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?