For example, to calculate the displacement of the particle in an harmonic oscillator we do:

$$x(t) = x_{\max} \cos(ωt+φ)$$

What do we find out taking the cosine of (ωt+φ)?

Example Graph:

Simple Harmonic Motion Graph


If you are learning physics at the algebra level, you have but no choice to take this for granted. The same applies across much of algebra-based physics. There's no apparent rhyme or reason for why things work the way they do. This makes algebra-based physics rather difficult because everything appears to be ad hoc.

The answer is simple if you are learning physics at the introductory calculus level. Sinusoids are the solution to the equation of motion. Calculus makes much of the apparently random stuff one has to learn in algebra-based physics suddenly make sense. Apparently distinct concepts are suddenly united, and solutions such as this that apparently were random plucked out of the clear blue sky suddenly have very solid explanations. That sinusoidal behavior was not plucked out of the clear blue sky. It was derived from the equations of motion.

This trend continues as one moves to ever more advanced physics. In a way, the physics gets easier with each step. On the other hand, the mathematics can become quite overwhelming.

  • $\begingroup$ Nice. I can actually prove to my algebra/trig based class that sinusoids are the correct solution for harmonic oscillators. Admittedly it takes more than thirty minutes and a long detour through an intermediate step that will confuse about a third of them to a level that requires a stop by the tutoring center or my office to get untangled, but I can do it. Or I can take 2 minutes to exhibits a couple of differentiations for the calculus class. Alas, some students will never finish (or even attempt) calculus and some that do will take the "easiest" physics class their major allows. $\endgroup$ – dmckee --- ex-moderator kitten Sep 20 '14 at 20:40
  • $\begingroup$ It kinda disturbed me that I couldn't fully understand what's going on. I'm glad it doesn't seem to be my fault. I'm going to keep up with the class and not learn any calculus until it's over, ~9 months. Hopefully I'll be able to have a way better understanding of physics then. Thank you for your answer. $\endgroup$ – rck Sep 20 '14 at 20:48
  • $\begingroup$ @rck - Learning physics without calculus is the hardest way to learn physics, and even then you're learning but a tiny fraction of physics. Concepts don't connect, everything's ad hoc. It might seem contradictory, but introductory physics with calculus is a far easier subject. The needed math is actually fairly simple. When I was 16 (back in the stone age; we had to fight dinosaurs to get to school), that basic calculus was crammed down my throat in the first two weeks of an eight week NSF summer program. That gave us the tools to proceed onward with digital electronics and nuclear physics. $\endgroup$ – David Hammen Sep 20 '14 at 21:23

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