1
LECTURE 2
Oscillations
Time variations that repeat themselves at regular intervals periodic or cyclic behaviour
Examples: Pendulum (simple); heart (more complicated)
Terminology:
Period: time for one cycle of motion [s]
Frequency: number of cycles per second [s-1 = hertz (Hz)]
How can you determine the mass of a single E-coli bacterium or a
DNA molecule ?
CP458
CP Ch 14
2
Signal from ECG
period T
Period:
time for one cycle of motion [s]
Frequency: number of cycles per second [s-1 = Hz hertz]
1 kHz = 103 Hz
106 Hz = 1 MHz
1GHz = 109 Hz
CP445
3
Brightness
Example: oscillating stars
Time
CP445
4
Simple harmonic motion SHM
x=0
spring
Fe
Fe  k x
restoring force
x
+X
• object displaced, then released
• objects oscillates about equilibrium position

• motion is periodic
• displacement is a sinusoidal function of time (harmonic)
T = period = duration of one cycle of motion
f = frequency = # cycles per second
• restoring force always acts towards equilibrium position
CP447
5
Motion problems – need a frame of reference
origin 0
equilibrium position
displacement
x [m]
velocity
v [m.s-1]
- xmax
Vertical hung spring: gravity
determines the equilibrium position
– does not affect restoring force for
displacements from equilibrium
position – mass oscillates vertically
with SHM
Fe = - k y
+ xmax
acceleration
a [m.s-2]
Force
Fe [N]
CP447
6
Connection SHM – uniform circular motion
w = dq/dt
w = 2p / T
amplitude A
A
T=1/f
f=1/T
w=2pf
w=2 p / T
One cycle: period T [s]
Cycles in 1 s: frequency f [Hz]
q
- -1
]
angular frequency w [rad.s
CP453
7
SHM & circular motion
Displacement is sinusoidal function of time
t

x  xmax cos  2p   xmax cos  2p f t   xmax cos w t 
 T
uniform circular motion
radius A, angular frequency w
v
A  xmax
q wt
x
2p
w  2p f 
T
x component is SHM:
x  A cos w t 
xmax  A
CP453
8
Simple harmonic motion
displacement
T
amplitude
xmax
T
time
T

Displacement is a sinusoidal function of time
 t 
x  x max cos2p  x max cos2pft  x max coswt 
 T 
By how much does phase change over one period?
CP451
9
Simple harmonic motion
x  xmax cos w t 
v   xmaxw sin w t 
a   xmaxw 2 cos w t 
a  w 2 x
equation of motion
substitute
xmax  A
m a  k x
k
1
w 
f 
m
2p
(restoring force)
k
m
 T  2p
m
k
oscillation frequency and period
CP457
10
Simple harmonic motion
a  w 2 x

At extremes of oscillations, v = 0
When passing through equilibrium,
v is a maximum
acceleration is p rad (180) out
of phase with displacement
2
v 2  xmax
w 2 sin 2 w t 
2
 xmax
w 2 (1  cos2 w t )
2
 w 2 ( xmax
 x2 )
2
v  w ( xmax
 x2 )
CP457
11
SHM
position x
10
0
-10
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
time t
60
70
80
90
100
velocity v
5
0
acceleration a
-5
1
0
-1
How do you describe the phase relationships between displacement, velocity and
acceleration?
CP459
12
Problem 2.1
If a body oscillates in SHM according to the equation
x  5.0cos(0.40 t  0.10) m
where each term is in SI units. What are
(a) the amplitude
(b) the frequency and period
(c) the initial phase at t = 0?
(d) the displacement at t = 2.0 s?
13
Problem 2.2
An object is hung from a light vertical helical spring that
subsequently stretches 20 mm. The body is then displaced and
set into SHM. Determine the frequency at which it oscillates.
14
Answers to problems
2.1
5.0 m
0.064 Hz
2.2
w = 22 rad.s-1
16 s
f = 3.5 Hz
0.10 rad
3.1 m