S.H.M Graphing Motion

Graphing position, velocity, and acceleration vs time

Position vs time: the graph of position verse time is a sine wave with a possible phase shift.  The phase shift is how much the ahead or behind the position is on the sine wave.

Consider this graph, if the "clock" is started at 0.05 (where the mass is at it's maximum stretch) seconds then there would be a phase shift of 90 degrees (or we could replace the sine function for a cosine). If the "clock" is started at 0.1  seconds (where mass moves down instead of up; left instead of right) then the phase shift would be 180 degrees (a negative sine function).

Velocity vs time: Consider the position verses time graph, at any point were the mass has reached the amplitude (maximum distance from from the equilibrium point) the speed of the mass at these point is zero. When the position is at zero then the speed is at a maximum (if you don't believe it, consider conservation of energy). This "shifts" the position graph by 90 degrees "creating" a cosine graph for velocity.  The other way of thinking about is velocity is the change in position with respect to time, the change in a sine wave with respect to time is a cosine graph.

Acceleration vs time:  The acceleration verse time graph is the easiest of the graphs to make.  The simple harmonic motion is based on a relationship between position and acceleration; x = - Ka. So the graph of position and acceleration should look alike, except for the negative sign.  In fact position and acceleration are the same shape just mirror copies of each other.

peak height:  It is important to note that the shapes of each graph is simular, that each graph has the same frequency, period and wavelength, but they don't have the same amplitude, for common simple harmonic motion, the height the peak  of the function (not to confused with the amplitude, amplitude refers to the height of the position graph alone, i'm talking about the height of the position, velocity and acceleration graphs) tend to get smaller and smaller starting with the position graph being the tallest and the acceleration being the shortest.

Graphing position, velocity, and acceleration with respect to each other

Graphing position to the other functions can be complicated and when tested on it, most student are unable to give the right answer.  First consider this, for simple harmonic motion position and acceleration are proportional. x = -k a this is a linear relationship so the graph is a line, the slope is negative so the line is heading down.

The graph is an example of a negative slope on a linear fit. This is what a position verse acceleration would be.

Position and acceleration verses velocity

The position and acceleration verse velocity graph look entirely different.  First off the straight line test fails when plotting position vs velocity or acceleration verse velocity.  Take the point were x and a are zero, there are two possible answers for the point (the speed maybe at maximum) the object could be moving down or up at the point.  That means the velocity can be a positive maximum or a negative maximum, two separate values. Looking a the points were the velocity is equal to zero, there are two possible answers, either at the top or at the bottom.  If you continue to plot data points graph that is developed is a ellisipe