Chap 15ha Oscillations
Chap 15ha Oscillations
Chap 15ha Oscillations
Frictionless surface
Frictionless surface
Position - xmax = A
Velocity - vmax = ωA
y = A sin(ωt )
x = A cos(ωt )
x= A { }
sin t
2π
cos T
- ϕ
A plus sign indicated the phase
is shifted to the left
x = Asin ωt - π2
x = A ( sinωt cos π2 - sin π2 cosωt )
x = A ( sinωt (0) - (1)cosωt )
x = -Acosωt
2 1.50
sin(ωt)
π 1.00
sin(ωt-δ)
ωt - = 0
2 0.50
π
ωt = -3.00 -2.00 -1.00
0.00
0.00 1.00 2.00 3.00
2 -0.50
π 1 1 T
t= ; = -1.00
2ω ω 2π
-1.50
π T T
t= = Time
2 2π 4
∑F x
= - kx = ma x
k
Classic form for SHM a x ( t ) = - x ( t ) = - ω2 x ( t )
m
1 2 1 2
Also, E = K ( t ) + U ( t ) = mv ( t ) + kx ( t )
2 2
At equilibrium x = 0:
1 2 1 2 1 2
E = K + U = mv + kx = mv
2 2 2
2π 2π
ω= = = 12.6 rads/sec
T 0.50 s
∑ F = Fmax = ma max = m A ω = (
mA(2πf ) 2
= 4π)2
mAf 2 2
ω 1.57 rads/sec
f = = = 0.250 Hz
2π 2π
2π 2π
The period of the motion is T= = = 4.00 sec
ω 1.57 rads/sec
xmax = A = 8.00 cm
vmax = Aω = (8.00 cm )(1.57 rads/sec ) = 12.6 cm/sec
amax = Aω 2 = (8.00 cm )(1.57 rads/sec ) = 19.7 cm/sec2
2
L
T = 2π
g
Solving for L: L= 2
= 2
= 0.25 m
4π 4π
θ
Lcosθ
L L
y = L(1 − cos θ )
y=0
I
T = 2π
MgD
I = Irod + Idisk
M = mrod + Mdisk
2
' b k m
ω = ω0 1- ω0 = τ= ; bc = 2mω0
2mω0 m b
2
EαA
x = Acos( ωt - δ )
F0 bω
A= tanδ =
m 2 (ω02 - ω 2 )2 + b 2 ω 2 m(ω02 - ω 2 )
Nov. 7, 1940
Nov. 7, 1940
http://www.youtube.com/watch?v=3mclp9QmCGs
x t dx dt
- = Constant - =0
λ T λ T
dx λ
= =λf =v Wave velocity
dt T
MFMcGraw-PHY 2425 Chap 15Ha-Oscillations-Revised 10/13/2012 61