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1st Pre-Lab Quiz
2nd Pre-Lab Quiz
3rd Pre-Lab Quiz
4th Pre-Lab Quiz
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Today’s Lecture:
• Brief logistics
• Experiment #1– Intro– Procedural comments– Experimental considerations
• Error Propagation:– Recap from last lecture– Intuitive view– Useful hints for experiment 1
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• Please consult course web page regularly.– This is where we post all relevant announcements– It is our ONLY way to communicate with you!
2BL.UCSD.EDU– Your lab sections and TAs schedule on web page
• Important:– Bring printouts of guidelines to Lab– Be on-time.
• Take quiz• Go to beach
Logistics etc…
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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 1: Measure Density of Earth
• Need to know: Mass, Volume– <Density> = Mass/Volume
• Volume: assume earth is a sphere (99.8% accurate)• Mass: acceleration of gravity on Earth’s surface (Galileo’s
experiment’s)• Two measurements:
– (a) Earth’s Radius Re - Watch a sunset– (b) Local acceleration of gravity g - Swing a pendulum
(a) Radius of Earth, direct measurements of: Length, Time, Angle.– Deduce: Radius of Earth
(b) Mass of Earth, direct measurements of: Length, Time.– Deduce: Mass of earth– Use Newton’s constant G = 6.67 × 10-11 N m2/kg2
==> Calculate average density
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Experiment 1: Basics
What is the radius of the earth?
What is the value of g?
Local g
Gravitational force
Measuring g & Re
==> Solve for
( M )
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What Element(s) make up the Earth• Assume most of earth’s
volume is one element.
• Assume earth is a sphere
• Need to know: Mass, Volume
Densities (g/cm3)
Which of these can be eliminated as the dominant constituent in the earth’s composition?
–Look it up!
–10% measurement needed to determine composition.
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Measure Earth’s Radius using t Sunset
Is this time delay measurable?
Sunset on earth’s surface (tangent)
Sunset at height h
How do we convert the distance to the horizon line, L, into the sunset time delay, t ?
Therefore,
The length of the earth’s
circumference, corresponds
to a time delay of
- height above the sea levelFrom right triangle
- distance to the horizon line
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Measure Earth’s Radius using t Sunset
Sunset on earth’s surface (tangent)
Sunset at height h
Looks doable!Have we forgotten something?
- our cliff
- height above the sea levelNow, is this time delay measurable?
- distance to the horizon line
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s
winter view
Correct for Latitude and Earth’s Axis
La Jolla latitude
Solar latitude varies.d = days since March 20 (or Sept 22).
This formula accounts for our latitude and for the This formula accounts for our latitude and for the angle of the earthangle of the earth’’s axis from the plane of its orbit.s axis from the plane of its orbit.
Set your mode to Rad as appropriate when using your calculators…
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“The Equation” for Experiment 1a
from previous page.
Time difference between the two sunset observers.
Season dependant factor slightly greater than 1.
Which are the variables that contribute to the error significantly?
What other methods could we use to measure the radius of the earth?
The formula for your error analysis..
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Procedural Comments
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Measuring the Height of the Cliff
h1
l
h1=l cos()
• Work in two groups: – Half on the cliff and – Half on the beach
• The group on the cliff will measure l, using a laser range finder.
• The group on the beach will measure . Note that it is measured relative to gravity!
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Your Height Above Sea Level on Beach
h2
• The formula we derived is for height above sea level.• The experimenter on the beach also views the sunset from
above sea level (at least her height…)• Check the error propagation for the sensitivity of the
measurement of the earth’s radius to the h2 measurement!
• h2 must be measured at the same approximate time as t.– Remember: tides change the sea level by 1 to 2 meters!
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Cliffs West of Muir Campus (not to scale)
Access is easy but wear walking shoes.
It may be cold in the evening.
h2
h1
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What phase of sunset would make the best reference point?
-Sun touching the horizon? -One half below the horizon?-Completely disappeared?
•Sunset is later than most lab sections --> you will have to return to observe and measure the sunset times.
•When you go to measure the time of sunset, go in groups larger than two and remember to:
• Synchronize your watches before you split• Agree on the exact phase of sunset you are timing!
Weather is important:Need a clear day!
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Next need to measure g
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Simple Pendulum - Reminder• Measure g by constructing a simple pendulum and measure its period, T.
• A simple pendulum is a mass on the end of a mass-less, perfectly flexible string, suspended from a rigid support.
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Measuring g with a Pendulum
Can we solve this differential equation?
Looks rather tough… Let’s simplify.
Harmonic oscillation with frequency ,
and period
when is smallSince:
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Experimental Considerations
What assumptions have been made?
• m is a point mass, so that l is measured from the point of suspension to the point mass. (The center of mass will be a good approximation to a point mass.)
• The mass of the “string” is negligible relative to m. (We can ensure this by choosing the correct material for the string and sufficient mass for m).
• The angle is small
• Rigid, stable support.
Your TAs will point these out to you during Lab
Important: Start calculations at home, before next week’s lab!
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Quick recap from last lecture…
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Error Propagation - SumWhat is the perimeter of this figure?
y
wz
xp = w + x + y + z
1. Estimate errors from x, y, z, w. - They all are likely to be on the order of precision of the ruler, 1/32” or ~0.75 mm.
2. Propagate these to compute the error on p.
How would you calculate the error on p?
You measure x, y, z, w and compute p.
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Error Propagation - Sum (cont)
y
wz
xp = w + x + y + zWe estimated the errors on x, y, z, and w as:
What is our estimate of error on p?
Since
But, since x, y, z and w are all error estimates ==> we do not know their signs! Therefore, we could do:
Worst case, if all have same sign!
We would normally use the rule of addition in quadrature:
Independent, random errors.
Consider:
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Another Example of Error Propagation - Product
• Take x, y and q as xbest, ybest, qbest • Measure (x+x) and (y+y)• Compute (q+q)
Subtract q from both sides of the above equation.(Notice partial derivatives)
(neglect xy)
q
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Another Example of Error Propagation - Product
Here again, we don’t know the signs.Sometimes these contributions will cancel, sometimes add up.
We can compute errors two ways:1) Maximum possible error
2) If the uncertainties x, y are independent & random:
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Fractional Errors
For products like q=xy, we can add the fractional errors on the measurements (x/x) to get the fractional error on the result (q/q) :
Simple Derivation
This also works for ratios like .
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Example
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Summary
Uncertainties in Sums and Differences:
Uncertainties in Products and Ratios:
General Rule:For independent random errors
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Back to Error Propagation…
30xbest
xbest - x xbest + x
qbest = q(xbest)
q(xbest – x)
q(xbest + x)
Error Propagation
31xbest
Error Propagation
xbest - x xbest + x
qbest = q(xbest)
q(xbest – x)
q(xbest + x)
q = q(xbest+x) – q(xbest)
q depends on:1) x2) q(x)
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q = q(xbest+x) – q(x)
Error Propagation - Why Partial Derivatives?
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Error propagation
For any function:
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Propagating Errors for Experiment 1
Formula for density.
Take partial derivatives and add errors in quadrature
shorthand notation for quadratic sum:
Or, in terms of relative uncertainties:
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Propagating Errors for Re
basic formula
Propagate errors (use shorthand for addition in quadrature)
Note that the error blows up at h1=h2 and at h2=0.
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hl
==> Find h.
always use radians when calculating the errors on trig functions
Given that:
Example
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Next Lecture
• Issues from last week
• Statistical Analysis:• Histograms and Distributions• The Limiting Distribution
• Introducing the Normal Distribution
Read Taylor Chap 5.