Schedule
Lecture Exp
Date
A
July 5
3
B
July 10
4
A
July 12
5
B
July 17
6
A
July 19
B
July 24
A
July 26
B
July 31
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
August 2
Final Exam
1
July 3
2
1
2
3
7
8
9
10
4
Assignment
Lab: -Prepare for Quiz #1
Taylor: -Read chapters 1-3, HW 1
Lab: -Analyze data for Exp #1
Taylor: -Read chapter 4, HW 2
Lab: -Prepare for quiz #2
Taylor: -Read chapter 5, HW 3
Lab: -Analyze data for Exp #2
Taylor: -Read chapters 6-7, HW 4
Lab:
Taylor:
Lab:
Taylor:
Lab:
Taylor:
Lab:
Taylor:
Lab:
8PM, York 2722
Physics 2BL Summer I 2012
-Prepare for quiz #3
-Read chapter 8, HW 5
-Analyze data for Exp #3
-Read chapters 9 & 12
-Prepare for quiz #4
-HW 6
-Analyze data for Exp #4
-Prepare for final exam
-Pick up graded work from
TAs
-Pick up final from LTAC
4th Quiz
1
Experiment 4 – Goals, Setup
(Calibrate a voltmeter)
Lecture # 7
Physics 2BL
Summer Session I 2012
Physics 2BL Summer I 2012
2
Outline
•
•
•
•
•
•
Issues from last week’s lab?
Quiz #4 Thursday (Complete prelab , HW)
Reminder about lecture 9
Recap from last lecture
Introduction to experiment #4
Homework
Physics 2BL Summer I 2012
3
Lecture 9 – QUESTIONS!
Lecture 9 will be a final review:
No lecture slides, bring your questions or
lecture will end early
UNACCEPTABLE QUESTIONS: “What is
going to be on the final?”
Physics 2BL Summer I 2012
4
Linear Fit: Finding A and B
• Need to minimize:
∑ [y − (A + Bx )]
2
j
j
• Looks nasty, not so
bad…
• See Taylor, example
8.1
∂
= ∑ y i − AN − B∑ x i = 0
∂A
∂
2
= ∑ x i y i − A∑ x i + B ∑ x i = 0
∂B
A=
B=
2
x
∑ i ∑ yi − ∑ xi ∑ xi yi
∆
N ∑ xi yi − ∑ xi ∑ yi
∆
∆ = N ∑ xi −
2
Physics 2BL Summer I 2012
(∑ x )
2
i
5
Uncertainty in Measurements of y
• Before, measure
several times and take
standard deviation as
error in x
• Can’t now, since yi’s
are different quantities
• Instead, find standard
deviation of deviations
σx =
σy =
Physics 2BL Summer I 2012
1 N
2
(
x
−
x
)
∑
i
N − 1 i =1
N
1
2
(y i − A − Bx i )
∑
N − 2 i=1
6
Uncertainty in A and B
• A, B are calculated
from xi, yi
• Know error in xi, yi ;
use error propagation
to find error in A, B
• A distant extrapolation
will be subject to large
uncertainty
Physics 2BL Summer I 2012
σA = σy
σB = σy
∑ xi
2
∆
N
∆
∆ = N∑ xi −
2
(∑ x )
2
i
7
Other Functions
• Convert to linear
• Can now use least
squares fitting to get ln
A and B
Physics 2BL Summer I 2012
y = Ae Bx
ln y = ln A + Bx
8
χ2 Test
Functional fit (i.e. linear)
(Measured)
Distribution fit
(Predicted from fit)
 y j − f (x j )
χ = ∑

σy
j =1 


N
2
2
n
χ =∑
2
i =1
(O
k
− Ek
)
2
Ek
d=n-c
d=N-c
χ~ 2 =
(
χ2
d
2
Pd χ~ 2 ≥ χ~0
)
(Larger table in Taylor)
Physics 2BL Summer I 2012
10
The Four Experiments
• Measure coulomb force and calibrate a voltmeter.
– Reduce systematic errors in a precise measurement.
Physics 2BL Summer I 2012
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Experiment 4
Construct a device to measure the absolute
value of voltage difference through a force
measurement
Method
• Use Torsional pendulum
– Measure mass, distance, and time in order to
obtain a measurement of an electric potential in
Volts
Physics 2BL Summer I 2012
12
Basic Equations
Q1Q2 Force between two point charges
F=
4πε 0r 2 (Coulomb’s law)
ε 0 = 8.85 × 10
−12
F Permittivity constant
m
F
Q1
E=
=
2 Electric field from a point charge, Q1
Q2 4πε 0 r
Coulomb force acting on a charge Q2
F = EQ2
1 ∞
Q1
Voltage – Potential energy per unit charge V =
Fdr
=
Q2 ∫r
4πε 0 r
Voltage difference ∆V =
∫
r2
r1
Edr = −
Q1
4πε 0 r1
+
Q2
4πε 0r2
Physics 2BL Summer I 2012
13
Parallel Plate Capacitor
Plug in numbers to get an
estimate for the force strength
Physics 2BL Summer I 2012
14
How to measure F = 0.1 g due to a potential difference?
Adjustment lever
Fixed capacitor plate
Moving
capacitor
plate with a
spacer
Attractive Torque: Fl
Restoring Torque: κθ
Electrical contact
to torsion fiber
Torsion fiber
Mirror for optical lever
Balance plate
κ - Torsion constant
θ - Deflection angle
Physics 2BL Summer I 2012
F = κθ l
15
Using force (torque) to construct a “Voltmeter”
1) Let torsion rod come to rest at equilibrium
2) Bring aluminum plate close to disk with stopper at zero voltage (leave a
very very small gap)
3) Apply voltage between plates
4) Increase torque from torsion fiber by twisting top end of fiber and determine
the angular displacement that causes the plate to move apart
Force by torsion balance: F = κθ l
Set equal and solve for V
1
A
ε
0
Electrostatic attraction: F =
V2
2
2 d
2
At twisting angle causing plates to just move apart: V = d
(at equilibrium)
lA
Moveable plate
Top View
θ
∆V
κθ
ε0
Balance (and
damping) plate
Fixed (Al) plate
Physics 2BL Summer I 2012
16
How to measure torsion constant, κ?
Period of torsion oscillations: T = 2π

m

Remember for a spring?  T = 2π k 


2π 2 I
κ= 2
T
I - Moment of inertia
Disks of radius R
I
κ
Weigh the support
beam and masses
separately!
m1 > m2
l1 < l2
2
m
m
R
2
2
2
2
I = (l1 + l2 ) + (l2 − l1 ) + m1l1 + m2 l2 + (m1 + m2 )
12
4
4
(Moment of Inertia in the lab is incorrect! Assumes m1 = m2)
Physics 2BL Summer I 2012
18
How to measure the period, T?
Torsion pendulum travels slowly difficult to tell when
the pendulum has reached max rotation amplitude (turn
around)
Reduce uncertainty in period by measuring T relative to
the equilibrium position (pendulum rotates faster)
Use mirror to reflect laser onto paper taped to cardboard
Let torsion rod come to rest at equilibrium (mark it)
Give torsion rod a small displacement from equilibrium
Start counting when laser passes dot
θ(t)
Count 5 periods,
repeat for 3-5
trials
Physics 2BL Summer I 2012
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Voltmeter Apparatus
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Capacitor
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Assembling & Testing the voltmeter
1.
2.
3.
4.
5.
6.
7.
Set up torsion pendulum and check for
conduction from the eyehook screwed into
2π 2 I
the piston to the disk with the stopper
κ= 2
T
Attach mirror to the support beam, set up
laser
Measure period and calculate torsion
constant
Add damping to the system to limit
unwanted oscillations (water bath)
Protect your setup from unwelcome wind
currents using poster board
Bring pendulum to equilibrium and place
fixed capacitor plate parallel to the moveable
plate, just barely touching the insulating dot
Record equilibrium position on the
protractor on the top
Physics 2BL Summer I 2012
θ0
23
Assembling & Testing the voltmeter
8.
9.
Apply 600 V across the capacitor
Apply a restoring torque (in the direction to pull
the plates apart) by rotating the adjustment lever
until plates separate (record angle)
10. Compute the voltage
V =d
2κ (∆θ )
lAε 0
∆θ = θ − θ0
∆V
θ
11. Turn off power supply, touch capacitor plates
together to discharge the plates
12. Find the angle at which the plates separate for
700-1000 V
13. Repeat the steps #6-11 5 times (to
determine/reduce uncertainties)
Physics 2BL Summer I 2012
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Analysis
Make a graph of Vcalc vs VPS:
– x-axis is the voltage read from the power
supply (600-1000 V)
– y-axis is the voltage calculated from the angle
of capacitor plate separation
=V(θ)
=A + Bx
Fit to straight line χ 2 = ∑  yi − f (xi )
σ yi 

2

Calculate χ
2
χ~ 2 =
χ2
d
Discuss goodness of fit
SDOM of V(θ)
Calculate probability of result
Physics 2BL Summer I 2012
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Homework
• Prelab problems for Exp 4
• Read Taylor Chapter 9, 12
• HW 6: Taylor problem 12.3, 12.14
• Review goals and questions from current
and previous labs
Physics 2BL Summer I 2012
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