The Period and Sinusoidal Nature of SHM

General comments concerning the importance of the following thought process - across all fields of science

 

"Mathematical and physical analogies that can be shown to be relevant to (given same results as) the current problem"

Sine wave for analyzing AC current, electrons as particles around the nucleus, motion on a table from above as discussed below, etc., etc.

The approach frequently taken by Newton, Feynman, Nash, and many others

Ø The circular motion of the red object as seen from the side is illustrated on the right.      Ø The same motion as seen from the side is shown on the far right      Ø The mathematical similarity of this situation to SHM is useful, although nothing is actually          rotating in SHM   Ø The two triangles containing theta are similar (see explanation below), therefore   Ø Sin q = v/vmax = (A2-x2)1/2/A, and solving for v   Ø  v =   vmax(1 - x2/A2)1/2   Ø Note that this is the same expression previously derived     Ø The point is that the projection onto the x axis of an object revolving in a circle has the same          motion as a mass at the end of a spring - we can use the analogy with accuracy

 

Geometry to Show that the Triangles are Similar (refer to the Prelimiaries section of this chapter if necessary)

 

Using the above similarity to derive the period of SHM

The maximum velocity is equal to the circumference of the circle divided by the period

V_max = 2pA /T = 2 pAf        ð T = 2p A/v_max

T_s = 2p(m/k)^1/2           THIS FORMULA FOR THE PERIOD OF A SPRING IS IN IN THE HANDOUT