Download Simple Harmonic Motion

Simple Harmonic Motion (SHM)
Ball in a Bowl:
SF
SF
FN
Fg FN
Fg

F 0
Stable Equilibrium
(restoring force, not constant force)
m
k
rest position
0
+x


 F  ma
 kx  mx
Hooke’s Law
(equation of motion)
“Guess” a solution:
x t   A sin  Bt  C 
Does it work?
 kA sin  Bt  C    mAB 2 sin  Bt  C 
Yes, if:
k  mB 2
x(t)
…A and C can take any value
A = amplitude
t
C = phase lag = 
Period = 2p/B
B = angular freq. = w
w2pf
So usually written…
xt   A sin wo t   
k
wo 
m
A and  from initial conditions.
Example: Mass (m) on spring (k) displaced x 1.2
= 1 cm at t=0:
y = -9.8427 + 16.35x R= 1
1
xt   A sin wo t   
x t   Aw o cosw o t   
x 0   A sin    1cm
p

A sin    1cm
2
x 0   Awo cos   0
p

2
Force [N]
0.8
0.6
0.4
0.2
0
-0.2
0.58
A  1cm
 k p
xt   1cm sin 
t 
 m 2
0.6
0.62
0.64
0.66
Displacement [m]
0.68
0.7
When a mechanical system is at a point of
stable equilibrium, a perturbation will result in
a restoring force that drives it back to
equilibrium. The resulting equation of motion
has the following form:
x  wo x  0
2
The solution is an oscillating trajectory at
frequency wo known as Simple Harmonic
Motion.