Simple Harmonic Motion (SHM) formulas

Angular Velocity and SHM:

w = rads/sec =  = 2pf = v_maxA_max  (where v_max is the maximum velocity of a pendulum/spring-mass and A_max is the maximum amplitude of that system) = q/t  where q is in radians.

w² = a_c/r where r = A_max.

----------------------------------------------------------------------------------------------------

How to determine the instantaneous position of a pendulum/spring-mass.  This position is often called y because the spring-mass usually bounces up and down along the y axis.

Y_o = A sin (wt + d)  where d can be in radians or degrees.  Even though w is in radians, since we usually are looking for Y_o, that means t = 0 seconds so it doesn’t matter if you define the phase constant (d) in radians or degrees.  But, whatever you decide, you need to set your calculator accordingly.

If the pendulum starts at +Amax, then d = p/2 or 90^o

If the pendulum starts at -Amax, then d = 3p/2 or 270^o

If the pendulum starts at 0, then d = 0 or 0^o if going up and p or 180^o going down

      Notice that w can be substituted for any of the equations for w above.  For example, if you are given a graph of SHM, you can determine the period of the oscillation and thus the frequency.  Notice that w = 2pf.  WOW! 

     What if you knew the spring constant and the mass?  w =

----------------------------------------------------------------------------------------------------------

How to determine linear (tangential) velocity of SHM

V_instant = wAcoswt  (if Y_o = 0)

V_max = wA = (A) = 2pfA