1. 1 4 th Pre-Lab Quiz

2. 2 Last two Lectures - Overview Today’s Lecture: Introduction of Experiment #4 Reminders from last lecture: –  2 Test of a distribution  Next Week (and last) Lecture: –Issues in Experiment #4 –Covariance and Correlation –Problems and examples before final

3. 3 Logistics During Thanksgiving week A05,A07,A08 need to be rescheduled: Make-up sessions on Monday Nov. 21 st Thanksgiving week is the last lab of the quarter The reports are due at the lab, during that week!

4. 4 End of quarter logistics Quiz #4 will be given this week Review Session will be held on: Sunday, Nov 27 th,i.e. the day before the final. Final: Two weeks from today, same time/place: Monday, Nov 28 th, 19:00. Solis 107 (this room).

5. 5 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.

6. 6 Experiment 4 Construct a device to measure the absolute value of a voltage through the measurement of a force Measure voltage difference with a standard meter Measure force by deflection =>Compare and calibrate the voltmeter 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 force and voltage. Q-Q V

7. 7 Review the Basic Physics Force between point charges - Coulomb ’ s low Permittivity constant Q-Q Electric field from a point charge Q 1 Voltage differenceVoltage - potential energy per unit charge Coulomb force acting on a charge Q 2

8. 8 The Parallel Plate Capacitor We use a parallel plate capacitor rather than charged spheres The weight of 0.1 g. from Gauss ’ s Law voltage difference Attraction force Plug in some numbers to get a feeling for it:

9. 9 Adjustment lever Balance plate Mirror for optical lever Torsion fiber Electrical contact to torsion fiber Fixed capacitor plate Moving capacitor plate with a spacer How do we measure an attraction force F = 0.1 g? - Torsion constant - Deflection angle

10. 10 Torsional oscillations - 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 ? Disks of radius R You want to weigh the support beam and disks separately - Moment of inertia Error Propagation…

11. 11 Experimental Technique - Equilibrium Electrostatic attraction: 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 At twisting angle that just causes plates to move apart: l balance and damping plate Force applied by torsion balance: Stable or un-stable? Equilibrium Solve:

12. 12 Power supply (battery) Capacitor plates Damping bath Mirror Torsion fiber Angle/torque adjuster hanger

13. 13 Lever to adjust angle Rotating support for torsion fiber Torsion fiber Protractor Electric connector

14. 14 Brass disk capacitor plate Hanger Aluminum capacitor plate Damping water bath

15. 15 Capacitor Hanger Mirror with laser beam Damping water bath

16. 16 1.Attach mirror to the support beam and set up the laser to project a spot on the wall. 2.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. 3.Measure the period of the oscillations by watching the spot on the wall and then calculate the torsion constant of the fiber. 4.Add damping to the system to limit unwanted oscillations (water bath) 5.Bring the pendulum to equilibrium and place the fixed capacitor plate parallel to the moveable plate, just barely touching the insulating dot. Assembling & Testing the calibrator - step by step (i)

17. 17 6.Apply 1000 volts across the capacitor. -The plates will clamp together. -Read “initial” angle 7.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-  8.Compute the voltage using: 9.Repeat several times (at different voltages 600-1000 V) -5 voltages -Do above procedure at least 3 times Assembling & Testing the calibrator - step by step (ii)

18. 18 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   Discuss goodness of fit Calculate probability of result.

19. 19 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 can ’ t eliminate them Water must be stable. Move slowly. Protect your apparatus from air currents. And your partners…

20. 20    Test Recap from last Lecture

21. 21 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 (k=1,2,...,n) and count the number of observations that fall into each bin (O k ). You also calculate the expected number of measurements (E k ), in the same bins, based on some hypothesis. Calculate: If  2<n, then the agreement between the observed and expected distributions is acceptable. If  2>>n, there is significant disagreement.

22. 22 Degrees of Freedom Number of degrees of freedom, d = number of observations, O k, minus the number of parameters computed from the data and used in the calculation. d=n ‐ c, –Where c is the number of parameters that were calculated in order to compute the expected numbers, E k. –It can be shown that the expected average value of  2 is d. Therefore, we defined “reduced chi ‐ squared”: If reduced chi-squared is <1, there is no reason to doubt the expected distribution.

23. 23 This term is the squared difference between observation and expectation. In computation of   the   term is divided by expectation.  is square root of expectation (E y =  y 2 ) Example: Application of    Test v 1 2 3 4 5 6 _ obs 91 137 111 87 80 94 exp 100 100 100 100 100 100  81 1369 121 169 400 36   i 0.81 13.7 1.21 1.69 4.0 0.36 Total  21.76 n dof 5 reduced  4.35  Die is tossed 600 times Expectation: each face has same likelihood of showing up Verification of expectation by computing the   10 10 10

24. 24 Application of    Test: Usage of Table D Just calculated: Total   21.77 n dof 5 Reduced   4.35 Die is loaded at 99.9% Confidence Level

25. 25 You can use: Your calculator Your textbook Your notes Principal Formulas and Tables On class web site - 2bl.ucsd.edu - Lecture notes - Experiment Guidelines - Homework solutions Don ’ t forget to bring Final Exam Bring your own calculator!! and make sure its charged… No Laptop computers

26. 26 Next Lecture… Discuss issues in Experiment #4 –Covariance and Correlation –Problems and examples before final