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

USING MATHEMATICA FOR EDUCATION IN PHYSICAL SCIENCES

Volna Marie, Latal Frantisek, Kainzova Veronika

Department of Experimental Physics, Faculty of Science, Palacký University Olomouc

Outline of the presentation

1. Palacky University (Faculty of Science) and Moravian University College Olomouc

2. Project Mathscience

3. Teaching sw Mathematica at Faculty of Science

4. Examples of experimental measurements and processing data in Mathematica

Measurement of gravity acceleration with the help of Kater’s reversible pendulum

Projectile motion

Palacky University in Olomouc, Faculty of Science

Characteristics

Palacky University is a state University consisting of 8 faculties

now more than 2,500 students in B.Sc. and M.Sc. programs at the departments of the Faculty of Science

provides professional education in mathematics, informatics, physics, chemistry, biology, geography, geology and ecology

trains also 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

Moravian University College Olomouc

Characteristics

private college of higher education

provides a Bachelor study program, which can be studied either in full-time (daily) or part-time (distance learning) regime

provides its graduates not only with theoretical knowledge but also with practical skills (practical training opportunities, environment experiences, cooperation with private companies, public sector)

MUCO was accredited with the 3-year Bachelor study program „Economics and Management” in 3 different fields

5 departments, our department - Department of Informatics and Aplied Mathematics

guaranteed disciplines

Priorities of MUCO in Science and Research

Participation in innovation processes, focusing on

scientific research

regional projects

cooperation with Palacky University in Olomouc

European Social Fund projects

Teaching sw Mathematica at Palacky University

Winter term in 1st year of study - seminar First introducing of Mathematica for students (2 hours)

Summer term in 1st year of study - a subject: Software Mathematica

Sylabus - A first Course in Mathematica (solving equation, creating plots, derivation, etc.)

Free License for the participants

Homeworks - Moodle

Report from the course laboratory fo Mechanics

Experimental measurement No1: 
Measurement of acceleration of gravity with the help of Kater’s reversible pendulum

Description of the Kater’s reversible pendulum

A Kater’s pendulum is a reversible freeswinging pendulum invented by British physicist and army captain Henry Kater in 1817 to measure the local acceleration of gravity.
The advantage of the Kater’s pendulum is that the pendulum’s centre of gravity and center of oscillation don’t have to be determined, allowing greater accuracy.

The pendulum consists of a rigid metal bar with two pivot points, one near each end of the bar.
It also has either an adjustable weight that can be moved up and down the bar to adjust the periods of swing.

Figure 2. The Kater’s reversible pendulum

In use, it is swung from one pivot, and the students measure the period, and then turned upside down and swung from the other pivot and measure the period. 
The movable weight is adjusted until the two periods are equal. 
At this point the period is equal to the period of an ‘ideal’ simple pendulum of length equal to the distance between the pivots. 
From the period and the measured distance between the pivots, the acceleration of gravity can be calculated with great precision.
For the calculation the acceleration of gravity we can use the equation for a simple pendulum (1).

T=2Π(Sqrt[(L/g)])

So by measuring the length L and period T of a pendulum, g can be calculated, using the equation:

g=((4(π^2)L)/(T^2)).

The experimental method for discovering the acceleration of gravity

1. Adjust 6 different position of the weight and measure the periods when the weight is up and down.
2. From the data built a graph which shows the dependence period on the distance of the weight. 
3. From the graph determine the distance, when the period of oscilation is equal in both of the position of the pendulum.
3. Adjust the distance of the weight and measure the period with the weight up and down.
4. Calculate the acceleration of the gravity

1. 	Adjust 6 different positions of the weight and measure the periods of swinging

The student measure the periods of swinging pendulum in 6 different positions.

In one positions, they measure the periods two times;  when the weight is up and down. In all positions they measure the periods 10 times.

The pedulum has to swing with the deviation 5° from the state of equilibrium to obtain the most accurate value.

The periods were measured by photogate and wrote in text file.

The data from text file were imported as one folder into sw Mathematica.

Data: Positions of the weight, Periods

SetDirectory[NotebookDirectory[]]

Positions={12.65, 11.885, 10.56, 9.10, 8.14,6.96};

DATA=Import["DATA.txt","Data"];

Dimensions[DATA]

Down1=Table[DATA[[i,1]],{i,15}];
Down2=Table[DATA[[i,2]],{i,15}];
Down3=Table[DATA[[i,3]],{i,15}];
Down4=Table[DATA[[i,4]],{i,15}];
Down5=Table[DATA[[i,5]],{i,15}];
Down6=Table[DATA[[i,6]],{i,15}];
Up1=Table[DATA[[i,7]],{i,15}];
Up2=Table[DATA[[i,8]],{i,15}];
Up3=Table[DATA[[i,9]],{i,15}];
Up4=Table[DATA[[i,10]],{i,15}];
Up5=Table[DATA[[i,11]],{i,15}];
Up6=Table[DATA[[i,12]],{i,15}];

datadown={{Down1, Down2, Down3,Down4, Down5, Down6}};

dataup={{ Up1, Up2, Up3, Up4, Up5, Up6}};

rowHeadings2={""};
colHeadings2={"Position 1 ","Position 2", "Position 3", "Position 4", "Position 5", "Position 6" };

TableForm[datadown,TableHeadings->{rowHeadings2,colHeadings2}]

TableForm[dataup,TableHeadings->{rowHeadings2,colHeadings2}]

2.	Built a graph

From the data Postions and measured periods, they built a graph to find the positions of weight where the periods will be equal in both of the position of the pendulum

For the graph dependence period on the distance  is needed;

calculate the average of the periods

create two lists of periods data : ListWeightDown and ListWeightUp.

AverageDown={Mean[Down1], Mean[Down2],Mean[Down3],Mean[Down4],Mean[Down5],Mean[Down6]};

AverageUP={Mean[Up1], Mean[Up2],Mean[Up3],Mean[Up4],Mean[Up5],Mean[Up6]};

ListWeightDown={{6.96, 2.16771},{8.14,2.15916},{9.1, 2.15236},{10.56, 2.13958}, {11.885, 2.13274},{12.65, 2.12759}};

ListWeightUp={{6.96, 2.25965},{8.14,2.20792},{9.1, 2.17556},{10.56, 2.12344},{11.885, 2.07803},{12.65, 2.05681}};

ListLinePlot[{ListWeightDown, ListWeightUp}, PlotRange->{{6,13},{2,2.6}},  PlotStyle->{{Red,Thick},{Orange,Thick}}, 
FrameLabel->{ "Distance (cm)", "Period (s)"},LabelStyle->Directive[20, Bold], ImageSize->800 , PlotLegends->{"weight up","weight down"}, Frame->True, GridLines->Automatic, GridLinesStyle->Dashed]

With the help of function Get Coordinates,  the value for equal periods is found.

ListLinePlot[{ListWeightDown, ListWeightUp}, PlotRange->{{9.9,10.05},{2.14,2.15}},  PlotStyle->{{Red,Thin},{Orange,Thin}}, 
FrameLabel->{ "Distance (cm)", "Period (s)"},LabelStyle->Directive[16, Bold], ImageSize->800 , PlotLabel->Style["Focused part of the Graph", FontSize->20 ],Frame->True, GridLines->Automatic, GridLinesStyle->Dashed,  PlotLegends->{"weight up","weight down"}]

distanceANDperiod={{9.961,2.145}};

3.	Adjust the distance (Subscript[a, 0]) and measure the period with the weight up and down.

Adjust the weight at distacce 9.96 cm and measure the period with the weight up and down.

10 periods of the pendulum are measured together by stopwatch.

dataperiodUp={21.47,21.41,21.30,21.47,21.31,21.54,21.32,21.30,21.50,21.50};

dataperionDown={21.41,21.50,21.44,21.40,21.44,21.44,21.59,21.47,21.47,21.50};

Data of the distance between two pivot points has to be find and calculate the average.

L={1.146, 1.146, 1.145, 1.145, 1.145, 1.145, 1.146, 1.145, 1.145, 1.146};

(OverBar[L])=Mean[L]

Time1=dataperiodUp/10;
Time2=dataperionDown/10;

TimeDown=Mean[Time1]
TimeUp=Mean[Time2]

For calculating acceleration of  gravity, the mean value of the Time 1 and Time 2 is used.

(OverBar[T])=((TimeDown+TimeUp)/2)

Finally, the acceleration of the gravity is calculated by the equation:

(OverBar[g])=((4 (π^2)(OverBar[L]))/((OverBar[T])^2))

4.	Calculate the measurement error

Students have to calculate how accurately calculated the acceleration of gravity.

The error of measurement is a error of indirect measurement. The gravity acceleration is function af two direct measured quantity, period and lenght.

g=g(l, T)

The measurement error is

σg=g(Sqrt[(((((σL)/(OverBar[L]))))^2)+(((((2σT)/(OverBar[T]))))^2)])

σL=MeanDeviation[L]/(Sqrt[10])

σT1=MeanDeviation[Time1]/(Sqrt[10])
σT2=MeanDeviation[Time2]/(Sqrt[10])

σT=(((Subscript[σT, 1])+(Subscript[σT, 2]))/2)

δg=(OverBar[g]) σg

σL=MeanDeviation[L]/(Sqrt[10])
σT1=MeanDeviation[Time1]/(Sqrt[10])
σT2=MeanDeviation[Time2]/(Sqrt[10])

σT=((σT2+σT1)/2)
δg=(OverBar[g]) σg

σg=(OverBar[g])(Sqrt[(((((σL)/(OverBar[L]))))^2)+(((((2σT)/(OverBar[T]))))^2)])

gravityAcceleration =((OverBar[g])±σg)

One example: Projectile motion

One of the experiments that our students must conduct in the laboratory is experiment projectile motion.
The projectile motion is an example of curved motion with constant acceleration. It is two dimensional motion of a particle thrown obliquely near the earth’s surface, and it moves along a curved path under the action of gravity only.
For this demonstration we have lab equipment from PHYWE.

Principle of experiment
A steel ball is fired by a spring at different velocities and at different angles.

Problems (tasks)
1.	To determine the range as a function of the angle of inclination.
2.	To determine the maximum height of projection as a function of the angle of inclination.

Theory
If we want to express the position coordinates as a function of time t, we must use the equations:

x(t) = (Subscript[v, 0])·cosϕ ·t

y(t) = (Subscript[v, 0])·sinϕ·t-(1/2)(gt^2) 		(1)

From this, the maximum height of projection h is obtained as a function of angle of projection ϕ:

h = ((Subscript[v, 0]^2)/(2g))·(sin^2)ϕ 			(2)

and the maximum distance s is:

s= ((Subscript[v, 0]^2)/g)·sin2ϕ			(3)

Real experiments

In traditional laboratory measurements students change the initial velocity of the projectile and change the angle of firing a projectile that ranges from 0 to 90 degrees.
In this experiment, students can only choose three different initial velocities ((Subscript[v, 01])= 2.52((m )/s), (Subscript[v, 02])= 3.55((m )/s), (Subscript[v, 03])= 5.25((m )/s)).

Plot123a:=ListPlot[Table[{k,((2.52^2)/9.81) *Sin[2*k*Pi/180]},{k,0,90,1}],AxesLabel->{"ϕ [∘]","s[m]"},PlotStyle->{Red}]

RealData123a:=ListPlot[{{0,0.00},{10,0.20},{20,0.43},{30,0.56},{40,0.62},{45,0.64},{50,0.62},{60,0.55},{70,0.41},{80,0.21},{90,0.00}},PlotStyle->Directive[PointSize[Large],Black]]

Plot123b:=ListPlot[Table[{k,((3.55^2)/9.81) *Sin[2*k*Pi/180]},{k,0,90,1}],AxesLabel->{"ϕ [∘]","s[m]"},PlotStyle->{Blue}]

RealData123a:=ListPlot[{{0,0.00},{10,0.20},{20,0.43},{30,0.56},{40,0.62},{45,0.64},{50,0.62},{60,0.55},{70,0.41},{80,0.21},{90,0.00}},PlotStyle->Directive[PointSize[Large],Black]]

RealData123b:=ListPlot[{{0,0.00},{10,0.42},{20,0.85},{30,1.10},{40,1.25},{45,1.29},{50,1.25},{60,1.11},{70,0.85},{80,0.45},{90,0.00}},PlotStyle->Directive[PointSize[Large],Black]]

Plot123c:=ListPlot[Table[{k,((5.25^2)/9.81) *Sin[2*k*Pi/180]},{k,0,90,1}],AxesLabel->{"ϕ [∘]","s[m]"},PlotStyle->{Green}]

RealData123c:=ListPlot[{{0,0.00},{10,0.92},{20,1.79},{30,2.40},{40,2.72},{45,2.82},{50,2.73},{60,2.41},{70,1.78},{80,0.91},{90,0.00}},PlotStyle->Directive[PointSize[Large],Black]]

Show[{Plot123a,Plot123b,Plot123c,RealData123a,RealData123b,RealData123c},PlotRange->Automatic,GridLines->{{0,10,20, 30, 40, 50, 60, 70, 80, 90},Automatic}]

Using software Mathematica students can determine maximum height of projection h as a function of the angle of inclination ϕ for the initial velocities.

Plotha:=ListPlot[Table[{k,((2.52^2)/(2*9.81)) *(Sin[k*Pi/180])^2},{k,0,90,1}],AxesLabel->{"ϕ [∘]","h[m]"},PlotStyle->{Red}]

Plothb:=ListPlot[Table[{k,((3.55^2)/(2*9.81)) *(Sin[k*Pi/180])^2},{k,0,90,1}],AxesLabel->{"ϕ [∘]","h[m]"},PlotStyle->{Blue}]

Plothc:=ListPlot[Table[{k,((5.25^2)/(2*9.81)) *(Sin[k*Pi/180])^2},{k,0,90,1}],AxesLabel->{"ϕ [∘]","h[m]"},PlotStyle->{Green}]

Show[{Plotha,Plothb,Plothc},PlotRange->Automatic,GridLines->{{0,10,20, 30, 40, 50, 60, 70, 80, 90},Automatic}]

Virtual experiments

Because our students have license software Mathematica, they can combine real measured data with virtual models.

Manipulate[
Module[
{g=9.81,x,y},
v0Max=20.;
α=(Pi/2)-(deg*Pi/180);
HMax=10;
y=H+Tan[α] x-(g/(2(v0^2) ((Cos[α])^2))) (x^2);
xRange=v0 Cos[α] ((v0 Sin[α]+(Sqrt[(v0^2) ((Sin[α])^2)+2 g H ]))/g);
xMax=(v0Max/g)(Sqrt[(v0Max^2)+2 HMax g]);
yMax=HMax+(((v0Max^2) )/(2g));
αOptimum=(180/Pi)*ArcSin[(Sqrt[((v0^2)/(2((v0^2)+H g)))])];
ar=(yMax/xMax);
Cliff=Graphics[{Black,Rectangle[{-1,-1},{0,H}]}];
yaxis=Graphics[{Black,Dotted, Line[{{0,H},{0,H+5}}],Line[{{0,H},{ar *8*Cos[α],H+8Sin[α]}}]}];
anglelabel=Graphics[{Black,
Circle[{0,H},4,{α+.05,Pi/2}],Text[Style["α",FontSize->14],{ar*3*Cos[1/2(α+Pi/2)],H+ar*3* Sin[1/2(α+Pi/2)]}]}];
Sea=Graphics[{ColorData["HTML","Green"],Rectangle[{-1,-1},{xMax,0.0}]}];
v0Vector=Graphics[{Thick,RGBColor[1,.47,0],Arrow[{{0,H},{(v0/3)ar Cos[α],H+(v0/3) Sin[α]}}]}];
Show[Plot[y,{x,0.000,xMax},
PlotRange->{{-0.5,xMax},{-0.5,yMax}},Frame->True,FrameLabel->{Row[{Style["x",Italic]," (m)"}],Row[{Style["y",Italic]," (m)"}]},
PlotLabel->Row[{"α = ",NumberForm[N[90-α/Degree],{5,2}],"°    |  ",Row[{(Subscript[Style["v",Italic], 0])}]," = ",NumberForm[N[v0],{5,1}]," m/s    |  ",Style["H",Italic]," = ",NumberForm[N[H],{5,2}]," m    |   optimum α = ",NumberForm[N[(90)-αOptimum],{5,2}],"°  |   range = ",NumberForm[N[xRange],{5,4},ExponentFunction->(Null&)]," m"}],ImageSize->500],Sea,Cliff,yaxis,anglelabel,v0Vector]],
{{deg,90-αOptimum,"angle α in degrees"},3,90,Appearance->"Labeled"},{{v0,v0Max,"(Subscript[v, 0])"},1/1000,v0Max},{{H,HMax},0,HMax},TrackedSymbols:>Manipulate]

THANK YOU FOR YOUR ATTENTION