Rotation; Angle and Revolutions?

Question

The tires on a new compact car have a diameter of 2.0 ft and are warranted for 58,000 miles.

Determine the angle (in radians) through which one of these tires will rotate during the warranty period.
?rad

How many revolutions of the tire are equivalent to your answer in (a)?
?rev

Answer 1

A tire will go one circumference C = 2pi R for each complete rotation. Thus, the number of rotations N = S/C = 58,000*5280/2pi; where S is the warranted distance = 58,000 miles*5280 ft/mi in feet and C = 2pi R = 2pi because R = D/2 = 2/2 = 1 ft. N is the number of rotations equivalent to THETA below.

Each rotation has angular length of 2pi; so the total number of angles the tire rotates through in 58,000 miles is THETA = 2pi N = 2pi*S/2pi R = S/R radians. THETA is the angle you are looking for.

BTW, your use of revolution is a misnomer. It connotes something going around an axis that is not its own axis. Like the Moon revolves around an axis that is at the center of mass for the Earth.

Rotation connots something going around its own axis. Like the Earth rotates around its axis one rotation about every 24 hours (a day). Thus you want to know how many rotations the tire made, not revolutions, because the tire is rotating around its own axis.