Oscillations
Different type of motion:
•Uniform motion
•1D motion with constant acceleration
•Projectile motion
•Circular motion
•Oscillations
Different type of oscillations:
•Periodic oscillations
A motion is called periodic when the system comes back
to the same physical conditions every time interval T.
•Harmonic oscillations (the simplest, the most
important type of periodic oscillations)
•Chaotic oscillations
Harmonic oscillations
Relation between circular motion and simple harmonic oscillations
y
R=A

2

 2f
T
  t  
x  A cos 
x
x max  A
x  A cost   
v  A sin t   
v max  A
a   2 A cost   
a max   2 A
a   2 x
Differential equation:
d 2x
2



x
2
dt
d 2x
2


x0
2
dt
t


T
x
x  xmax
v0
a  amax
A=xmax
x   xmax
v0
a  amax
x0
v  vmax
a0
t
x0
v  vmax
a0
v
t
a
t
Hook’s law and simple harmonic motion
Hook’s law:
Newton’s second law:
Simple harmonic motion:
F  kx
ma  kx
F  ma
a   2 x
k
a x
m
k
 
m
2
T
2

 2
m
k
Example: Two identical masses hang from two identical springs.
In case 1, the mass is pulled down 2 cm and released.
In case 2, the mass is pulled down 4 cm and released.
How do the periods of their motions compare?
A. T1 < T2
B. T1 = T2
C. T1 > T2
Period is independent of amplitude!
This is in fact general to all SHM (not only for springs)!
Example: A 3.00-kg block is attached to the end of a spring
with k = 1000 N/m. The spring is compressed 25.0 cm and
released at t = 0. What is the velocity of the mass 10.0 s later?
m  3.00kg
k  1000n / m
x0   0.25m
k
 18.26s 1
m

v0   0
v10.0 s   ?
xt   A cost   
vt   A sin t   

cos   1
 0.25m  A cos 
0  A sin  


sin    0
A  0.25m

v10s    1000 / 3s 1  0.25msin 1000 / 3s 1  10s   
v10s   1.62m / s
x=0
x = -25 cm

 
Energy in the simple harmonic motion
F   kx
U  12 kx 2
k  m 2
x  A cost   
v  A sin t   
U
E
–A
K
E  12 mv 2  12 kx 2
A
x
K
1
1
K (t )  mv 2  mA22 sin2 (t )
1
m 22A 2 sin22 t 
2



2 1 2
1 
U  12 UkA
(t ) cos
kx 
t 
kA cos (t )
2
2
2
2
E  12 kA2  12 m 2 A 2
2
U
t
t
E
t
Total mechanical energy is constant through
oscillation: conservation of energy!
Example: A mass m = 5 kg oscillates at the end of a spring of constant
k = 2000 N/m. At t = 0, its acceleration is maximum. How long will it
take before the potential energy reaches its next maximum?
At t = 0: Max |a| → Max |x| → Max U
U
To get to the next peak in
U, it takes half a period
t
T
1 2
m


 0.16 s
2 2 
k
x
T
t
Oscillations about an equilibrium position
When a 1D system is released near a stable equilibrium point, the
motion is periodic: oscillations between two turn-around points x1
and x2.
If not too far from the minimum,
the curve is approximately a
parabola. (Taylor’s expansion of
U(x) up to the quadratic term).
U(x)
E
x1
SE
x2
Forbidden region Oscillations Forbidden region
x