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