Equations for simple harmonic motion

In this video, we'll go through how to model an object's position as it undergoes simple harmonic motion using sine or cosine functions.

Simple harmonic motion is defined as periodic motion caused by a restoring force which is proportional to the objects displacement from equilibrium (more on that in this video:____________________). This definition may seem arbitrarily complicated, but there are good reasons for separating this type of motion. Important for us in this video is that the position, velocity, and acceleration of any object undergoing simple harmonic motion can be described using the sine and cosine functions. Other types of repeated motions will not (necessarily) be possible to model with simple equations like this.

A general form frequently used here for the sine function is:

x(t) = A*sin (wt)

Where
x = the position at some moment in time,
A = the amplitude of the motion,
w (omega) = the angular frequency of the motion, and
t = the time

*note that there is sometimes added a "plus phi" inside the parenthesis. This shifts the whole graph left or right. Since we can decide when to start our time measurements, we'll just choose to have zero time correspond with a moment when the oscillator is passing through equilibrium and moving in the positive direction. Everything in this video is done using the sine function, but works equally well with cosine. The only difference is the starting position. If your oscillator is starting at its maximum displacement at time zero (and you don't have the option to change the meaning of time zero), cosine would be a logical choice.

In this video, I'll show you how to determine the values for the coefficients to tailor this equation to the oscillator you are dealing with. Once you have an equation like this, you can do all sorts of nifty tricks with it like taking the derivative with respect to time to find a velocity function, and once more to find the acceleration function.

You could also substitute these equations into some of the energy equations. For example, replace the speed term in the kinetic energy equation to get an equation for kinetic energy as a function of time.

Видео Equations for simple harmonic motion канала Bennett Science