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When I was working on my dynamics problems in college, I always thought that there should be an easier way. We were always solving for problems at an instant. And then there's those dreaded "instant center of rotation" problems. I had an idea. One that I couldn't see we were not taking advantage of. That is to add the graphs of the circular functions (sine, cosine, tangent) of each rotational member and get the shape, distance, angle, and position of the force diagram. For example if one of the members is 90 degrees while the other is 30 degrees and they both rotate around a central axis we would just add their circular functions and get there angle at any given position. The overall geometry of the two's path may be elliptical. This could also work for non circular orbits as long as the circular function could be defined.

The reason I don't think this has been done, if not already done by computers, is the fact that the shape and angle are useful, but different lengths change the amount of force and velocity.

So we would add together the graph of the sine of all the members, out of phase by the degrees separating them, and the total combined graph is what we are searching for. This method solves the problem of calculus graphically. Another use is finding the triangles of our members knowing only their length and the angle between them.

This short essay is not to prove anything. It is just a thought of how we can more easily describe dynamics problems. Why don't we take advantage of this in dynamics class?