What is the difference between a graph of linear motion and a graph of harmonic motion?

1 Answer
Mar 10, 2018

Linear motion can be represented by a displacement-time graph with an equation of #x=vt+x_0# where #x=text(displacement), v=text(velocity), t=text(time), x_0="initial displacement"#, this can be interpreted as #y=mx+c#.

Example - #x=3t+2#/#y=3x+2# (initial displacement is 2 units and every second displacement increases by 3):
graph{3x+2 [0, 6, 0, 17]}

With harmonic motion, an object oscillates around an equilibrium point, and can be represented as a displacement-time graph with either the equation #x=x_text(max)sin(omeg+s)# or #x=x_text(max)cos(omegat+s)#, where #x=text(displacement), x_text(max)=text(maximum displacement), omega=text(angular velocity), t=text(time), s=text(phase shift)#. This equation is similar to #y=acos(bx+c)# or #y=asin(bx+c)#.

Example - #x=3cos(10t-1)#/#y=3cos(10x-1)# (harmonic motion with a maximum displacement of 3 units, an angular velocity of #10text(rad s)^-1#, and a phase shift of 1 #text(radians)#:
graph{3cos(10x-1) [-10, 10, -3, 3]}