Simple Harmonic Motion and
Damping
& Uncertainty Analysis Review
Lecture # 5
Physics 2BL
Winter 2011
Outline
•
•
•
•
•
Significant figures
Gaussian distribution and probabilities
Experiment 3 intro
Physics of damping and SHM
Experiment 3 objectives
Significant Figures
What is the correct way to report the
following numbers: (Justify your answer)
(a) 653 ± 55.4 m
(b) 256.55 ± 27 kg
Yagil
p. 287 Taylor
Yagil
Yagil
Example problem
Measure wavelength λ four times:
503 ± 10 nm
491 ± 8 nm
525 ± 20 nm
570 ± 40 nm
Should we reject the last data point?
570 – 500 nm
tsus = ∆λ =
= 1.73 σ
2
2
6 + 40 nm
Prob of λ outside ∆λ =
Yagil
Example problem
Measure wavelength λ four times:
503 ± 10 nm
491 ± 8 nm
525 ± 20 nm
570 ± 40 nm
Should we reject the last data point?
570 – 500 nm
tsus = ∆λ =
= 1.73 σ
2
2
6 + 40 nm
Prob of λ outside ∆λ = 100 % - 91.6 % = 8.4 %
Total Prob = N x Prob = 4 * 8.4 % = 33.6 %
Is Total Prob < 50 % ? Yes, therefore can reject data point
The Four Experiments
• Determine the average density of the earth
– Measure simple things like lengths and times
– Learn to estimate and propagate errors
• Non-Destructive measurements of densities, structure–
– Measure moments of inertia
– Use repeated measurements to reduce random errors
• Test model for damping; Construct and tune a shock
absorber
– Damping model based on simple assumption
– Adjust performance of a mechanical system
– Demonstrate critical damping of your shock absorber
– Does model work? Under what conditions? If needed, what more needs
to be considered?
• Measure coulomb force and calibrate a voltmeter.
– Reduce systematic errors in a precise measurement.
Experiment 3
• Goals: Test model for damping
• Model of a shock absorber in car
• Procedure: develop and demonstrate critically
damped system
• check out setup, take data, do data make
sense?
• Write up results - Does model work under all
conditions, some conditions? Need
modification?
Simple Harmonic Motion
• Position oscillates if
force is always
directed towards
equilibrium position
(restoring force).
• If restoring force is ~
position, motion is
easy to analyze.
FNet
FNet
Springs
• Mag. of force from
spring ~ extension
(compression) of
spring
• Mass hanging on
spring: forces due to
gravity, spring
• Stationary when
forces balance
FS = −kx
FG = −mg
FG = FS
mg = kx
m12
x = x1
m2
x = x2
Simple Harmonic Motion
x = x0
3
dipslacement
• Spring provides
linear restoring force
⇒ Mass on a spring
is a harmonic
oscillator
2
x =0
1
0
-1
F = −kx
d2x
m 2 = −kx
dt
x(t) = x 0 cos ωt
-2
-3
0
10
20
30
40
50
60
time
T=
2π
ω
ω=
k
m
Damping
• Damping force opposes
motion, magnitude
depends on speed
• For falling object,
constant gravitational
force
• Damping force
increases as velocity
increases until damping
force equals
gravitational force
• Then no net force so no
acceleration (constant
velocity)
Terminal velocity
• What is terminal velocity?
• How can it be calculated?
Falling Mass and Drag
At steady state:
From rest:
Fdrag = F gravity
bvt = mg
y(t) = vt[(m/b)(e-(b/m)t – 1) + t]
Terminal Velocity
For velocity:
.
y(t) = vt[1 - e-(b/m)t]
Experimental Setup for Falling
Mass and Drag
How do you measure velocity?
Damped Harmonic Motion
Damped SHM
• Consider both
position and velocity
dependant forces
• Behavior depends
on how much
damping occurs
during one
‘oscillation’
⎛ b ⎞ ⎛ k
b2 ⎞
⎟.
x = x 0 exp⎜−
t ⎟ cos⎜ t
−
⎝ 2m ⎠ ⎜⎝ m 4m 2 ⎟⎠
d 2x
dx
m 2 = − kx − b
dt
dt
⎛ b ⎞ ⎛
k
b2 ⎞
⎟
x = x0 exp⎜−
t ⎟ exp⎜⎜ it
−
2 ⎟
⎝ 2m ⎠ ⎝ m 4m ⎠
or
⎛⎛⎛ b
22
⎞ ⎞⎞
b
k
b
k
⎜
exp⎜⎜⎜−−
±+
−− ⎟⎟ t⎟⎟t
xx== xx00 exp
22
⎜⎜ 22m
4m
4m mm⎠ ⎟⎠⎟⎠
⎝⎝⎝ m
Relative Damping Strength:
Weak damping
b2
k
−
m 4m2
⎞
⎟⎟ .
⎠
3
dipslacement
⎛
⎛ b ⎞
t ⎟ cos ⎜⎜ t
x = x 0 exp ⎜ −
⎝ 2m ⎠
⎝
k
b2
<<
m
4m2
2
1
0
6
-1
4
weak damping
2
-2
0
0
-2
2
4
6
(underdamped)
-4
-6
time
8
10
0
100
200
300
400
time
500
600
Relative Damping Strength:
Strong damping
b2
k
>>
4m 2
m
strong damping
(overdamped)
3
dipslacement
2
⎛ b
⎞
b
k
⎟t
+
−
x = x0 exp⎜ −
2
⎜ 2m
⎟
m
m
4
⎝
⎠
2
1
0
0
5
10
15
time
20
25
Relative Damping Strength:
Critical damping
b2
k
=
2
4m
m
critical damping
⎞
⎟t
⎟
⎠
3
dipslacement
2
⎛ b
k
b
+
−
x = x0 exp⎜ −
2
⎜ 2m
m
m
4
⎝
2
1
0
-100
bcrit = 2 mk
0
100 200 300 400 500 600 700
time
Comparison of the various
types of damping
dipslacement
3
overdamped
critically
damped
2
1
0
-1
underdamped
-2
-100
0
100 200 300 400 500 600 700
time
Plotting Graphs
Give each graph a title
Determine independent and dependent variables
Determine boundaries
Include error bars
Experimental setup
spring
photogate
scale (mm)
cylinder
photogate
timer
piston
valve
holes
Experiment 3: achieve critical
damping
• Show/test method
– Determine spring constant, predict critical
damping coefficient
– Determine how damping coefficient
depends on air flow (valve position)
• easy at terminal velocity
• how do you know it’s vterminal?
– Set damping to critical level
Demonstrate critical damping:
show convincing evidence that
critical damping was achieved
• Demonstrate that damping is critical
– No oscillations (overshoot)
– Shortest time to return to equilibrium
position
Error propagation
(1) kspring = 4π2m/T2
σkspring = εkspring * kspring
εkspring = εm2 + (2εT)2
(2) kby-eye = m(g∆t*/2∆x)2
σby-eye = εby-eye * kby-eye
εby-eye =
(2ε∆t*)2 + (2ε∆x)2 + εm2
Remember
• Prepare for Exp. 3
• Homework Taylor #8.6, 8.10
• Read Taylor through Chapter 9