Aerodynamic Forces

Primer on Vectors

*NB – a review of this article by my partner, who studies Physics, indicated that I had used the wrong units to represent forces – they should be Newtons (N). It is a bit difficult to rectify at this point, so please read ‘kg’ as ‘N’. The concept is the same otherwise.

Vectors are a representation of the direction and magnitude (strength). These vectors can be re-arranged in any way around the diagram as long as the direction and magnitude is preserved (or compensated for). By doing that, these forces can be composed and decomposed into smaller components using Pythagoras’ Theorem (typically the horizontal and vertical components). Suppose I have a ball with two strings where I’m exerting a 3kg force downwards and a 4kg force sideways:

The vectors can be arranged such that they are ‘tip to tail’ so that we can resolve (i.e., combine) the vectors using Pythagoras’ Theorem. This example can be looked at a different way – suppose if I walked 4km to the east and 3km to the south, how far am I from the beginning using the shortest distance. This distance is the what we are after.

Using Pythagoras, we get:

For the mathematically pedant, we obviously reject the negative answer from the square root as we can’t really have negative distances (in this example anyway). Also, some variants of the formula use ‘h’ instead of ‘c’…these mean the same thing. As for the newly determined resolved vector, suppose if I pulled on the object in that direction with that much force the effect on the object would be exactly the same as the original example.

The calculations can also go the other way – we can break down a vector into many components. The direction and number of these components is completely up to you, but typically the horizontal and vertical axes are considered. There are two ways to approach the actual calculations. One is to draw exact and straight lines, as shown in the diagrams, and work in a manner that could be described as working backwards from the resolution method described above. The other is work the numbers out mathematically – but this requires us to know the angle between the original vector and one of the horizontal or vertical planes. It also requires a knowledge of trigonometry.

Resolved vector and 'component' lines

The first method involves mapping out the proposed component lines and then drawing a vector along that component line until it is in line with the original vector.

Vector components

The mathematical method is possible if we know the vector magnitude and the angle between itself and one of the new vector components. Suppose if the magnitude of the original vector was 6kg and the angle between it and the vertical was 30 degrees, we would initially work out the magnitude of the vertical vector component (we’ll define this as ‘x’):

Then, using Pythagoras, we’ll find out the horizontal component (we’ll define this as ‘y’):

Therefore, the horizontal component of this vector is 3 kg and the vertical component of the vector is kg.