this quantity is always positive. And inside I have–

I’m also going to pull the minus sign out. And I have cosine theta of t i

hat plus sine theta of t j hat. Now what we have here is

the unit vector r hat t. r hat has a cosine adjacent in

the i hat direction and a sine component in the

H Hut direction. So our acceleration–

## One Reply to “9.1 Uniform Circular Motion”

never seen that invisible kinda black board ever.

well,it is super cool.