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Analysis: Adding forces Part 1

Now that you have internalised the concept of the Free Body Diagram, you can reveal forces.
This week you’ll be using the laws of physics to design a safe suspension system for a loud speaker. Two wires will share the load. They will be at an angle. How they share the load isn’t obvious. Here we’ll show how it works by analysing the experiment on adding forces, where two force transducers, also at angles, share the load. How’s your trigonometry? You really only need to know the definitions of tangent, sine, and cosine, but we’ll briefly mention the cosine rule and the sine rule along the way. Back to our experiment. Here are the two linked chains of rubber bands supporting the load. What is their combined effect? Or put another way, what share of the load does each one take?
This is where the trigonometry comes in. Forces are vectors, which means they have a direction as well as a magnitude or size. The bold red arrows in the previous video were vectors. For example, the magnitude of the weight was w, and its direction was straight down. This red arrow represents a general case of a vector. Its magnitude is given by its length. Its direction is given by the angle to a datum or reference line. In our experiment, the datum was the support beam, but it could have been any fixed line.
Now, two forces acting at a point add according to the parallelogram rule of vector addition. This means that we complete the parallelogram defined by the two forces like this. By add, we mean the resultant of adding the forces. The resultant is shown as a black arrow, has the same effect as the two separate forces, shown as red arrows, acting together. The resultant acts as a point where the separate forces are joined. If you know the magnitude and direction of the two components, you can find the magnitude and direction of the resultant.
For example, if you know the angle theta shown on the diagram and you know the length of each vector, A and B, then you have all the information you need to get the length of the resultant by the cosine rule. And you could find the other angles by chasing around the diagram with the help of the sine rule. Don’t worry. We won’t be using the cosine rule or the sine rule in these activities. You can work it in reverse. For example, if you know the magnitude and direction of the resultant, and you know the directions of the two components, then you have all you need to find the magnitudes of the two components using the sine rule.
Luckily, there is a simpler version. If the angle between the components is 90 degrees, the parallelogram becomes a rectangle, and everything is simpler. These are rectangular components. It’s simpler because the sine rule and the cosine rule aren’t needed anymore. You can use basic trigonometry to relate the various angles and magnitudes. Pythagoras Theorem comes in useful too. In two dimensions, the list of quantities is magnitude of vector 1, magnitude of vector 2, magnitude of the resultant, angle of the resultant. If you know any two of them, you can get the other two. For example, if you know the magnitudes of the two components, you can find the magnitude of the resultant and the angle of the resultant.
Or if you know the magnitude of the resultant and its angle to the x-axis, you can find the components Fx and Fy. You’ll be using this a lot. There is something special about a right angle. In the next video, you will compare the results of our experiments with theory to see how this all works out.

Now that you have internalised the concept of the Free Body Diagram, you can reveal forces.

What can you do with them?

We’ll find many ways to manipulate forces in the next few weeks. Here we’ll start by ‘adding’ them. By ‘add’ we mean finding out how we can combine forces and replace their combined effect by a single force.

There is important terminology here; two or more ‘components’ can be combined to give a single ‘resultant’. And it works the other way round too; a single ‘resultant’ can be decomposed into (usually two) components.

Forces are ‘vectors’ which means that there are particular rules for manipulating them. This video gives you the general procedures. The next video applies these procedures to the configuration of our experiment.

Talking points

  • How did you get on with the trigonometry?
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Through Engineers' Eyes: Engineering Mechanics by Experiment, Analysis and Design

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