Chapter 13: S.H.M

Simple harmonic motion is the projection of uniform circular motion on a diameter.

x = A Cos(t + ) v = = - A Sin(t + ) dxdt

a = = = - 2 A Cos(t + )dvdt

= = 2 f 2T

d2xdt2

F = -mg Sin -mg

x = L

F - x mgL

F = -kx k = mgL

= max Cos(t + )

km = =

gL

Fdamping = -bv

ma = -kx - bv

kx = 0d2xdt2

m dxdt

b+ +

= b2m

`= km

b2

4m2-

km

b2

4m2-f = 1

2`2

=

Guess the Solution x = Ae-t Cos(`t)

A : Underdamped B : Critical DampingC : Overdamping

b2 << 4mk b2 = 4mkb2 > 4mk

t