Asked by: Karl from Liverpool
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
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.
David G. Simpson, PhD
NASA Goddard Space Flight Center