Lecture 2

Revision as of 18:11, 30 September 2016

1.) Review and revisit last lectures content Modeling (heuristics [wiki: are simple, efficient rules, learned or hard-coded by evolutionary processes, that have been proposed to explain how people make decisions, come to judgments, and solve problems typically when facing complex problems or incomplete information], use triangles to build a structure) is intrinsic to the way we think. Show examples from http://c2.com/cgi/wiki?HeuristicRule deciding to eat at restaurant B rather than restaurant A only because B has more cars in its parking lot When the level in the fuel tank drops to 1/2 tank or less, always filling up the fuel tank at the very next re-filling station.

Does your model of the way the world works match the data? Does sacrificing people lead to good weather (Mayan model)? Say you collected data which supported this model and tried to publish the data. Why would it be difficult to get it accepted? There is no scientific model of the universe that explains the results. Therefore people would have trouble believing it. We need a model to justify how the world behaves. For example, the blood vapor contains particulates upon which water condenses to form raindrops.

A current controversy involves a propulsion method that doesn’t emit anything. The model is that the group velocity is less on the cone side where reflection leads to a smaller momentum exchange. http://emdrive.com/ Note the fringe theory warnings on this site https://en.wikipedia.org/wiki/RF_resonant_cavity_thruster

Why isn't modelling a familiar topic in what you have studied at Mines? You spend most of your time learning about complicated mathematical models rather than thinking critically about these models. For example, what are the assumptions of the model and when are they valid?

In all lab expts I expect you to formulate a model and see if the data supports that model.

How do you make an argument that the model is valid? Does that sound like what you would hear in an english class where you are asked to make an argument about an interpretation of a novel. Does anyone want to give an example of what they had to do in an English or history class?

Example: You collect data which is presented in this graph

http://wwwold.mpiwg-berlin.mpg.de/projects/DeptII_Sepkoski_Extinction/Extinction.jpg Here is the source of the data https://www.mpiwg-berlin.mpg.de/en/research/projects/deptii_sepkoski_extinction

What do you think? What questions do you have?

By the end of this class I hope that the first question you have is what is the error in the data? Maybe it's just noise. How do you check if it’s noise? REPEAT the expt. However, you can’t do that in this case. How do determine the error? Use multiple ways of measuring the data to check it. How well you know the assumptions in each measurement method and their validity will determine your estimation of the error.

Assume that these extinction events are valid. What model is supported by the data? Periodic volcanic eruptions, asteroids that intersects the Earth's orbit periodically due to a Nemesis star or planet X orbiting every 27 million years, dark matter, etc. Would these models be accepted compared with the sacrifice model?

Some people think the data is valid but that no model is appropriate because the random events lead to this distribution. Then if you were to repeat the expt it would not look the same. This again focuses on our nature to come up with models of what we observe even if the data is random.

3.) Error analysis (see section 3.7 of Baird)

-Working eqn using e/m as an example.

-Add errors in quadrature. Review the focal length example.

-Do error of the product of two quantities. Z = xy. Use percent error.

-Do error of the sum Z = x + y

-What’s the error in the mean value of n voltage measurements? Vmean =Sum[Vi,{i,1,n}]/n. This is a sum. How does error in this sum depend on errors in the individual terms of the sum. Use the heuristic that to understand need to simplify the notation: use (V1+V2+V3+ )/n rather than sum notation. Assume dV1 = dV2 = dV3 . . . and the error in dV3/n is just due to dV3 and not n since n is an integer with no error.

delVmean=Sum[dVmean/dVi delVi, {i,1,n}]/n where dVmean/dV1 = 1/n, dVmean/dV2=1/n , etc. So, delVmean=Sum[delVi/n, {i,1,n}] . Assume delVi = delV is constant, that is every measurement of Vi has the same error. Adding the errors in quadrature we get delV=Sqrt[Sum[(delV/n)^2, {i,1,n}] but summing identical terms n times give a factor of n so delV=Sqrt[(delV)^2/n] = delV/Sqrt[n].

The error in the mean value is the error in an individual measurement divided by the square root of the number of measurements.

4.) Data visualization

You want to devise detectors to measure the energy of high energy particles (gamma rays etc).

-The first is an ionization device (charged cylindrical cap) which outputs a voltage spike (ac coupled) when a gamma rays creates a spark (avalanche breakdown) in the device. Sketch the device. Would the voltage output be linearly related to the energy of the incident particle? What model might describe the device? Turns out low energy particles are more likely to ionize the inert gas atoms. The spark is a good indicator that a particle has entered the detector but not of its energy.

-Next you use a material that emits visible light when the particle traverses it. Any idea how this works? This is like the e/m expt where the electron strikes multiple He atoms, each time exciting the atom which then fluoresces. However, in this case the number of photons emitted is related to the energy of the incident particle. A gamma ray gives its energy to an electron via the Compton effect and this electron travels through the material losing energy. How do you measure this flash of light? Use the photoelectric effect. The light free electrons from a surface and these are made to go through a resistor to measure a voltage. The more photons the more electrons the more voltage = IR.

The other is a bolometer. It measures the change in resistance when heat is absorbed. The first detector consisted of two strips of platinum covered with black paint. One was the sensor and the other a calibrator. A difference in resistance between these is measured with a bridge circuit. “By 1880, Langley's bolometer was refined enough to detect thermal radiation from a cow a quarter of a mile away,” (detects differences of 0.00001 C) from wikipedia. How would you make that measurement? (move the cow or put up a reflector for the IR).

The two devices measure high energy particles from the same sources, each of which emits a different energy particle, with the following tabular results for the detector voltage output. Your assignment is to come up with a way to visualize this data. Talk about sig. Figs. in the output not exceeding the error.

If you graph the two outputs vs particle energy on one graph can see the two are similarly calibration. Are the points on this graph within error. If you graph output 1 on the x-axis vs output 2 on the x-axis you would like it to fit within error a line at 45 degrees. If there is a curve upwards then the two outputs are not proportional. Which detector is more likely to be valid? How do you determine the repeatability (reliable)?