FDC
Wood Burners and Banked Turns
on November 27, 2019

This is, I have to admit, a blog only tangentially related to aviation, but I came across a mathematical jollity which I just had to share.

# Woodburner

My brother-in-law lives in a cottage in Devon, and wanted to fit a second stove to keep his house warmer. The chimney goes up from the stove, through the wall and then up the outside of the house. In the spirit of de-identification, and because I don’t have a photo of his house, here is a typical installation.

The hole in the wall is circular in this case, but we wanted to feed the pipe through the wall at an angle (in fact, at 45deg, but let’s not get ahead of ourselves).

Now, as you will remember from your work on conic sections at school, the intersection of a plane and a cone (of which a cylinder is a special case) at an angle is an ellipse. The length of the minor axis of the ellipse is clearly the diameter of the pipe, but the length of the major axis is greater by an amount which depends on the angle of the pipe.

If the pipe goes straight through the wall (as illustrated above) we can say the angle between the pipe and the perpendicular is 0deg, and the hole in the wall is a circle. As the angle, let’s call it ‘theta’, changes, so the ratio between the minor and major axes changes as:

At 45deg, this makes the ratio 1.414 (square root of 2). Even more excitingly, for this special case the linear eccentricity is the same as the semi-minor axis length. That means that you can mark the ellipse for a 7in pipe by sticking nails into the wall 7in apart, then draw the ellipse with a piece of string and a pencil.

# Banked Turns

Well, the title of this blog gave it away, didn’t it? The same formula which computes how to put a pipe through a wall at an angle determines the normal acceleration to maintain level flight in a turn.

For an aircraft in a steady balanced turn, the component of the lift which keeps the aircraft in the air must equal the weight, giving:

At zero bank angle, the lift equals the weight, but as the angle increases in either direction the lift has to increase to maintain level flight. What is not obvious from this mild-mannered equation is how quickly it increases at higher angles. We often come across quadratic functions, where something is the square of something else. I have drawn a quadratic curve that passes through 1 at zero “bank angle” and 1.414 at 45 to compare with this 1/cos function.

See how the blue quadratic curve behaves gently as angles increase towards 90deg, but from 60deg upwards the 1/cos function shown in orange rapidly increases. This is why fighter pilots have to wear “g” suits to stop the blood draining into their legs and leading them to pass out.

# Conclusion

At high angles of bank, the force needed to keep level flight increases rapidly. Curiously, if you put a pipe through a wall at a steep angle, the length of the hole in the wall increases just as rapidly.