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2nd Pre-Lab Quiz

3rd Pre-Lab Quiz

4th Pre-Lab Quiz

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Today’s Lecture:

• Issues from last week lab sections ?

• Logistics

• Introduction to experiment #2– Strategy– Non-destructive measurement of variation– Discussion of data Interpretation & Error Analysis

• Compatibility of measured results– T-score– Confidence Level

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Logistics

• During Thanksgiving week A05,A07,A08 need to be rescheduled: Make-up sessions on Monday Nov. 21st Thanksgiving week is the last lab of the quarterThe reports are due at the lab, during that week!

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The Four Experiments

• Determine the average density of the earthWeigh the Earth, Measure its volume

– Measure simple things like lengths and times– Learn to estimate and propagate errors

• Non-Destructive measurements of densities, inner structure of objects– Absolute measurements Vs. Measurements of variability– Measure moments of inertia– Use repeated measurements to reduce random errors

• Construct, and tune a shock absorber– Adjust performance of a mechanical system– Demonstrate critical damping of your shock absorber

• Measure coulomb force and calibrate a voltmeter.– Reduce systematic errors in a precise measurement.

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Experiment 2

• Perform a simple, fast, and non-destructive experiment 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%.

Sufficient accuracy High speed

The problem can be solved by measuring the mass and moment of inertia of the balls.

Note, we only need to measure the variation in thickness

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Racquet Balls

d

R

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.

R

r

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Reminder- What is “moment of inertia?”

• Moment of inertia (I), also called mass moment of inertia or the angular mass, (SI units kg m2), is the rotational analog of mass. That is, it is the inertia of a rigid rotating body with respect to its rotation.

• Moment of Inertia is a measure of resistance to change of angular velocity.

• Where– m is the mass,– and r is the (perpendicular) distance of the point mass to the

axis of rotation.

Slide stolen from Mike Riley

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moment of inertia for spheres and cylinders

• Based on dimensional analysis alone, the moment of inertia of a non‐point object must take the form:

Where:– M is the mass– R is the radius of the object from the center of mass (in some cases, the

length of the object is used instead.)– k is a dimensionless constant that varies with the object in

consideration.• Inertial constants are used to account for the differences in the

placement of the mass from the center of rotation. Examples include:– k = 1, thin ring or thin‐walled cylinder around its center– k = 2/5, solid sphere around its center– k = 1/2, solid cylinder or disk around its center

Slide stolen from Mike Riley

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What’s the message?

• 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.

• Learn to distinguish between sources of uncertainties.Error Propagation is not a mathematical exercise!

need to look, analyze and understand the data to do these

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Basic Strategy - Overview

• Measure a “time” t.– Repeat measurements to determine error on t.

•– Propagate error on t to get error on I.

•– Propagate error on I to get error on q.– The actual value of q is not interesting in its own right … … it matters only for comparison between racket balls.

>> Determine how many repeated measurements are needed to get the desired accuracy.

• Calculate the desired quantity q (thickness) from I.• Calculate the Moment of Inertia, I, from t.

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Rolling ball

R

w

Parameters you need to measure once and use for all tests:

- distance, x, between the starting point and the 2nd photo-gate;- distance, x1, between the starting point and the 1st photo-gate;

- height difference, h, between the two photo-gates:- width of the groove, w, on the rail.

“Rolling radius” of the ball, R’, is different from the actual radius of the ball, R.

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Energy conservation.

Rolling radius.

Newtonian Mechanics for uniform acceleration.

v

v/2

Plug.

How to use the measured parameters (time and geometry) to calculate I?

Solve for I, we get:

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Repeating Measurements• The errors on rolling time 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 then use the repeated measurements to reduce the random error on mean time.

Mean time for one ball after n measurements, t1, t2, …, tn, -

our best estimate of the actual time.

Standard deviation, SD, – a measure of scatter of the individual data points, t1, t2, …, tn, and our

estimate of error of an individual measurement

Standard deviation (error) of the mean (SDOM) – our estimate of error of the calculated mean value, based on the random uncertainties of a large number of individual measurements.It can be reduced to minimum by repeating the measurements.

How many times should you measure to get the SDOM to 30% (0.3) of the SD?

Key is reproducibility!!!

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Measuring Variation in Balls

• 1. Measure rolling time of one ball many times to determine the measurement error in t, measurement

• 2. Measure rolling time of many, different balls to determine the total spread in t, total

• 3. Calculate the spread in time due to ball manufacture, manufacture, by subtracting the measurement error

• 4. Propagate error on t into error on I and then into error on thickness d

variation in t variation in I variation in dt I d

Next week

Relating <t> to the thickness

• We now know how to analyze the timing of the balls.• We next need to relate this to the thickness of the

balls.• We need this in order to understand how a 10%

variation in thickness (as claimed by the manufacturer) turns into a X% variation in <t>. I.e. we need to determine X here.

• We will find that the math involved is too painful to do analytically.

• We will thus learn how to do this numerically.

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“Normalized” moment of inertia

We define “normalized” moment of inertia:

We measured the moment of inertia of the ball(s), I, Defined as:

And finally:

Side-Bar:• We measure a time and compute the moment of inertia.• The Moment of Inertia is a useful intermediate quantity.• We have computed I, but have specified accuracy needed on thickness• We need to compute thickness and its associated error.• Do it numerically for the ball

• Don’t try to solve the 5th order equation…

•

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We measure and want to know, how much the error of influences the error of

Typical value for the balls are d = 4.5 mm and R = 28.25 mm, r = 23.75 mm:

Let’s take d = 4.4 mm; we obtain for it r = 23.85 mm and

We have and

To get the error of d down to 5%, we need to know with a precision of 0.75%

That is for the fractional errors

Propagate Error from I to d - Numerically

Make a small “perturbation”…Make a small “perturbation”…

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We needed to propagate errors for a complex function

The numerical method we used was to perturb the argument near a known value, to calculate the change in the function and the relative values of the two. It can be applied in general and is exactly the same as calculating:

We needed to know and plug in the actual approximate values of d and R.

Propagate Error from I to d - Summary

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Propagate Error from Time to I

We have got and

We want to evaluate in terms of (because it is t that we measure)

Go straight to:

Finally, we obtain:

The error analysis indicates that to obtain d with a precision of 10%, we need a precision of ~0.4% in t.

Multiply by t/t &Rearrange…Multiply by t/t &Rearrange…

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We are only trying to find differences between balls, therefore, many errors can be ignored.

Only the measured rolling times are important.

We must measure one ball many times to determine the measurement accuracy.

We must measure many balls of each type experimentally to determine the spread in thickness.

Propagate error on I into error on thickness.

In practical terms:as long as the setup and the intended ball starting point do not change, only random errors of photo-gates time are important.

To make a conclusion, whether the thicknesses of different balls are uniform within 10%, it is desirable to measure individual shell thicknesses to ~1% error.

==> An error of 3% in ball thickness, d, implies that error of average time, t, as small as 0.04%!

Summary

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Continuing from last lecture…

Quick reminder…

T-scoreCompatibility of measurementsConfidence level

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=

X -

X +

= 0.68

Gauss distribution: the meaning of

The area under a segment from X -

to X+ accounts for 68% of the total area under the bell-shaped curve.

That is, 68% of the measured points fall within from the best estimate

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tt

What about the probabilities to find a point within 0.5 from X, 1.7 from X, or in general t

from X ? To find those probabilities we need to calculate

Unfortunately, we cannot do it analytically and have to look it up in a table

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t = 1.47

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Compatibility of a measured resultt-score

• Best estimate of x:

• Compare with expected answer xexp and compute t-score:

• This is the number of standard deviations that xbest differs from xexp.

• Therefore, the probability of obtaining an answer that differs from xexp by t or more standard deviations is:

Prob(outside t) = 1-Prob(within t)

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“Acceptability” of a measured resultConventions

< 5 % - significant discrepancy, t > 1.96

< 1 % - highly significant discrepancy, t > 2.58

boundary for unreasonable improbability

erf(t) – error function

If the discrepancy is beyond the chosen boundary for unreasonable improbability, ==> the theory and the measurement are incompatible (at the stated level)

• Large probability means likely outcome and hence reasonablediscrepancy.• “reasonable” is a matter of convention…

• We define:

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Example: Confidence LevelTwo students measure the radius of a planet. • Student A gets R=9000 km and estimates an error of = 600 km • Student B gets R=6000 km with an error of =1000 km• What is the probability that the two measurements would disagree by more than this (given the error estimates)?==> Define the discrepancy 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 observed standard deviations from the expected value of q:

• Now we look at Table A ==> 2.56 corresponds to 98.95% So, The probability to get a worse result is 1.05% (=100-98.95)We call this the Confidence Level, and this is a bad one.

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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 the measurements were normally distributed, with a center at the accepted value of 9.8, 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 required probability is 0.27%.

Slide stolen from Jim Branson

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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 the measured value ?

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%.

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Example: Confidence Level (from one of your colleagues)

• 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 the measured value ?

• Looking in table A for 6.74, we read 99.999%.

• This is the probability to be more than 6 sigma away so the C.L. is 100% - 99.9999% = very very small number.

Try it with an uncertainty of 2000km

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The Normal Distribution - Example

The grades of students in a course where found to be normally distributed with a mean of 80 points and standard deviation of 5 points. There were 275 students in the course. How many students would you expect to have scores:

a) Between 75 and 85 points?b) Higher than 90 points?c) Between 70 and 90 points?d) Bellow 65 Points?

Solution:a) This grades range is: +/- 1 therefore, 68% of the scores should be within this range

275*0.68 = 187 students.b) Grades higher than 90 points are +2 and above the average. Therefore:

c) Scores between 70-90 points are within +/-2 of the mean. Therefore 95.45% of scores are expected to be within this range:

d) Scores bellow 65 points are more than -3 bellow the average score. We know that +/-3 is expected to contain 99.7% of the scores. The negative tail (outside this range) is expected to contain half as many:

Note: one sided ---> 0.5

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Next Week’s Lecture:

• Issues from last week’s lab-sections

• Rejection of Data

• The Principle of Maximum Likelihood– The weighted average– Introduction to fitting

Read Taylor Chap 7