S.H.M Position, Velocity, and Acceleration

x = A sin(ωt + φ)

v = Aω cos(ωt + φ)

a = -Aω^2 sin(ωt + φ)

x = Position

(from equilibrium)

v = Velocity

a = Acceleration

ω = Natural frequency

φ = Phase swift

Simple Harmonic Motion

Some of the important things about you need to know about simple harmonic motion.

1. Restorative force. In order for a object to undergo simple harmonic motion, there must be a restorative force.  A restorative force that pushes the object to an equilibrium point. The spring force is the example of a restorative force, a mass hooked to a spring is pulled down, the spring exerts a force on a mass upwards.  Therefore when you let go of the spring accelerates the mass up towards the equilibrium, the mass then moves beyond the equilibrium because of its momentum, then spring changes the direction of the force to downward.  The mass is always being pulled towards the equilibrium, this creates the motion that behaves like

x = -Ka

where k is some contant

restorative forces are spring, electrostatic and gravitational forces.

2. The relationship between position, velocity, and acceleration.  Because of the restorative creates the x = -ka relationship, when an object is at a peak on the position graph then the acceleration of the object is in a through and vise verse. So a graph of position verses acceleration is a linear graph with a negative slope (the magnitude of the slope is the natural frequency squared).  The graphs of velocity verses position or acceleration is a circular plot, why is this so? plot it out for yourself and figure out why.

Position

Velocity

Acceleration