Calculus-Based Physics by Jeffrey W. Schnick - HTML preview

PLEASE NOTE: This is an HTML preview only and some elements such as links or page numbers may be incorrect.
Download the book in PDF, Kindle for a complete version.

Chapter 18 Circular Motion: Centripetal Acceleration

Starting with equation 18-1:

s = r θ

we take the derivative of both sides with respect to time:

ds

= r

dt

dt

and then rewrite the result as

.

.

s = r θ

just to get the reader used to the idea that we represent the time derivative of a variable, that is

the rate of change of that variable, by the writing the symbol for the variable with a dot over it.

Then we rewrite the result as

.

v = r θ (18-2)

to emphasize the fact that the rate of change of the position-on-the-circle is the speed of the

particle (the magnitude of the velocity of the particle). Finally, we define the variable w

.

(“omega”) to be the rate of change of the angle, meaning that w

θ

is

and w is

. It should be

dt

clear that w is the spin rate for the imaginary line from the center of the circle to the particle.

We call that spin rate the magnitude of the angular velocity of the line segment. (The expression

angular velocity, w, is more commonly used to characterize how fast and which way a rigid

.

.

body, rather than an imaginary line, is spinning.) Rewriting v = r θ with θ replaced by w

yields:

v = r w (18-3)

1

13