The project is co-financed by the European social fund and the state budget of the Czech Republic

APPLICATION OF THE MATHEMATICA SOFTWARE FOR ANALYZING LARGE AMOUNTS OF DATA IN TEACHING PHYSICS

František Látal & Marie Volná

Department of Experimental Physics, Faculty of Science, Palacky University in Olomouc, Czech Republic

Stucture of the presentation

Palacky University

Project ICTEDUCA

Experimental system Vernier

Physical experiments and analyzing data in sw Mathematica

Palacky University

Palacky University

- one of the oldest universities in Central Europe,
	- 400-year tradition of higher education it is nowadays a renowned centre for teaching and research,
	- attract students from other European and overseas countries,
	- with almost 23,000 undergraduate students on eight faculties, it provides quality education and excellent background for research in the wide range of academic disciplines.

Faculty of Science

- more than 2,500 students in Bachelor and Master programme,
	- obtain proffesional education in mathematics, physics, biology, chemistry, geography, geology and ecology,
	- doctoral programmes at various departments  is available for selected graduates in Master’s programmes,
	- trains future secondary school teachers of sciences, mainly in full-time courses, which can be combined with courses offered by the Faculty of Physical Culture and the Philosophical Faculty.

Introduction - Project ICTEDUCA

Exact Title
	Modern ICT Tools in Fields of Science and Economy and Their Presentation
	
Institutions
	Palacky University in Olomouc, Faculty of Science and Moravian College Olomouc
	
Project Duration
	June 2009 - May 2012
	
Main Goals
	- to improve the training of science teachers and specialists using ICT tools,
	- to support students in presentations of their research results (multimedia classroom),
	- to provide students with high quality mathematical software,
	- to provide students a study text about mathematical software.
	
Target Group
	Students of the Faculty of Science - study programmes in various fields of mathematics, physics, chemistry, biology, geography
	Moravian college - students of economy

Using software Mathematica at Faculty of Science

The software Mathematica is used in several subject in Faculty of Science.

Thought the project ICTEDUCA, new subjects was created.

Presentation in Physics discipline, Biology, Chemistry, Geography, Matematics ,

Presentation in English,

Mathematical software in sciences.

acquaint with background of Mahtematica,

used in mathematical calculations,

work with vectors, matrix,

basic work with 2D plots,

command Plot and its parameters,

3D plots,

animations,

using Mathematica in physics tasks and analyzing data,

possibility of using in chemistry (specially modelation chemical compound) or  using in geography.

Mathematica is usig in other subject in our deparmatment, in subject Computers in physics teaching, Modern tools ICT in teaching, others subject.

Knowleges of sw Mathematica students use in subject Practical of school experiments, where they analyze experiments in sw Mathematica.

Laboratory measurement with Vernier

A dynamic way to teach science.

New school experimental system for elementary school, middle school and high school.

Senzors and probes for Biology, Chemistry, Physics and Engineering
LabQuest Interface System

accelerometer, motion detector, EKG senyor, Spirometer, Photogate, PH senzor

LabQuest is a standalone computer interface for Vernier sensors

colour touch screen, high capacity battery (for several hours of intensive operation),

weight 350 g, durable desigh and water resistan, use in temperature 0 - 70 °C

measurement outside

40 megabytes internal memory can be expanded using SD or USB flash drive

Connectors - 2 digital and 4 analog sensors for Vernier

Physical experiment 1

At our Department of Experimental Physics students must attend subjects, where they carry out physical experiments.

In the laboratory we have many different experiments, but recently we started using Vernier measuring system.

Data is collected using sensors by Vernier but processing and analysis of the results are carried out in software Mathematica.

For this presentation we have prepared five examples of usage of software Mathematica in our physics laboratory.

Import["C:\\Users\\Franta\\Desktop\\USA\\exp1\\f1.jpg", "Image"]

Description of the image: kettle with hot water, 1 - Thermo mug (keeps liquids either hot or cold) without cap, 2 - Thermo mug with cap,
3 - mug from porcelain, 4 - mug from glass.

Hot water (approximately heated at 90° C) was poured to each mug. Actual temperature for each mug was measured every 5 seconds.

This measurement took 20 minutes and at the end of the measurement we had approximately one thousand values (4 x 240)​​.


Task for students: Students measured only 20 minutes. What will be the temperature in each cup in one hour?

Import["C:\\Users\\Franta\\Desktop\\USA\\exp1\\T1.txt", "Elements"]

TEMP1 = Import["C:\\Users\\Franta\\Desktop\\USA\\exp1\\T1.txt", "Data"]

Obtained package involving data that need to be processed.
First, we must check the dimension and then if the data are in the form of numbers. Since, sometimes problems can appear when you import the data from Excel into Mathematica.

Dimensions[TEMP1]

MatrixQ[TEMP1]

MatrixQ[TEMP1,NumericQ]

The same procedure we did with other data for other mugs.

TEMP2 = Import["C:\\Users\\Franta\\Desktop\\USA\\exp1\\T2.txt", "Data"];

Dimensions[TEMP2]

MatrixQ[TEMP2]

MatrixQ[TEMP2,NumericQ]

TEMP3 = Import["C:\\Users\\Franta\\Desktop\\USA\\exp1\\T3.txt", "Data"];

Dimensions[TEMP3]

MatrixQ[TEMP3]

MatrixQ[TEMP3,NumericQ]

TEMP4 = Import["C:\\Users\\Franta\\Desktop\\USA\\exp1\\T4.txt", "Data"];

Dimensions[TEMP4]

MatrixQ[TEMP4]

MatrixQ[TEMP4,NumericQ]

Now it is necessary to analyze the measured data.

Chart1 = ListPlot[TEMP1, PlotMarkers->{Automatic,4},PlotStyle->Red,PlotRange->{{0,20},{50,95}},PlotLabel->StandardForm["Thermo mug without cap"],AxesLabel->{Time"(min)",Temperature "(°C)"}]

Chart2 = ListPlot[TEMP2, PlotMarkers->{Automatic,4},PlotStyle->Blue,PlotRange->{{0,20},{50,95}},PlotLabel->StandardForm["Thermo mug with cap"],AxesLabel->{Time"(min)",Temperature "(°C)"}]

Chart3 = ListPlot[TEMP3, PlotMarkers->{Automatic,4},PlotStyle->Green,PlotRange->{{0,20},{50,95}},PlotLabel->StandardForm["Mug from porcelain"],AxesLabel->{Time"(min)",Temperature "(°C)"}]

Chart4 = ListPlot[TEMP4, PlotMarkers->{Automatic,4},PlotStyle->Orange,PlotRange->{{0,20},{50,95}},PlotLabel->StandardForm["Mug from glass"],AxesLabel->{Time"(min)",Temperature "(°C)"}]

Show[ListPlot[TEMP1, PlotStyle->Red],ListPlot[TEMP2, PlotStyle->Blue], ListPlot[TEMP3, PlotStyle->Green], ListPlot[TEMP4, PlotStyle->Orange],PlotRange->{{0,20},{50,95}},PlotLabel->StandardForm["Blue - Thermo mug with cap, Red - Thermo mug without cap, Orange - Mug from glass, Green - Mug from porcelain"],AxesOrigin->{0,50},AxesLabel->{Time"(min)",Temperature "(°C)"}]

As follows, we wanted to find out what the temperature will be in each cup in one hour.

FindFit[TEMP1,a1 × (E^(-b1 × x)),{a1,b1},x]

a1 = 92.21316438071848;

b1 =0.018227171249700035;

Chart1Fit = Plot[a1 × (E^(-b1 × x)),{x,0,60}, PlotStyle->Red, PlotRange->{{0,60},{20,95}},PlotLabel->StandardForm["Thermo mug without cap"],AxesLabel->{Time"(min)",Temperature "(°C)"}]

The same procedure we did with other data for other mugs.

FindFit[TEMP2,a2 × (E^(-b2 × x)),{a2,b2},x]

a2 = 93.3631728777392;
b2 = 0.00802528559371246;

Chart2Fit = Plot[a2 × (E^(-b2 × x)),{x,0,60}, PlotStyle->Blue, PlotRange->{{0,60},{20,95}},PlotLabel->StandardForm["Thermo mug with cap"],AxesLabel->{Time"(min)",Temperature "(°C)"}]

FindFit[TEMP3,a3 × (E^(-b3 × x)),{a3,b3},x]

a3 = 84.31543232470034;

b3 = 0.019278586210833306;

Chart3Fit = Plot[a3 × (E^(-b3 × x)),{x,0,60}, PlotStyle->Green, PlotRange->{{0,60},{20,95}},PlotLabel->StandardForm["Mug from porcelain"],AxesLabel->{Time"(min)",Temperature "(°C)"}]

FindFit[TEMP4,a4 × (E^(-b4 × x)),{a4,b4},x]

a4 = 86.35951420080076;

b4 = 0.02119229561370266;

Chart4Fit = Plot[a4 × (E^(-b4 × x)),{x,0,60}, PlotStyle->Orange, PlotRange->{{0,60},{20,95}},PlotLabel->StandardForm["Mug from glass"],AxesLabel->{Time"(min)",Temperature "(°C)"}]

Show[{Chart1Fit, Chart2Fit,  Chart3Fit, Chart4Fit}, PlotLabel->StandardForm["Blue - Thermo mug with cap, Red - Thermo mug without cap, Orange - Mug from glass, Green - Mug from porcelain"]]

Physical experiment 2

Description of the image: (set) two very close but different signals from the datalogger and microphone sensor.

Sound waves are described by frequency, wavelength and amplitude.

When two waves of slightly different frequencies combine, they produce a wobbling sound called beats.

Tasks for students: Measure beats of sounds. Analyze measured data by software Mathematica.  Create models of beats.

Beats=Import["C:\\Users\\Franta\\Desktop\\USA\\exp2\\exp2.txt","Data"]

We measured 5 seconds with a frequency of 1000 samples per second, which means that 5000 values are presented in this table.

ListLinePlot[Beats]

We created beats with software Mathematica.

Two waves with close value of frequency.

Plot[Sin[880×Pi×x],{x,0,0.02},AxesLabel->{Time"(sec)",Amplitude}]

Plot[Sin[892×Pi×x],{x,0,0.02},AxesLabel->{Time"(sec)",Amplitude}]

Two waves with close value of frequency in one graph.

Plot[{Sin[880×Pi×x],Sin[892×Pi×x]},{x,0,0.02},AxesLabel->{Time"(sec)",Amplitude}]

By summing up two waves with slightly different frequencies we get beats.

Plot[Sin[2×440×Pi×x]+Sin[2×446×Pi×x],{x,0,1},AxesLabel->{Time"(sec)",Amplitude}]

Using the function Manipulate is demonstrated interactive changes of one frequency.

Manipulate[Plot[Sin[2×440×Pi×x]+Sin[2×a×Pi×x],{x,0,0.5}],{a,441,450}]

Physical experiment 3

Import["C:\\Users\\Franta\\Desktop\\USA\\exp3\\perioda.jpg", "Image"]

Description of the image: mathematical pendulum of length of 84,2 cm and photogate.

This measurement took 2 minutes and we measured the oscillation period of mathematical pendulum.


Tasks for students: Students had to calculate the acceleration of gravity in our laboratory using WolframAlpha.

PEN = Import["C:\\Users\\Franta\\Desktop\\USA\\exp3\\exp3.txt", "Data"]

Firstly we calculated the average oscillation period.

PEN1 = Mean[PEN]

freq = 1/1.8407666551724138

We found information about the mathematical pendulum on WolframAlpha.

WolframAlpha["small\[Hyphen]oscillation pendulum"]

WolframAlphaQueryResults

Physical experiment 4

Import["C:\\Users\\Franta\\Desktop\\USA\\exp4\\liss.jpg", "Image"]

Description of the image: A Blackburn pendulum is a device for illustrating harmonic motion. 

This pendulum oscillates (simultaneously) in two perpendicular directions with different periods.

Students can display curve using the Blackburn pendulum in the laboratory or they can create Lissajous curves using software Mathematica.

Lissajous curves are the family of curves described by the parametric equations.

ParametricPlot[{x =Sin [2*Pi*t + Pi/2],y = Sin [6*Pi*t ]},{t,0,2*Pi}]

Manipulate[ParametricPlot[{x =Sin [a*Pi*t + c],y = Sin [b*Pi*t ]},{t,0,2*Pi}],{a,1,3},{b,1,5}, {c,0,2*Pi}]

Physical experiment 5

Damped harmonic oscillator

Realize experiment by spring and weight, measured by LabQuest and senzor Vernier GoMotion

In the real world is valid a dampen force, that implicate decreased amplitude of oscillation.

These equations describe motion of damped harmonic oscillator

y=A×sin(ωt+φ)

A=(Subscript[A, 0])×(e^(-bt))
(Subscript[ω, t])=(Sqrt[(ω^2)-(b^2)])=(Sqrt[((((k/m)))^2)-(b^2)])
b is a damping coefficient.

Task for students

	- find coefficient b from measured data
	- modelating damped oscilation

Experiment data

Parameters of measurement: 1000 samples per a 1 minute
10 minuten ⟶10,000 data

Exp5=Import["D:\\WolframT\\Math\\exp5\\exp5position.txt","Data"]

Dimensions[Exp5]

MatrixQ[Exp5]

MatrixQ[Exp5,NumericQ]

The next graph is made from measured data.
It shows, how is the amplitude changed during the measurement

Chartexp5a=ListLinePlot[Exp5, AxesLabel->{Time"(min)", Amplitude"(m)"}, AxesOrigin->{0.,0.293}]

From previous graph students made new data. With drawing tools and get coordinate, students get new data of decreace amplitude.

mark several points - CTRL C, and CTRL V - to get the data measure 5

Measure5={{0.009518,0.3661},{0.03171,0.3656},{0.0502,0.3649},{0.06869,0.3649},{0.08718,0.3642},{0.1094,0.3641},{0.1316,0.364},{0.15,0.3635},{0.1648,0.363},{0.187,0.3626},{0.2092,0.362},{0.2314,0.3615},{0.2499,0.3611},{0.2684,0.3608},{0.2906,0.3606},{0.3091,0.3604},{0.3312,0.3596},{0.3497,0.3591},{0.3682,0.3589},{0.3904,0.3584},{0.4126,0.358},{0.4274,0.3577},{0.4496,0.3573},{0.4681,0.357},{0.4903,0.3564},{0.5125,0.3562},{0.5309,0.3559},{0.5494,0.3557},{0.5679,0.3557},{0.5938,0.3555},{0.6123,0.3552},{0.6382,0.3549},{0.6456,0.3545},{0.6678,0.354},{0.69,0.354},{0.7084,0.3539},{0.7343,0.3536},{0.7491,0.3531}}

Students created this plot from date Measure 5

Chart5exp5b=ListPlot[Measure5]

Then they fit data Measure 5, by means of function Find Fit and find damped coefficient

FindFit[Measure5, a5 Exp[b5 x],{a5,b5},x]

k5a={23.927,23.357,22.295}

Studnets determine the spring coefficient from experiment and calculate damped frequency to make a model

k5=Mean[k5a]

m5=0.8016

ω5=k5/m5

(Subscript[ω, t])=(Sqrt[28.9334-(0.0312594^2)])

Finally student modelate the damped oscillation by means of function Manipulate.

Manipulate[Plot[0.07295 (E^(-b5×t))×Sin[2×5.37889×t],{t,0,600}],{b5,0.001,0.01}]