A different kind of lab course'

1
Physics 2BL

• A different kind of lab course.
• No cookbook
• Student is responsible for measurement and
  presentation
• Discussion in Lecture and with Lab TAs
• Read syllabus
• procedure
• grades (65 labs, 20 exam, 10 homework, 5
  in-lecture lab)
• schedule
• description of experiments
• Come to lecture.

2
Four Easy Experiments

• Measure the density of the earth.
• Measure simple things like lengths and times.
• Learn to estimate and propagate errors.
• Measure moments of inertia.
• Use repeated measurements to reduce random
  errors.
• Design, build, and test shock absorber.
• Get the bugs out of your mechanical system.
• Measure coulomb force and calibrate voltmeter.
• Reduce systematic errors in a precise
  measurement.

3
Measurement Errors

• A measurement is useless unless we know its
  error.
• What weight can a bridge support ?
• Is crime going down?
• Are two experimental results consistent?
• Are data consistent with the laws of physics?
• Does cold fusion produce power?

What is the important scientific difference
between global warming and cold fusion
experiments?

• Is the expansion of the universe accelerating?

• Will the ozone layer disappear?
• Is the earths climate warming due to CO2?

4
How Accurate Are Your Measurements?
"We had planned to approach the planet at an
altitude of about 150 kilometers (93 miles). We
thought we were doing that, but upon review of
the last six to eight hours of data leading up to
arrival, we saw indications that the actual
approach altitude had been much lower. It appears
that the actual altitude was about 60 kilometers
(37 miles). We are still trying to figure out why
that happened," said Richard Cook, project
manager for the Mars Surveyor Operations Project
at NASA's Jet Propulsion Laboratory. "We believe
that the minimum survivable altitude for the
spacecraft would have been 85 kilometers (53
miles)."
Wrong Units
Mars Climate Orbiter lost
5
Measurement Errors

• We define two types of errors.
• Random Errors
• Can be reduced by repeated measurements.
• Can also be estimated by repeated measurement.
• Example measure the fraction of people over 6
  by surveying 100 people from around the world.
• Systematic Errors
• Harder to estimate and reduce.
• Calibration errors, neglecting small corrections,
  or mistakes. (cold fusion, global warming)
• Example measure the fraction of people over 6
  by surveying a group from France.

6
Error Propagation
What is the perimeter of this figure?
How would you estimate the error on x?
How would you calculate the error on p?

• Measure w, x, y, and z.
• Compute p
• Propagate errors from w,x,y,z to p

7
Example of Error Propagation

• Suppose that q xyz. We wish to determine q by
  measuring x, y, and z. The error on q, (dq), can
  be calculated from the errors on the measured
  quantities.

What is an RMS error?
8
Another Example of Error Propagation
9
Root Mean Square (RMS) Errors
What do we do when the partial derivative is
negative?
s (sigma) RMS error Standard Deviation
10
Why Use Partial Derivatives
1) It worked for the qxy case.
2) It makes sense graphically.
sx
x0
11
Fractional Errors are Sometimes Useful
For products like qxy, we can add the fractional
errors on the measurements to get the fractional
error on the result.
12
Example of Error Propagation
13
Example Errors on Trig Functions
Always use radians when calculating the errors on
trig functions.
Plug whatever you want into cosine, but, sq must
be in radians. So if sq is 2 degrees, convert it.
14
Averaging Data

• Random Errors can be reduced by repeated
  measurements.
• The best estimate of the true value of a measured
  quantity is the average (mean).

• We can also estimate the RMS error from the set
  of measurements.

• We can then compute the error on the mean which
  decreases with the number of measurements.

If sx is 1 mm, how many times must I measure to
get a 0.2 mm error on the mean?
15
How to Write Errors

• We typically are estimating error, implying that
  we do not know them precisely.
• Even when we measure errors by repeated
  measurements, other systematic effects may limit
  our accuracy.

L30.563 m sL0.005 m
L30.5627 m sL0.0013 m
16
Four Easy Experiments

• Measure the density of the earth.
• Measure simple things like lengths and times.
• Learn to estimate and propagate errors.
• Measure moments of inertia.
• Use repeated measurements to reduce random
  errors.
• Relative measurement
• Absolute measurement
• Design, build, and test shock absorber.
• Get the bugs out of your mechanical system.
• Measure coulomb force and calibrate voltmeter.
• Reduce systematic errors in a precise
  measurement.

17
Experiment 1 Measure Density of Earth

• Two measurements
• (a) Earths Radius Re . (challenging
  measurement)
• (b) Local acceleration of gravity g. (fairly
  easy)
• Use Newtons constant G6.67 X 10-11 N m2/kg2
• Calculate average density r and determine which
  elements constitute the major portion of the
  earth.
• Aim for 10 or better error on r.

What is the value of g?
What is the radius of the earth?
18
Whats the Point
This is an experiment in which you measure
everyday quantities like length and time for
which you should be able to estimate errors. You
should understand basic lab techniques for
measuring and error estimation. You should
understand how errors on measured quantities can
be propagated to the desired physical
quantities. Its an experiment about optimizing
measurement technique, error estimation, and
error propagation.
19
What Element(s) make up the Earth

• Assume most of earths volume is one element.
• Top 100 km is rock.
• 10 measurement needed to determine composition.
• What can you deduce from your measurement.

Where do all these earth elements come from?
20
Measure Earths Radius using Dt Sunset
How fast does the earth rotate?
View from North Pole
Assume we are at equator
21
q Increases at Earth Rotates
Earth makes (nearly) one rotation per
day. Angular frequency is 2p radians per day.
w (omega) earths angular frequency.
Why do weather systems swirl?
22
Can We Make this Measurement?
What time delay do you expect for a 200m height?
What important flaws are there in this
calculation?
23
Correct for Latitude and Earths Axis
Why does the climate change from summer to winter?
24
Measuring the Height of the Cliff

• The formula we derived is for height above sea
  level.
• Strings, protractors, and rulers will be
  available.
• Be sure to understand how well heights must be
  measured before you do the experiment.
• Each pair of experimenters should get their own
  measurements.

Should q be small or large?
What is the most important error due to string
sag?
Where should you measure the string angle?
25
Your Height Above Sea Level on Beach

• The experimenter on the beach also views the
  sunset from above sea level.
• When you check the error propagation you will
  find that the measurement of the earths radius
  is quite sensitive to the h2 measurement.
• h2 must be measured at the same approximate time
  as Dt.
• Tides change the sea level by 1 to 2 meters.

h2
Should the student on the beach measure t near
the ocean or near the string?
What is the effect of waves in the ocean?
26
Cliffs West of Muir Campus
At the bottom of the asphalt road is a reasonable
place to measure.
Must return there at sunset. Access is fairly
easy but wear walking shoes. It may be cold in
the evening.
27
The Equation for Experiment 1a
Which are the variables that contribute to the
error significantly?
28
Propagating Errors for Re
Note that error blows up at h1h2 and at h20.
If the cliff is 30m high, what is a good h2 to
use?
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Weather plays a role. Completely clear days are
best.
32
Other Methods?
What other methods could we use to measure the
radius of the earth?
33
Measuring g with a Pendulum
What other methods could we use to measure g?

• Period can be measured with electronic timer over
  one cycle of with a stopwatch over many cycles.
• Frictional forces play a role for light weights.
• Small oscillations are good.
• Heavy weights may cause coupling to other
  oscillators like unstable stand.
• Short strings may cause moment of inertia to
  become important.

34
Propagating Errors for Experiment 1
What do you expect to limit the accuracy of your
experiment?
35
Example Average Re from 38 Students
36
Systematics from h1, h2, and Dt Measurements
h1 varies with tide but we dont see any
systematic effect. Small h2 seems to be a bad
idea.
37
Systematics from Dt
There is a clear trend that larger Dt gives
smaller R.
What do you think causes this?
38
Probability Distributions

• Assume the true value of x is 5.5 m. We make
  repeated measurements of x with an error of 2.5
  m.
• What do we expect the distribution of
  measurements to look like? This depends on the
  probability distribution.
• This is one possible example.

What is the probability to measure x5?
What is the probability to measure x between 6
and 10?
Can you think of something with a flat
probability distribution?
39
Standard Normal (Gaussian) Distribution

• Many probability distributions, including errors,
  approach the Normal distribution.
• Biological parameters
• Test scores
• Any combination of random variables (Central
  Limit Theorem).
• Normal distribution has average and standard
  deviation as parameters.

40
The Normal Distribution

• X and s are parameters of the Normal
  distribution.
• X is the true mean of the distribution.
• The RMS width of the distribution is s.
• x is the independent variable.
• P is the probability density to measure x.
• The combination of 10 uniform distributions is
  almost identical to a Normal Distribution.

What are the units of P(x)?
41
Histograms of Measurements

• Divide x axis into bins of equal size.
• Plot the number of entries in each bin.
• Expect statistical fluctuations for small numbers
  of measurements.
• For large numbers of measurements, the
  distribution should approach P(x).
• Usually P(x) is close to a normal distribution.

42
Histograms of Measurements

• As we accumulate measurements, we may fill in a
  histogram.
• Comparing with the normal distribution is often
  useful.
• Plot your histograms by hand.
• A typical error on the number of entries in a bin
  is the square root of the number of entries.

43
Histogram Re Measurements
n

• We choose bins of about one half sigma that are
  easy to use.
• Plot normal distribution (with measured mean and
  sigma) for reference.

Measured Re
Do you think any of the measurements should be
dropped?
Is this histogram consistent with a normal
distribution?
44
Probability Integrals of Normal Distribution
Probability Table from Taylor probability to be
within t standard deviations of the mean.
What is the probability to be more than 1 s off
the mean?
The normal dist. cannot be integrated
analytically. We must use tables or numerical
functions (Prob., Error function). We will use
this for rejection of data and calculation of
confidence levels.
45
2BL Student Survey
From the home office in La Jolla CA
From the home office in La Jolla CA
Top 10 Reasons 2BL Students Don't Like Physics
Top 10 Reasons 2BL Students Like Physics
1) Its difficult 2) The math 3) Too much
derivations 4) Too much work 5) Labs 6)
Homework 7) No, I'm serious, I like everything
about it. 8) But I do like physics. 9) Very
confusing lectures 10) Unorganized lab
instruction
1) Its interesting 2) Its logical 3) Like the
math 4) Interesting, good profs 5) Practical
6) Its reality 7) Hands on experiments 8) Its
better than Chemistry 9) Allows me to maintain
4.0 GPA 10) Quizzes are cake
46
More on Prob
What was the probability to get an A before grade
inflation?
47
Example Confidence Level
A student measures g, the acceleration of
gravity, repeatedly and carefully, and gets a
final answer of 9.5 m/s2 with a standard
deviation of 0.1 m/s2. If his measurement were
normally distributed, with a center at the
accepted value of 9.8 and with sigma 0.1, what is
the probability of getting an answer that differs
from 9.8 by as much as (or more than) his result.
Its three standard deviations off the mean.
Looking up the probability,
we see that 99.73 are within 3 sigma, so, the
probability is 0.27.
48
Example Confidence Level

• The Confidence Level is the probability to get a
  worse result than you measured.
• What is the probability to be further off the
  correct radius of the earth than was our average
  shown on the right?

If we histogram the CL from many good
experiments, what would the distribution look
like?
What are the units of t?

• Looking in table A for 1.07, we read 71.54.
• This is the probability to be less than 1.07
  sigma away so the C.L. is 100 - 71.54 28.46.

49
Example Confidence Level

• Two students measure the radius of a planet.
  Student A gets R9000 km and estimates an error
  of s 600 km. Student B gets R6000 km with an
  error of s 1000 km.
• What is the probability that the two
  measurements would disagree by more than this
  (given the error estimates)?
• Define the quantity q RA-RB 3000 km. The
  expected q is zero.
• Use propagation of errors to determine the
  error on q.

• Compute t the number of standard deviations
  from the expected q.

• Now we look at Table A. 98.95 should be
  within 2.56 s. So the probability to get a worse
  result is 1.05. We call this the Confidence
  Level, and this is a bad one.

50
Rejection of Data

• Rejecting data in an unwarranted fashion can bias
  your measurements.
• If there is suspicion of a mistake, data should
  be rejected without looking at the value
  measured.
• If only the measured value is suspicious, we
  should develop a prescription for data rejection.
  We will use one called Chauvenets Criterion.
• Data are rejected if we expect less than 0.5
  measurements with a deviation from the mean as
  large or larger than the one in question.
• The criterion should be reapplied after the worst
  case is rejected.

What would you call a follower of Chauvenet?
51
Example Chauvenets Criterion
A student makes 14 measurements of the period of
a torsion pendulum. She gets the following
measurements, all with the same estimated error.
T 2.7, 2.3, 2.9, 2.3, 2.6, 2.9, 2.8, 2.7, 2.8,
3.2, 2.5, 2.9, 2.9, and 2.3 Should any of these
measurements be dropped?

• Add up all the periods and divide by 14 to get
  the average, T2.7 seconds.
• Compute the standard deviation from the data,
  s0.27 seconds.
• The measurement furthest from the mean is 3.2
  seconds giving t0.5/0.271.85.
• Look up the probability to be further off,
  P6.43.
• Multiply by the number of trials to get the
  expected number of events that far off,
  nexp(14)(0.0643)0.9
• Do not drop this measurement (or any other).

Assume the student made a 15th measurement but
her partner bumped the pendulum during the
measurement. She got a period of 2.8 seconds.
Should she drop this measurement?
52
Student Data on Re

• Find the measurement furthest from the mean.
• Calculate the number of sigma its off.

• Look put the probability to be 2.14 or more sigma
  off. Its 100-96.763.24.
• Compute the number of events expected more than
  2.14 sigma away.

Keep all measurements
53
Reminder of Lab Procedures

• You need 2 quadrile ruled lab books.
• Always record your data in the lab book, (not
  scrap paper) as you work. Never erase data.
• Work with exactly one partner. (One group of 3
  is permitted in a lab.)
• Do not use any data from other partnerships.
  Measure everything yourself.
• Do not fudge data.
• Put calculations and lab report in your lab book.
• Make your report complete yet concise.

54
Experiment 2

• Devise a simple, fast, and non-destructive method
  to measure the variation in thickness of the
  shell of a large number of racquet balls in
  shipments arriving at a number of stores, to
  determine if the variation in thickness is much
  less than 10.
• Devise a method to measure the thickness and
  density of the inner and outer cylinders without
  damaging them so that rods outside 5 tolerance
  will not be used in a machine.

!
Relative
Enough accuracy with better speed
Absolute
best accuracy
55
Whats the Point

• This is an experiment about using repeated
  measurements to determine the accuracy of a
  measurement technique.
• Experimental methods can be modified and improved
  in light of the result of repeated measurements.
• We should learn to use averages to improve the
  accuracy of our results.

56
Racquet Balls
The counterfeiters make balls with the same mass
and the same average moment of inertia, I, but
have worse quality control on the thickness, d,
and hence on I. We are looking for a larger
spread in d implying a spread in I.
57
The Rods
The rods have materials of two densities. The
radii can be measured with a caliper. The total
mass can be measured. What we are checking is
the density. We want our measurement to be
accurate, even if it is time consuming.
58
Basic Strategy

• Measure a time t.
• Repeat measurements to determine error on t.
• Calculate the Moment of Inertia, I, from t.
• Propagate error on t to get error on I.
• Calculate the desire quantity q (thickness or
  density) from I.
• Propagate error on I to get error on q.
• The actual value of q is not important except as
  a check.
• Determine how many repeated measurements are
  needed to get the desired accuracy.
• Which method is better (rods) or faster (balls).
• Try out your method.

Why is the value of q not too important?
59
Moments of Inertia

• Both problems can be solved by measuring the mass
  and moment of inertia of the objects.
• For the balls, we only need to measure the
  variation in thickness but do it quickly.
• For the rods, we need absolute measurements but
  have more time.

We want R-r to much less than 10.
We want both densities to 5.
You should propagate errors to these quantities
in your proposal.
60
Two Methods of Measurement

• Roll object down incline and measure time.
• Deduce I using conservation of energy.
• Photogate timers will work well.
• Cylinder may be difficult to roll straight for a
  long distance. Perhaps a short distance will do.
• Use Torsion pendulum and measure period.
• Deduce I from T and M.
• Stopwatch and/or timers should be employed.
• Can the balls be done as rapidly as with the
  rolling technique? Accuracy is important to
  reduce repeated measurements.

What will limit the accuracy of each method?
61
Rolling Objects
How can you control the effect of nonzero x1?
Energy conservation.
Rolling radius.
For uniform acceleration.
Plug.
Play.
Solve for I.
62
The Torsion Pendulum
Torque equation gives diff. eq. in q.
Compute the period. T.
Solve for I.

• Calibrate the restoring torque constant k using a
  solid cylinder for which I can be computed.
• Minimize the wobble of the pendulum since this
  couples it to other modes and changes the period.

Will this work for the ball?
What will cause wobble?
When can we retie the string?
How will k depend on the length?
63
Repeating Measurements

• The errors on rolling time and pendulum period
  will likely be bigger than the smallest division
  on your timer.
• You will need to measure repeatedly to find out
  what the error is.
• You may also have to measure repeatedly to reduce
  the random error.

Measure times or periods many times (using the
same object) to find the width of the error
distribution.
How many times should you measure to get s to
30?
64
Propagating Errors

• In both methods, we measure a time and compute
  the moment of inertia.
• The Moment of Inertia I is a useful intermediate
  quantity, attainable in either method.
• We have computed I, but have specified accuracy
  needed on thickness and density.
• We need to compute thickness and density and
  their errors.
• Do it numerically for the ball. Dont try to
  solve 5th order equations, even on your fancy
  calculator.
• Plot the equations, on paper, by hand, and try to
  understand what you are doing.
• Lecture notes should be helpful.

65
The Balls

• We are only trying to find differences between
  balls, therefore, many errors can be ignored.
• Only the measured rolling time or pendulum period
  are important.
• We must measure one ball many times to determine
  the measurement error.
• We must measure many balls of each type
  experimentally determine the spread in thickness.
• Propagate error on I into error on thickness.
• There are physical limits on I that can play
  havoc on your calculations.

Now calculate the spread in time due to ball
manufacture by subtracting the measurement error.

66
Balls How Well Must We Measure

• We must estimate the error on the thickness.
• I measured the thickness to be d4.5 mm and
  radius to be R28.25 mm for one ball.

Can you use this 6.826 for your error analysis?
To get a 10 error on the thickness, we need
1.46 on the moment of inertia.
Remember fractional error for square.
To get a 1.46 error on I, we need a 0.73 error
on the period of the torsion pendulum or about
0.36 on the rolling time (see next slide).
67
Computing the Moment of Inertia
Fifth order equation is best dealt with
numerically.
0.841
How would solving this equation for z(I) help
you?
68
Propagate Error from Time to I
From previous page
Compute derivative
Can you explain this big factor in the error?
Propagate error
Work out fractional error numerically.
We must measure t with a fractional error half of
that needed for the period of the torsion
pendulum.
69
The Rods

• We need an absolute measurement of the cylinder
  radii and densities so all errors are important.
• Measure R, r, (Mm).
• Measure I by torsion pendulum or rolling.
• Minimize wobble and/or rolling error to get
  consistent measurement of I.

70
Consider Both Rolling and Pendulum

• Determine how well you can measure the rolling
  time and the pendulum period by repeating the
  measurements.
• Propagate the error and determine which methods
  meet the accuracy requirement and how many repeat
  measurements are needed to do it.
• After practicing the measurement techniques,
  determine how quickly a technician could perform
  the required measurement.
• Describe the method that you designed.

71
Try Out Your Method

• Choose your preferred methods for balls and rods.
• Now, measure the spread in thickness for a large
  enough group of balls.
• Check a group of rods to see if they are within
  tolerances.
• Do you need to modify your method?

How many balls do you need to try the method?
How many rods?
72
Summary of Error Propagation
73
Experiment 3
Construct and test a critical damping system for
a spring.

• A shock absorber damps oscillations of springs.
• If overdamped, hard jolts are transmitted to the
  mass and recovery is very slow.
• If underdamped, the spring will go through many
  oscillations before returning to equilibrium.
• If the damping is just right, we call it
  critically damped. It reduces shocks and returns
  to equilibrium as quickly as possible.

74
Whats the Point
The point of this experiment is to design
something using mechanical systems to solve a
problem. There are experimental difficulties to
overcome. You need to understand damped
oscillators. Overall, this is a fairly easy
experiment. I dont much like it but the
engineers tend to.
75
The Damped Oscillator
Gravity changes equilibrium position but nothing
else.
Write the differential equation in y and solve it
using an exponential function.
What would you adjust to optimize the ride of a
car?
damped oscillator frequency
76
The Equipment

• Use two different springs.
• Mass can be varied by hanging weights.
• Plexiglas cylinder used for damping.
• Need a just right fit. Adjust using tape.
  Need accuracy to about one half inch of tape.
• Keep it clean.
• Valve (plus holes) adjustment of air flow out of
  cylinder.

77
Damping Depends on Air Flow
Dominated by air flow
Dominated by friction
Is there any reason to repeat measurements here?
The valve is good for fine adjustment.
78
Construct and Test a Critical Damping System

• Measure the spring constant k.
• Compute the damping coefficient b needed for
  critical damping.
• Use terminal velocity measurements to determine b
  as a function of the number of holes covered and
  valve position.

• Test Fbv assumption.
• Test spring plus shock absorber and optimize.

79
Terminal Velocity

• Terminal velocity reached when drag force equals
  force of gravity.
• We use this to measure b.

80
Checking the Oscillator

• Good taping and a clean cylinder is crucial to
  good operation for terminal velocity and
  oscillation.
• Critical damping means no real oscillation but so
  does overdamping.
• Critical damping will have the smallest b with no
  oscillation.
• There may be a small jump at the end of the
  oscillation even for critical damping.

81
The Principle of Maximum Likelihood
Is L a Probability?
The best estimate Parameters of P(x) are those
that maximize L.
Why does max L give the best estimate?
82
Using the Principle of Maximum Likelihood
Assume X is a parameter of P(x). When L is
maximum, we must have
Lets assume a Normal error distribution and find
the formula for the best value for X.
Q.E.D. the mean
Under what conditions is min c2 the same as max
L?
83
Fitting Data

• We often fit data using a maximum likelihood
  method to determine parameters.
• For errors which are Normal, we can just minimize
  the c2.

• We will use this method to fit straight lines to
  data.
• More complex functions can be fit the same way.

To determine a straight line we need 2
parameters. How do we get 2 equations?
84
What is the Error on the Mean
Formula for mean of measurements. (Weve shown
that this is the best estimate of the true x.)
Now (simply) use propagation of errors to get the
error on the mean.
What would you do if the xi had different errors?
We got the error on the mean simply by
propagating errors.
85
Weighted Averages
We can use maximum Likelihood (c2) to average
measurements with different errors.
We derive the result that
From error propagation, we can determine the
error on the weighted mean.
What does this give in the limit where all errors
are equal?
86
Example Weighted Average
Suppose 2 students measure the radius of Neptune.
Student A gets r80 Mm with an error of 10 Mm
and student B gets r60 Mm with an error of 3 Mm.
What is the best estimate of the true radius?
What does this tell you about the importance of
error estimates?
87
Another Way to Compute s
88
Least Squares Fit to a Line
Assume all points have the same error and that
errors are normal.
Follow notation of book.
89
Deducing Errors

• As in the case of experiment 2, sometimes we
  cant estimate the errors too well beforehand.
• In that case we compared many measurements to the
  mean and computed the RMS.
• In this case we will compare to the line.

Note that now we use (n-2) because we have fit 2
parameters.
Where did we use (n-1)?
90
Motivation for the (n-2)

• Averages (1 parameter)
• no deviations from one point
• reduced deviations from two points

• Straight lines (2 parameters)
• no deviations from two points
• reduced deviations form three points

91
What About Errors on x

• In most fits, we have y errors only.
• If we also have errors in x, then we have to
  adjust the y errors to include the effects of
  errors on x.

This is not used in conjunction with the
determination of sy as shown on the previous
slide, since it will already be included in that
method.
92
Deriving the Parameters
93
Experiment 4
Construct a device to measure the absolute value
of a voltage through the measurement of a force.
The actual measurements you will make will be of
mass, distance, and time but the result will be a
measurement of an electric potential in Volts.

• Measure voltage difference with a standard meter.
• Measure force by deflection.
• We can calibrate the voltmeter.

94
Review the Basic Equations
This is voltage difference between r and
infinity. To get the voltage difference between
the two point charges, use superposition.
What problems will we have with this design?
95
The Parallel Plate Capacitor
We suggest the use of a parallel plate capacitor
rather than charged spheres.
The weight of 0.1 g.
96
Calibrate a Voltmeter

• Set up the apparatus.
• Keep table dry.
• Strip wire well.
• Measure the spacer.
• Measure k .
• Make the plates parallel for spacer in contact.
• Find Voltage that just causes plates to move
  apart.
• Try calibration at about 1000 Volts.
• Now get several measurements at lower voltage.
• Water must be stable.
• Move slowly.
• Protect your apparatus from air currents.
• How well could you check systematic differences
  in voltmeters this way?

97
Experimental Technique

• It is hard to get the plates stuck together
  because of the jolt you create when you let go.
• Because of the small forces involved, the
  apparatus is very sensitive to
• flow in the water
• air currents
• vibrations
• We can get these to a minimum but we cant
  eliminate them.
• Note that the spacer goes in and out of contact
  just before the voltage is lowered to point of
  release.

Can you measure the effect of vibrations?
98
Equilibrium Positions
Spacer can balance negative force when in contact.
99
The Chi-square Test

• We have used c2 minimization to fit data.
• We can also use the value of c2 to determine if
  the data fit the hypothesis.
• On average, the c2 value is about one per degree
  of freedom.

• The number of degrees of freedom is the number of
  measurements minus the number of fit parameters.

• We will use the c2 per degree of freedom to
  compute a probability that the data are
  consistent with the hypothesis. (table D)
• This probability of c2 is like the confidence
  level.

100
Probability of Chi-square
101
Some Examples

• If we use m measurements to take a (weighted)
  average, then compute the c2 based on the
  deviations from the average, we have fit one
  parameter. (ndofm-1) This is the origin of the
  (n-1) in our calculation of s.
• For the straight line fit we have two parameters.
  This is the origin of the (n-2) in computing s
  for the line.
• If we fit a parabola or Gaussian there are 3
  parameters.

102
c2 Example
103
Review

• RMS errors
• Propagation of errors
• Adding errors in quadrature
• Averages
• Histograms
• Probability distributions
• Normal distribution
• Confidence Levels
• Chauvenets criterion
• Principle of Maximum Likelihood

• Weighted averages
• Best fit straight line
• Chi-square
• Degrees of freedom
• Probability of chi-square

104
Exam Problems
123
Secret number
Bring your calculator.
Errors to one digit implies n-1 not important.
105
Exam Problems
106
Exam Problems
The 1 CL corresponds to 2.58s
The standard ball has an average time of 1.490
seconds. Balls with a CL higher than 1 should
have
So the balls with a moment of inertia in
agreement with the standard ball at the 1 CL or
better have times 1.489, 1.491, and 1.493.
107
Exam Problems
108
Exam Problems
Turn page for solution.
109
Solution
110
Exam Problem
111
Data Sheet
112
Exam Problems
113
Exam Problems
114
How Often do Killer Asteroids Hit Earth