Velocity, Acceleration, and Net Force in SHM

     Because we have already covered (see previous blog posts) the causes of SHM and aspects of graphs related to SHM, we can now put the two together. 

    To set the foundation, we can use the mass hanging on a spring as an example. The position of the mass relative to the position of equilibrium can be modeled by a sinusoidal function, as proved in previous posts, and included below: 

Position Time Graph

    To recap, the amplitude is the average of the maximum and the minimum, the period is the time from one peak to the next, and the x-intercepts are points of equilibrium. Because the x values represent time, we can find velocity using d/t, or the slope of this graph. Because the slope is changing, we can measure average values for the slope over long intervals...

Actually, we can't.

    Using average values over long intervals means a resultant piecewise function. To get around this, we can decrease the time interval over which we take the average slope, until we get one instant in time, for which the slope would be directly tangent to the position time graph. The result is given below:

    

Velocity vs Time Graph

    Some important things to note: the phases of either graph are not lined up. Instead, where the spring is at its maximum or minimum, we have x intercepts (signifying that velocity is 0, the system is instantaneously at rest), and that velocity is at its maximum/minimum when the mass moves through the point of equilibrium, which will be further corroborated when we explore energy in these contexts. Using the same technique as before, we can find the graph of acceleration vs time: 

Acceleration vs Time Graph

    Some important things to note HERE: the acceleration and velocity do not have phases that are lined up. When the velocity is at its maximum, acceleration is instantaneously 0, and when the velocity is 0, the acceleration is inciting the most change in velocity, which follows logically, but is modeled ideally in these sinusoidal graphs. 

    Now that we have noted what we needed to regarding the relationships between the graphs and what that looks like in simple harmonic motion, we can begin to understand what they look like as equations. Assuming the position of the block starts at the maximum, it can be modeled by this equation

x(t) = Acos(2pi/T * t)

    To put it to words, position as a function of time is dependent on the cosine of the time for a given period, and varies with the amplitude. 2pi/T makes it so that the cosine function has a period of T as opposed to its default period of 2pi, which is just a horizontal dilation. From here, we know that velocity is an upside down sin function where the period is the same, written:

v(t) = (-vmax)sin(2pi*t/T)

    Where here, the amplitude is equal to the maximum velocity, and the period is the same, because it is the derivative of position. Now, we know the relationship between accleration and velocity is similar, and thus a negative cosine graph (inverted sin with a shift, and the same period), so:

a(t) = (-amax)cos(2pi*t/T)

    But also, we can use Newton's second law and Hooke's law to show the behavior or net force in SHM. If

F = ma

F = -kx

ma = -kx

a = -kx/m

-kx/m = (-amax)cos(2pi*t/T)

    These mathematical relations between functions for displacement, velocity, acceleration, and net force, will be incredibly useful in the near future as we explore further into SHM.