circular motion

Question

Suppose a piece of dust is stuck to a CD such that it does not slip even when the CD is spinning.

a) if the constant spin rate of the CD is 460rpm, then what is the angular distance in radians, is traveled by the piece of dust in 3.3 minutes?

b) if the dust is 4.7 cm away from the center, then what is the total distance in meters, by which it traveled during the time interval of part a?

Answer 1

This is an exercise in unit conversion:

You need to know that 1 revolution is 2π radians, and

1 cm is 0.01 m.

Let

ω = 460 rpm be angular velocity,

t = 3.3 minutes be time interval,

r = 4.7 cm = 0.047 m be the radius of motion.

a)

By definition of angular velocity, the angle traversed during time t is

α = ωt = 460×3.3 = 1518 revolutions.

One revolution is 2π radians.

Therefore

α = 1518×2π = 9637.9 radians

b)

For an angle α expressed in radians, the distance travelled is

s = rα = 0.047×1518×2π = 448.28 m