Oscillations
I. Periodic motion
A. Also called Simple Harmonic Motion (SHM), oscillations, and vibration
B. Circular motion, bouncing motion, yo-yos, springs, and pendulums are
examples of periodic motion
II. Simple Harmonic Motion (SHM)
A. The period of oscillation is the time it
takes to complete one oscillation
2 1
1. T 


f
2. Defines the length of 1 complete
cycle
B. The frequency of oscillation is the
number of oscillations per unit time
1
1. f 
T
2.   2 f  angular velocity = angular frequency
3. units of f are in Hertz (Hz) where 1 Hz = 1 sec-1
C. The amplitude of oscillation is the maximum displacement that occurs in any
cycle
D. The phase angle or phase constant of an oscillation is the relationship
between the displacement and velocity at an instant in time
1. phase = t + 
2.  is the phase angle (phase constant) in radians determined from the
initial conditions in the problem
Two oscillators with  and  are in phase if  =  , and they are out
of phase if  ≠  .
4. Two oscillators with f1 and f2 may be out of phase at time t1, but in
phase later at time t2. The blinkers on different cars are an example.
5. Resonance is an enhancement of an object’s amplitude when it is in
phase with another oscillator.
E. SHM occurs whenever acceleration is proportional to displacement but
oppositely directed. This is the necessary and sufficient condition for SHM!
2.5
2
1.5
1
x
0.5
0
0
2
4
6
8
10
12
14
16
18
20
-0.5
-1
-1.5
-2
-2.5
t
III. Mathematics of SHM
A. x  A cos( t   ) represents displacement of an oscillator from equilibrium.
Oscillations are just as well described by the sine function with a phase angle

of
radians
2
B. x  A cos( t   )
1. v 
dx
dt
dx
  A sin(  t   )
dt
v   A sin(  t   )
dv
2. a 
dt
dv
  A 2 cos( t   )
dt
a   A 2 cos( t   )
a   2 x
3. max [sin(  t   )]  max [cos( t   )]  1
a. v max  A
b. a max   2 A
c. xmax, vmax, amax are out of phase
1. xmax, vmax are 90° out of phase
2. vmax, amax are 90° out of phase
3. xmax, amax are 180° out of phase
d. Graphically
C. Determining Phase Constant
1. xi  A cos 
dx
2. vi  @ t  0
dt
vi   A sin 
v
3. tan    i
 xi
D. Determining Amplitude of Oscillation
1. xi  A cos 
2. vi   A sin 
2.5
2
1.5
1
x, v, a
0.5
0
0
2
4
6
8
10
12
14
16
18
20
x(t)
v(t)
a(t)
-0.5
-1
-1.5
-2
-2.5
t
v
3. xi2  ( i ) 2  A 2 cos 2   A2 sin 2   A2

v
4. A  xi2  ( i ) 2

5. Independent of frequency and period
E. Properties of SHM
1. sufficient and necessary condition for SHM is acceleration be
proportional to displacement and oppositely directed
2. displacement, velocity, and acceleration vary sinusoidally with time
but are always out of phase
3. frequency and period are independent of amplitude and vice-versa
IV. Block-Spring System
A. Equilibrium Position
1. x = 0
2. position of block at rest when spring is neither stretched nor
compressed
B. Restoring Force
1. Fs = -kx =ma
k
2. a   x
m
3. negative sign indicates that restoring force is always directed toward
equilibrium point and is therefore opposite of displacement
C. Mathematics of Block-Spring System
d 2x
1. a  2   2 x
dt
k
2. a   x
m
k
3.  2 
m
2
m
4. T 
 2
 no dependence upon A

k
1 
1 k
5. f  

 no dependence upon A
T 2 2 m
D. Proof that Block-Spring System is SHM
k
1. x   x is a differential equation
m
k
k
k
2. x  x  0 has solution x  C1 cos(
t  C2 )  C1 sin(
t  C2 )
m
m
m
3. If block is released from x = A at t = 0 and if C2 =  is chosen 0 at x =
A, then C1 = A (see special case #1)
k
4. x  A cos(
t )  A cos( t )
m
E. Special Case #1
1. x  A cos( t   )
2. Suppose a block attached to a spring is pulled a distance A from its
equilibrium position. When the block is released, xi = A and vi = 0
3. At t = 0, xi = A if and only if  = 0
4. v   A sin(  t   ) v i  0 if and only if   0
5. a   2 A cos( t   )
6. x  A cos( t   )
v   A sin(  t )
a   2 A cos( t )
7. At t  0, a   2 A, v  0, x  A
F  ma   kx
k
 kA  m( 2 A)   2 
m
k

m
F. Special Case #2
1. x  A cos( t   )
2. Suppose block is given initial velocity v = vi at xi = 0
3. v   A sin(  t   )
4. At t = 0, x = 0 and therefore  = +90° or -90°. Since v = + vi at t = 0, 
= -90°
5. x  A cos( t  90  )  A sin(  t )
v   A sin(  t  90  )  A cos( t )
dv
  A 2 sin(  t )
6. a 
dt
7. vi  A
v  vi cos( t )
a   vi cos( t )
H. Examples 16.1 and 16.2
V. Energy of Block-Spring System
1
A. KE  mv 2
2
1. v   A sin(  t   )
1
2. KE  mA2 2 sin 2 ( t   )
2
1
B. U  kx 2
2
1. x  A cos( t   )
1
2. U  kA2 cos 2 ( t   )
2
k
3.  2   k  m 2
m
1
4. U  mA2 2 cos 2 ( t   )
2
C. E = KE + U
1
1
mA2 2 sin 2 ( t   )  mA2 2 cos 2 ( t   )
2
2
1
1
2. E  mA2 2  kA2
2
2
2
3. E  A
Conservation of Energy
1
1. At x = 0, U = 0 and E  KE  kA2
2
1 2
1
1
2. E  mvmax  mA2 2  kA2
2
2
2
At arbitrary x
1. KE + U = Etotal
1
1
1
2. Etotal  mv 2  kx 2  kA2
2
2
2
k 2
3. v  
( A  x 2 )   ( A 2  x 2 )
m
4. v = vmax at x = 0 and v = 0 at x = +A and –A
See Figure 16.6
Example 16.3
1. E 
D.
E.
F.
G.
VI. Pendulums
A. Simple Pendulum
1. SHM if  is small
2. driven by gravity
B. Forces
1. Fr  T*  mg cos  0 (radial force)
2. Ft  restoring force  mg sin   ma  m
3. s  L
s  L
s  L
d 2s
(tangential force)
dt 2
4. Ft  mg sin   mL
g
5.    sin 
L
6. For small , sin   tan    if  is in radians (small angle
approximation)
g
7.     0
L
8. Solution to differential equation is SHM
a.    max cos( t   )
g
L
2
L
 2
c. T 

g
1. All simple pendulums of same length have same period if
 is small
2. no dependence upon anything except gravity and length
3. pendulum clocks (metronomes) make very good
timepieces
C. Physical Pendulum
1. Depends upon moment of inertia
2. Need not be fixed at cm or cg
3. τ  Iα  d  Fg
I  dmg sin 
I  mgd (small angle approximation)
b.  
mgd
 0
I
4.    max cos( t   )
 
mgd
I
2
I
 2
6. T 

mgd
D. Torsional Pendulum
1. Horizontally oriented and fixed at cm
2.    where  is a constant called the
torsional constant
3. Torque is perpendicular to angular
displacement
4.     I  I

    0
I
5.    max cos( t   )
5.  
6.  
7. T 

I
2

 2
I

8. Determine  for a material experimentally
9. No small angle approximation needed as long as elastic limit is not
exceeded
E. Examples 16.4 – 16.6
VII. Comparing SHM to Uniform Circular motion
A. Uniform Circular Motion is SHM in 2
Dimensions
1. Reference circle viewed from side is
a projection along its diameter
2. Projection along x-axis is 90° out of
phase with projection along y-axis
B. Angular velocity in circular motion = angular frequency of oscillation
C. Radius of circle = amplitude of oscillation
D. phase angle = initial angle made with projection
E. Velocity
1. vc  r  A on circle
2. v proj  vcircle sin 
3.    t  
4. v proj   A sin(  t   )
F. Acceleration
1. ac  r 2   A 2
2. a proj  acirc cos
3. a proj   A 2 cos( t   )
1.5
1
0.5
0
0
5
10
15
20
25
30
35
40
x
VIII. Damped Oscillations
A. Ideal systems oscillate indefinitely
B. Retarding (damping) forces (usually
friction) reduce amplitude of
oscillation
1. Let R = -bv be a damping force
2. b = damping coefficient
Frestore  kx  bv  ma
3.  kx  bx  mx
-0.5
-1
-1.5
mx  bx  kx  0
4. Solution to the differential equation is
-2
a. x  Ae
 bt
2m
t
cos( t   )
k
b
 ( )2
m 2m
C. When retarding force is much smaller than restoring force, oscillations are
damped, oscillatory character is preserved, and motion ultimately ceases
D. Amplitude decays exponentially and is bounded by the “envelope” function
b.  
bt
x  Ae 2m
E. The natural (resonance) frequency is the natural vibrational frequency of an
k
object or wave. Most objects have one natural frequency given by  0 
m
k
b
b
 ( ) 2   02  ( ) 2
m 2m
2m
2. If Rmax = bvmax < kA then the system is underdamped
3. If b = 2m0 then the system is critically damped
4. If Rmax = bvmax > kA and b > 2m0 then the system is overdamped
5. Whether system is underdamped, overdamped, or critically damped,
the energy of the system eventually goes to zero
2
1
F. E  2 kAdamp
1.  
E  12 k ( Ae
bt
2m 2
)
bt
E  12 kA2 e m
G. Example 16.7
IX. Forced Oscillations
A. Work is done on a system to produce oscillations
B. If work = energy lost then amplitude remains constant
C. Example is child on a swing
D. Suppose that F  Fext cos( t ) where  is the driving frequency
1. mx  bx  kx  Fext cos( t )

2. Assume steady state such that work = E
3. Solution to differential equation is x  A cos( t   )
Fext
k
m
4. A 
and  0 
m
( 2   02 ) 2  (b ) 2
m
E. Resonance
1. If    o then A gets very large
2. 0 is the resonance (angular) frequency of the object

3. energy transfer is most favorable at resonance
a. x  A cos( t   )
b. v   A sin(  t   )
c. P  F  v  Fv cos 
d. If F and v are in phase then  = 0° and P is max
4. As b  0, A  , and    0
5. In practice there is always some friction so b ≠ 0
F. Quality Factor
1. depends upon b
2. depends upon how close  is to 0
3. when driving force is removed, A  Amax e
4. A 
Amax
Q
in
cycles

e
t
Q
5. Q = quality factor
G. Examples of Resonance
1. Electric circuits (AC circuits)
2. Tacoma Narrows Bridge (video clip)
3. Jupiter and Asteroids (Kirkwood gap)
4. Moons of Jupiter (Io volcanism)