Damped Harmonic Motion
Linear damping:
Fd  bv
F  bv  kx  ma
dx
d 2x
 b  kx  m 2
dt
dt
d 2 x b dx k

 x0
2
m dt m
dt
d 2x
dx
2

2



0x  0
2
dt
dt
k
  02
m
x(t )  A(t ) cos( ' t   )
A(t )  A0 e t  A0 e b / 2 m t
 '   02  2 
k  b 


m  2m 
2
b
 2
m
Three types of solutions:
1. Underdamping
b  2
x  A(t )cos(t  )
x(t)
b1 > b2
T 1 > T2
with
km

A(t )  Ae
(b /2m )t
 
k  b2
m 4m 2
Similar to ideal SHM except:
•
period of oscillation lengthened
•
exponential decay in amplitude
A(t)
t
A(t)
x(t)
2. Critical damping

b  2 km

 
k  b 2  0!!
m 4m 2
No oscillations!
t
3. Overdamping
b  2
km

No oscillations either. The decay is
slower than with critical damping.
x(t)
b  2 km
b  2 km
b  2 km
t
Forced Oscillations and Resonance
Damping is
always present
To keep a system going, we need
to apply a driving force.
Fext  Fmax cos( d t )
For a driving force of the form:
the amplitude is
A

Fmax m
2
0

  b
2 2
d
m
2
d
Resonance
When
d
k
 0
m
amplitude, A is very large, even for small Fmax
(without damping, ie b = 0, A → ∞).
 0 - natural frequency
Fmax
d 2 x b dx k

 x
cos d t 
2
m dt m
m
dt
x(t )  A0 e  b / 2 m t cos( ' t   )  A cos(d t   )
Waves
There are many examples of waves:
- Seismic waves
- Ripples on a pond
- Sound
- Electromagnetic wave including light
Types of Waves
•
A wave can have a short duration – just
a quick pulse; these are called “wave pulses”
•
A wave can also repeat; these are called
continuous waves
•
y ( x, t )  A cos(t  kx)
A transverse wave is a wave in which the oscillations are transverse to the
direction the wave travels
A longitudinal wave is a wave in which the oscillations are along the direction the
wave travels
Wavelength and Speed of Waves
v   / T  f
λ - wave length
T - period
f - frequency
The “speed” of a wave is the rate of movement of the disturbance.
It is not the speed of the individual particles!
The speed is determined by the properties of the medium
Sinusoidal waves
y ( x, t )  A cos(t  kx)
λ
T
y
y
t
  2 / T
k  2 / 
v   /T   / k
x
Speed of waves in strings
m  mass
v
F
m

l

l  length
F  tension
v  speed
The speed of the wave increases if we increase the tension (F) in the string,
and decreases if we increase the linear mass density (μ) of the string
Example: A metal string has a mass of 0.020 kg, a length of 40 cm, and is under 80 N
tension. 1. What is the speed of a wave in this metal string?

m 0.020kg
kg

 0.05
l
0.40m
m
v
F


80 N
m
 40
0.05kg / m
s
2. What happed if we increase a tension up to 120 N?
v
F


120 N
m
 49
0.05kg / m
s