Lecture Exp
1
2
3
4
5
6
7
8
9
10
1
2
3
4
Date
July 3
A July 5
B July 10
A July 12
B July 17
A July 19
B July 24
A July 26
B July 31
August 2
Lecture Topics
Course Overview
Discussion of Exp 1 – Goals, setup
( Deduce mean density of the earth )
Measurements, uncertainties.
Statistical Analysis
Intro to error propagation
Discussion of Exp 2 – goals, setup
( Deduction of mass distribution )
Histograms & distributions
The Gaussian Distribution,
Maximum likelihood,
Rejected data, Weighted mean
Discussion of Exp 3 – goals, setup
( Tune a shock absorber )
Fitting
Chi-squared test of distribution
Discussion of Exp 4 – goals, setup
( Calibrate a voltmeter )
Chi-squared
Covariance and correlation
Final Exam Review
Final Exam
8PM, York 2722
Lab:
Taylor:
Lab:
Taylor:
Lab:
Taylor:
Assignment
-Prepare for Quiz #1
-Read chapters 1-3, HW 1
-Analyze data for Exp #1
-Read chapter 4, HW 2
-Prepare for quiz #2
-Read chapter 5, HW 3
Lab: -Analyze data for Exp #2
Taylor: -Read chapters 6-7, HW 4
Lab:
Taylor:
Lab:
Taylor:
Lab:
Taylor:
-Prepare for quiz #3
-Read chapter 8, HW 5
-Analyze data for Exp #3
-Read chapters 9 & 12
-Prepare for quiz #4
-HW 6
Lab:
Taylor: -
-Analyze data for Exp #4
Lab: -Prepare for final exam
-Pick up graded work from
TAs
-Pick up final from LTAC
Physics 2BL Summer I 2012
3 rd Quiz
4 th Quiz
1

Physics 2BL Summer I 2012 2

• Issues from last week’s lab?
– Organize your lab according to the rubric
– (TAs shouldn’t have to search for your measurements, calculations, etc)
• Quiz #3 Thursday (Complete prelab , HW)
• Introduction to experiment #3
• Physics of damping and simple harmonic motion
• Homework
Physics 2BL Summer I 2012 3

• Error propagation
δ x tot
=
( δ x random
)
2 +
(
δ x systematic
)
2
δ q
=


∂ q
∂ x
δ x


2
+


∂ q
∂ y
δ y


2
+
...
+


∂ q
∂ z
δ z


2 x i x
±
± σ
σ x x
– Short-hand notation
δ q
=



∂ q
 ∂ x
δ x



⊕

 

∂ q

∂ y
δ y 



⊕
• Normal distribution
...
⊕



∂ q
 ∂ z
δ z



G
X ,
σ
=
σ
1
2
π e
− ( x
−
X
)
2
2
σ 2 t
= x
A
− x
B
σ
A
2 + σ
B
2
• Chauvenet’s criterion x
1
,..., x
N t sus
= x sus
σ
− x x
n = N * Prob(|t| ≥ t sus
)
If n < 0.5, the reject x sus
Physics 2BL Summer I 2012 4

• Principle of maximum likelihood
– L = P(x
1
)P(x
2
)…P(x
N
)
P
X ,
σ
– Prove that the mean maximizes the likelihood when errors are equal
X
=
σ
= x
1
2
π e
−
( x
−
X
)
2
2
σ 2
δ
X
= σ x
– Prove that the weighted mean errors are different
• Minimize chi-squared
X
= x wav
=
∑
∑ x i w i w i
2
= i
N
=
1


 x i
−
i
X



2
Physics 2BL Summer I 2012
δ
X
= σ wav
=
1
∑ w i w i
=
1
σ i
2
=
δ x
1
( ) i
2
6

• Construct and tune a shock absorber
– Adjust performance of a mechanical system
– Demonstrate critical damping of your shock absorber
Physics 2BL Summer I 2012 8

• Goals: Test model for damping
• Model of a shock absorber in car
• Procedure: develop and demonstrate critically damped system sense
• Write up results - Does model work under all conditions, some conditions? Need modification?
Physics 2BL Summer I 2012 9

Construct and test a critical damping system for a spring
• A shock absorber consists of a spring and a piston moving inside a cylinder with a damping liquid.
• The springs give you a gentle ride, so you do not feel every little dip and bump.
• However, if not substantially damped, the spring will go through many oscillations before returning to
• On the other hand, by damping the springs too much (over-damping), you destroy any positive effect of the springs and make the suspension exceedingly hard.
• If the system is just right, we call it critically damped. It reduces shocks and returns the system to equilibrium within optimal (minimum) time.
Physics 2BL Summer I 2012 10

(quick reminder)
∆ x
At Equilibrium mg
− k
∆ x
=
0 , mg
= k
∆ x
At strain/elongation
F net
= mg
− k ( x
+ ∆ x )
= − kx
Newton + Hooke ma
= − kx ; m d
2 x dt
2
+ kx
=
0
Equilibrium length
Simple Harmonic Oscillation x
= ∆ x
+ x
0 cos
( )
Amplitude
Physics 2BL Summer I 2012 11
Stolen From S. Bodde

Mass falls, gains velocity until the drag force balances the gravitational force: mg
− bv t
=
0 , v t
= mg b
Terminal velocity
Newton’s 2 nd
− b y &
+ mg
= m & y &
Velocity as a function of time:
+ m y & & y &
( )
=
0 y &
= v t



1
− e
− b m t



Time constant
Physics 2BL Summer I 2012 12
Stolen From S. Bodde

Damping Force :
• viscous friction damping
• proportional to velocity
Newton
+
Hooke
+ Damping Force: Sum of Forces
F damp
= − bv , F net
= − kx
− bv , ma
= − kx
− bv
Equation of motion for damped oscillator:
Exponential decay m d
2 x dt
2
+ b dx dt
+ kx
=
0
Solution: x
= x
0 e
−
( b
2 m
± i
ω
) t
Oscillatory?
where:
ω = b
2 k m
−
4m
2
Physics 2BL Summer I 2012 13

Case a: k m
>> b
2
4m
2
−
2 b m t
t t ) )
Under Damped
General Solution: x
= x
0 e
−
2 b m t
± i
ω t
ω = k m
− b
2
4m
2 x
0
What is the physical meaning?
Passing over a bump would result in oscillations that would continue for many cycles, making for an uncomfortable and unsafe ride.
− x
0
≈ k m
Physics 2BL Summer I 2012 14

Case c: b
2 k m
<
4m
2
Over Damped
−

 b
2 m
− b
2
4 m
2
− k m


 t
General Solution: x
= x
0 e
−
2 b m t
± i
ω t
ω = k m
− b
2
4m
2
What is the physical meaning?
• No oscillatory term
• Exponential decay as if there are no springs
Physics 2BL Summer I 2012 15

Case b: b
2 k m
=
4m
2 x
= x
0 e
− b
2 m t
Critically Damped x
0
General Solution: x
= x
0 e
−
2 b m t
± i
ω t
ω = k m
− b
2
4m
2
(a) Under-damped
(b) Critically damped
(c) Over damped b crit
=
2 mk
− x
0
Physics 2BL Summer I 2012 16

• Mass hanging on a spring: For a given combination of spring and mass, calculate the damping required for critical damping
• Mass free-falling through a tube: Determine terminal velocity
(and b) as a function of holes open. Find out how many holes need to be open for critical damping to be reached according to b crit
=
2
• Put the above two together and test for critical damping
• Adjust the damping (# holes open) to provide critical-damping, and compare: by how much the calculated value differs from the final adjusted value .
Physics 2BL Summer I 2012 17

Physics 2BL Summer I 2012 18

How to measure the spring constant?
Measure oscillation period using a stop watch:
=
2
T
= k m k
= m



2
T 


2 m
Upon measuring the mass of the piston, m , the period of the pendulum T , you can compute the spring constant, k , and then calculate the damping coefficient, b , needed for critical damping: b
=
2 mk
Physics 2BL Summer I 2012 19

(How to measure the damping coefficient, b ?)
We let the piston drop in the cylinder without spring.
>> But with a safety wire attached to it!
After a short while a terminal velocity is reached and the net force and acceleration become zero
(drag/damping force is equal to gravity force): mg
− bv t
=
0 , b
= mg v t damping tube to get a good graph!
v t
=
∆ h
∆ t v
∆ h Photogate h h ∆ h - Height of the wide part of piston.
Physics 2BL Summer I 2012 20

Dominated by compression of air under the piston b b
=
2 mk
Dominated by friction piston and the cylinder walls n- open holes
You need to adjust n so it satisfies b
=
2 mk
For fine adjustment you want to use the valve.
Physics 2BL Summer I 2012 21

Valve plus holes adjustment of air flow out of cylinder → adjustment of b
δ y/y
•Do NOT overstretch spring
•Plexiglas cylinder used for damping.
•Need a “ just right ” fit. Match # on piston to # on damping tube.
•Keep it stable (clamp to table!)
Physics 2BL Summer I 2012 22

Construct and Test a Critical Damping System
• Measure mass m, period T
• Compute the spring constant k.
v
• Compute the damping coefficient b needed for critical damping b
=
2 mk h
Terminal velocity is reached when
• • Use terminal velocity measurements to determine b as a function of the number of holes covered and valve position.
– Make sure to drop mass over the maximum range in heights!!!!
b b
=
2 mk
• Test spring plus shock absorber and optimize .
dominated by air flow
Physics 2BL Summer I 2012 n-holes dominated by friction
23

• Good taping of holes is crucial to good operation for terminal velocity and oscillation.
• Critical damping means no real
– How can you tell the difference ??
• Critical damping will have the smallest b with no oscillation.
>>> Demonstrate to your TA!
Physics 2BL Summer I 2012 24

– Shortest time to return to equilibrium position
Physics 2BL Summer I 2012 25

Give each graph a title
Determine independent and dependent variables
Determine boundaries
Include error bars
Physics 2BL Summer I 2012 26

Prelab problems for Exp 3
HW5: Taylor problems 8.6, 8.10, 8.24
Physics 2BL Summer I 2012 28