Oscillations

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Oscillations
oscillations_02
Time variations that repeat themselves at
regular intervals - periodic or cyclic
behaviour Examples Pendulum (simple) heart
(more complicated) Terminology Amplitude
max displacement from equilibrium position m
Period time for one cycle of motion s
Frequency number of cycles per second s-1
hertz (Hz)
SHM
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
voltage ?
time ?
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
Example oscillating stars
Brightness
Time
CP445
4
oscillations_02 MINDMAP SUMMARY
Reference frame (coordinate system, origin,
equilibrium position), displacement (extension,
compression), applied force, restoring force,
gravitational force, net (resultant) force,
Newtons Second Law, Hookes Law, spring constant
(spring stiffness), equilibrium, velocity,
acceleration, work, kinetic energy, potential
energy (reference point), gravitational potential
energy, elastic potential energy, total energy,
conservation of energy, ISEE, solve quadratic
equations, SHM, period, frequency, angular
frequency, amplitude, sine function (cos, sin),
phase, phase angle, radian, SHM circular motion

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Simple harmonic motion SHM
x 0
spring
restoring force
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
• amplitude max displacement from equilibrium
  position

CP447
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Click the image to view the animation of the two
objects executing SHM. By viewing the
animation You should have a better understanding
of the following terms SHM periodic
motion Equilibrium position Displacement Amplitude
Period Frequency
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Motion problems need a frame of reference
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
CP447
8
Connection SHM uniform circular motion
1 revolution 2? radians 360o
Angles must be measured in radians
rad.s-1
Angular frequency
CP453
9
SHM circular motion
1
0
-1
0 2? 4? 6?
Displacement is sinusoidal function of time
uniform circular motion radius A, angular
frequency ?
x component is SHM
x
X
CP453
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Simple Harmonic Motion
Displacement is a sinusoidal function of time
T
amplitude
displacement
time
T
T
By how much does phase change over one period?
CP451
11
Simple Harmonic Motion
x
k
force
m
X
x 0
angular frequency, frequency, period
CP457
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Simple harmonic motion
acceleration is ? rad (180?) out of phase with
displacement
CP457
13
Describe the phase relationships between
displacement, velocity and acceleration? What are
the key points on these graphs (zeros and
maximums)?
CP459
3 4 5 6 7 8
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Problem solving strategy I S E E Identity
What is the question asking (target variables) ?
What type of problem, relevant
concepts, approach ? Set up Diagrams
Equations Data
(units) Physical
principals Execute Answer question
Rearrange equations then substitute
numbers Evaluate Check your answer look at
limiting cases sensible ?
units ?
significant figures ?
PRACTICE ONLY MAKES PERMANENT
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• Problem 1
• If a body oscillates in SHM according to the
  equation
• where each term is in SI units. What are
• the amplitude?
• the angular frequencies, frequency and period?
• the initial phase at t 0 ?
• the displacement at t 2.0 s ?

use the ISEE method
9 10 11
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Solution 1 Identify / Setup SHM
Execute
(a) amplitude A xmax 5. 0 m (b)
angular frequency ? 0.40 rad.s-1
frequency f ? / 2? 0.40 / (2?) Hz
0.064 Hz period T 1 / f
1 / 0.064 s 16 s (c) initial phase angle
? 0.10 rad (d) t 2.0 s x 5
cos(0.4)(2) 0.1 m 3.1 m
Execute
OK
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Problem 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.
use the ISEE method
12 13 14
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Solution 2 Identify / Setup SHM
k ? N.m-1
x 20 mm
f ? Hz
m
x 20 mm 20?10-3 m
Execute
Execute
OK
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What are all the values at times t T/4, T/2,
3T/4, T ?
Problem 3
t x V a KE PE
0 A 0 - ?2 A 0 ½ k A2
T / 4
T / 2
3T / 4
T
0
-A
A
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Problem 4 A spring is hanging from a support
without any object attached to it and its length
is 500 mm. An object of mass 250 g is attached to
the end of the spring. The length of the spring
is now 850 mm. (a) What is the spring
constant? The spring is pulled down 120 mm and
then released from rest. (b) What is the
displacement amplitude? (c) What are the natural
frequency of oscillation and period of
motion? (d) Describe the motion on the object
attached to the end of the
spring. Another object of mass 250 g is attached
to the end of the spring. (e) Assuming the
spring is in its new equilibrium position, what
is the length of the spring? (f) If the
object is set vibrating, what is the ratio of the
periods of oscillation for the two
situations?
use the ISEE method
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Solution 4 Identify / Setup L0 500 mm 0.500
m L1 850 mm 0.850 m m1 250 g 0.250
kg ymax 120 mm 0.120 m k ? N.m-1 f1 ? Hz
T1 ? s m2 0.500 kg L2 ? m T2 / T1 ?
L0
L1
L2
m1
equilibrium position y 0
ymax
F k (L1 L0)
m2
m1 a 0
F FG
FG m g
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Execute
(a) Object at end of spring stationary F FG
? k (L1 - L0) m g k m g /(L1 L0) k
(0.250)(9.8) / (0.850 0.500) N.m-1 k 7.00
N.m-1
(b) (c) (d) Object vibrates up and down with
SHM about the equilibrium position with a
displacement amplitude A ymax
0.120 m
Evaluate
(e)
Again F m g k y ? y m2 g / k
m2 0.500 kg k 7 N.m-1 y (0.5)(9.8) / 7 m
0.700 m L2 L0 y (0.500 0.700) m
1.20 m
(f)
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Problem 5 A 100 g block is placed on top of a
200 g block. The coefficient of static friction
between the blocks is 0.20. The lower block is
now moved back and forth horizontally in SHM with
an amplitude of 60 mm. (a) Keeping the
amplitude constant, what is the highest frequency
for which the upper block will not
slip relative to the lower block? Suppose the
lower block is moved vertically in SHM rather
than horizontally. The frequency is held constant
at 2.0 Hz while the amplitude is gradually
increased. (b) Determine the amplitude at
which the upper block will no longer maintain
contact with the lower block.
use the ISEE method
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Solution 5 Identify / Setup
FN
1

• m1 0.1 kg m2 0.2 kg
• ? 0.20
• A2 60 mm 0.06 m
• max freq f ? Hz
• SHM
• amax A ?2 A(2 ? f)2 4 ?2 f 2 A

2
m1 a1y 0
Ff ? N ? m g
FN m g
FG
FN
1
2
m1
FG
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Execute (a) max frictional force between
blocks Ff ? m1 g max acceleration of block 1
a1max Ff / m1 ? g max acceleration of
block 2 a2max a1max ? g SHM a2max 4
?2 f2 A f 2 Hz ? g 4 ?2 f2 A f ??
g / (4 ?2 A) ?(0.20)(9.8)/(4)(?2)(0.06)
Hz 0.91 Hz
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Execute (b) max acceleration of block 1 (free
fall) a1max g max acceleration of block
2 a2max a1max g SHM a2max 4 ?2 f2
A g 4 ?2 f2 A A g / (4 ?2 f 2) (9.8) /
(4)(?2)(22) m 0.062 m Evaluate