## CIRCULAR MOTION

Suppose that an object moves around a circular path at a constant rate,
and returns to its starting point in a
time T. T is called the PERIOD of the circular motion. The particle has
travelled a distance s = 2(pi)r.
The particle's speed v is the distance s travelled per unit time.
s r

v = - = 2(pi)x -

T T

So that for a given T, v increases as r increases.
For the Earth's rotation, T = 1 day. What is T for the motion of
the Earth around the Sun?

#### EXAMPLE

A car is moving at 50 mph. Its wheels have a diameter of 2 feet. How
fast do the wheels rotate? (Find
the period of the motion.)
To obtain the speed, imagine covering the tire with wet paint.
As the car moves, it lays down a paint line on the road. When the
wheel turns once, the length of the paint line laid down equals
the circumference of the wheel, and the time taken is the period
T. Thus:

Speed of car = speed of a point on the rim of the wheel

2(pi)r (pi)d

v = ------= -----

T T

So:

miles miles 5280 ft 1 hr 1 min pi × 2 ft

50 ----- = 50 -----x ------- x ------ x ------ = --------

hr hr 1 mile 60 min 60 sec T

We want to find T, so we rearrange this equation to get T on the left
hand side.
pi x 2 x 60 x 60

T = ---------------- secs

50 x 5280

pi x 3

= ------ = about 1/12 s

5 x 22

Equivalently, the wheels rotate twelve times per second.

If you were driving a truck with 3 ft diameter wheels, would they
rotate faster or slower than this car's
wheels when going at 50 mph?