CMS - Preparing for Physics at Fermilab

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Last two Lectures - Overview

• Todays Lecture
• Introduction of Experiment 4
• Reminders from last lecture
• c2 Test of a distribution
• Next Week (and last) Lecture
• Issues in Experiment 4
• Covariance and Correlation
• Problems and examples before final

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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 quarter
• The reports are due at the lab, during that week!

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End of quarter logistics

• Quiz 4 will be given this week
• Review Session will be held on
• Sunday, Nov 27th ,i.e. the day before the
  final.
• Final
• Two weeks from today, same time/place
• Monday, Nov 28th, 1900. Solis 107 (this room).

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The Four Experiments

• Determine the average density of the earth
• Weigh 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 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
• gtCompare and calibrate the voltmeter

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Review the Basic Physics
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The Parallel Plate Capacitor
We use a parallel plate capacitor rather than
charged spheres
Plug in some numbers to get a feeling for it
The weight of 0.1 g.
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How do we measure an attraction force F 0.1 g?
Electrical contact to torsion fiber
Adjustment lever
Fixed capacitor plate
Torsion fiber
Moving capacitor plate with a spacer
Mirror for optical lever
Balance plate
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- Distance from the suspension to the disk is
measured with a ruler
- Deflection angle is measured with a protractor
How do we measure the torsion constant ?
Torsional oscillations
- Moment of inertia
Disks of radius R
You want to weigh the support beam and disks
separately
Error Propagation
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Experimental Technique - Equilibrium
1. Adjust the fiber so that plates just touch
each other (with spacer) at zero voltage 2. Apply
voltage between the plates 3. Increase torque
from the fiber by twisting the top end of the
fiber and determine the twisting angle that just
causes plates to move apart
Stable or un-stable?
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Angle/torque adjuster
Torsion fiber
hanger
Mirror
Power supply (battery)
Damping bath
Capacitor plates
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Electric connector
Rotating support for torsion fiber
Protractor
Lever to adjust angle
Torsion fiber
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Hanger
Damping water bath
Aluminum capacitor plate
Brass disk capacitor plate
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Hanger
Mirror with laser beam
Damping water bath
Capacitor
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Assembling Testing the calibrator - step by
step (i)

• Attach mirror to the support beam and set up the
  laser to project a spot on the wall.
• Bring the torsion pendulum to equilibrium so that
  it is not moving.
• - Mark the position of the laser spot on the
  wall with a piece of tape.
• Measure the period of the oscillations by
  watching the spot on the wall and then calculate
  the torsion constant of the fiber.
• Add damping to the system to limit unwanted
  oscillations (water bath)
• Bring the pendulum to equilibrium and place the
  fixed capacitor plate parallel to the moveable
  plate, just barely touching the insulating dot.

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Assembling Testing the calibrator - step by
step (ii)

• Apply 1000 volts across the capacitor.
• The plates will clamp together.
• Read initial angle
• Apply a known torque (in the direction to pull
  the plates apart) by rotating the torque
  adjustment lever, until plates separate
• Read final angle
• Compute Angle difference- ??
• Compute the voltage using
• Repeat several times (at different voltages
  600-1000 V)
• 5 voltages
• Do above procedure at least 3 times

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Analysis

• Make a graph of your data where
• x-axis is the voltage read from the power supply
  (600-1000V)
• y-axis is the calculated voltage from the
  torsional pendulum
• Fit to straight line
• Calculate c2
• Discuss goodness of fit
• Calculate probability of result.

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Experimental Hints

• 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

Water must be stable. Move slowly. Protect your
apparatus from air currents. And your partners
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c2 ?Test
Recap from last Lecture
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The Chi-Squared Test for a distribution

• You take N measurements of some parameter x which
  you believe should be distributed in a certain
  way (e.g., based on some hypothesis).
• You divide them into n bins (k1,2,...,n) and
  count the number of observations that fall into
  each bin (Ok).
• You also calculate the expected number of
  measurements (Ek), in the same bins, based on
  some hypothesis.
• Calculate
• If ?2ltn, then the agreement between the observed
  and expected distributions is acceptable.
• If ?2gtgtn, there is significant disagreement.

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Degrees of Freedom

• Number of degrees of freedom, d number of
  observations, Ok, minus the number of parameters
  computed from the data and used in the
  calculation.
• dn-c,
• Where c is the number of parameters that were
  calculated in order to compute the expected
  numbers, Ek.
• It can be shown that the expected average value
  of ?2 is d.
• Therefore, we defined reduced chi-squared
• If reduced chi-squared is lt1, there is no reason
  to doubt the expected distribution.

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Example Application of c2 Test
Die is tossed 600 times
Expectation each face has same likelihood of
showing up
v 1 2 3 4 5
6 _
Verification of expectation by computing the
obs 91 137 111 87 80 94
c2
exp 100 100 100 100 100 100

D2
81 1369 121 169 400 36
s
10 10 10 10 10 10
c
2
This term is the squared difference between
observation and expectation. In computation of
c2 the D2 term is divided by expectation. s is
square root of expectation (Ey sy2)
0.81 13.7 1.21 1.69 4.0 0.36
i
Total
c2
21.76
ndof 5
c2
reduced
4.35

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Application of c2 Test Usage of Table D
Just calculated Total ?c2 21.77 ndof
5 Reduced ?c2 4.35
Die is loaded at 99.9 Confidence Level
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Final Exam
You can use Your calculator Your textbook Your
notes Principal Formulas and Tables
No Laptop computers

• On class web site - 2bl.ucsd.edu
• Lecture notes
• Experiment Guidelines
• - Homework solutions

Bring your own calculator!! and make sure its
charged
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Next Lecture

• Discuss issues in Experiment 4
• Covariance and Correlation
• Problems and examples before final