The relationship between rate of turn, TAS and radius

[mchristlieb]

There seem to be some formula and the most common is that:

rate of turn is proportional to TAS / radius.

But the exact relationship seems a little more fraught.

For a circle of circ 'd' (NM) and an aircraft of speed 'v' (Kts) the time taken to complete the circle t = d/v

So the frequency of turning 1/t (per hour) = v (TAS) / d

Multiplying both sides by 360 we get the rate of turn (degrees per hour) = 360 v / d

But d = 2 pi r (radius in NM)

So rate of turn (deg /h) = 360 v / 2 pi r

We don't need to worry about NM because they cancel top and bottom but we can convert the units of time to min

rate of turn (deg / min) = 360 v / 2 x 60 pi r = 3 v / pi r

And again convert units of time to s

rate of turn (deg / s) = 3 v / 60 pi r = v / 20 pi r

From the airline interview example, a std rate turn at 180 kts has a radius of 1 NM

There fore the rate of turn = 180 / 20 pi = 9/pi ~ 3 deg per s.

This all seems to work - but I can't find this formula anywhere else except my derivation.

[John CTKI]

If you take a radius and place it on the preriphery of the circle it will form an arc. That arc has a name: it's called a radian and there are 2 pi radians in 1 revolution. See diagram. If we divide one revolution of 360 degrees by 2 pi we get 57.296 degrees. This means a radian has a value of 57.296 degrees. Engineers find the radian a more useful way of expressing angles than degrees because there is a direct link between angular velocity (radians / min), the radius and the peripheral speed.

Radians / min is a ratio between the peripheral speed and the radius of the gear wheel for instance. In other words, radians./ minute = peripheral speed: radius or radians / min = peripheral speed / radius.

So for an aeroplane flying in a circle the rate of turn (radians/min) becomes TAS / radius of turn.

Take the example of an aeroplane at 180 kts doing a rate 1 turn and you want to find the radius or turn.

180 kts = 3 miles a minute. therefore in 1 minute the aircraft will have flown half a revolution or pi radians.

Put the numbers in the formula, pi (rad/min) = 3 (NM/min) / radius (NM)

Transpose the formula to find radius. Radius = 3 (nm/min) / pi

Radius = 0.9549 Nm.

[mchristlieb]

Thank you, John.