## My friend and I are having a debate on wind turbines or anything that travels in a circular motion. One of us says that the very tip of the turbine blade is traveling faster than the inner part that is close to the axis. We both agree that it has the same rpm, but what part travels at a faster mph?

When a rigid object like a turbine blade rotates, there is a simple relationship between the “rotational” velocity w and the “linear” velocity v: it’s just v = r*w, where r is the distance from the rotation axis. In other words, the greater the distance from the rotation axis r, the greater the linear velocity v.

A point exactly on the rotation axis will not be moving at all, since at that point r = 0 and therefore v = 0. The part of the turbine farthest from the rotation axis will be traveling the fastest.

Wind turbines found near Halifax, UK.
Image Credit: K Ali via flickr. Rights Information.

You may have experienced this effect on a rotating merry-go-round: if you’re standing at the very center, you’re not moving at all. You’ll be moving fastest if you’re out at the edge of the merry-go-round.

As another example, you may have seen ice skaters join hands to form a straight line, but they make the line move in a circle. The skater at the end of the line that forms the center of the circle doesn’t have to move at all, but the skater at the other end of the line must move very fast in order to maintain the straight line.

If you want to find the speed, v, of the turbine blade in miles per hour (mph) at a distance r (in feet) from the rotation axis and a rotation speed, w, in revolutions per minute (rpm), there’s a simple formula: v = π/44 * r * w = 0.07140 * r * w. (The factor of π/44 = 0.07140 is there to get the units of rpm, mph, and feet to work together correctly.)

For example, if the rotational speed of the turbine blade is w = 50 rpm and the turbine blade has a length (from the rotation axis to the end) of r = 20 feet, then the speed of the turbine blade varies from 0 at the rotation axis to v = 0.07140 * (20 feet) * (50 rpm) = 71.4 mph at the very end of the blade.

Halfway between the rotation axis and the end of the blade, we would have r = 10 feet, and so the speed, v, of that part of the blade would be just 35.7 mph — half the speed found at the end of the blade.