Since we started out this term exploring differential equations lets start our discussion of periodic motion at that point. Here is a roughly 3 min video clip which constructs the differential equation for a mass on a spring (horizontal) using slightly different language than we do but it should make sense.
Differential equation for a horizontal spring system from Newton's 2nd Law:
Note where Walt's language and symbols are different than ours and any other confusions you have. He is quite cavalier about certain parts. You might think a little about the signs on his terms and the care we have tried to exercise about vectors -- see if you can convincingly show that his signs are correct and why -- think about his implicit choice of coordinate system. Check the units on each term.
Now take a look at this physical demonstration (2 min) of the generation of the actual plot of the motion of a spring.
Does it matter that the spring is now vertical instead of horizontal? See if you can produce the differential equation for a vertical spring following the model in the first clip. How is this differential equation the same or different from the horizontal spring?
Trial Solution: You will notice in this next clip (5 mn) that Walt uses the language of "trial solution" in the same way that we do.
Notice how he approaches this problem as a model for how we will do the same. I suspect there will be aspects of his discussion about the phase angle that will raise questions. If you don't ask I will assume that this all made sense.
In the trial solution there are still some constants that are undetermined (including the phase angle). In this next clip (3.5 min) Walt uses the initial conditions (a special case of boundary conditions) to determine their values. This is just like we did with the boundary conditions for our loan discussions.
You will find this process of using boundary conditions to determine constants in solutions to differential equations common in math, physics, engineering, and other disciplines. Make sure you understand.
Simple harmonic motion (often abbreviated SHO) has a natural connection to circular motion that Walt explores in this clip (3.5 min) which provides a connection which you will explore in more depth in your dynamics class in the spring. It's worth getting a jump on those ideas here. The second clip (1 min) is a direct illustration of this connection.
The mathematics that connects circular motion and linear oscillations
Demonstration of connection between circular motion and linear oscillations
