Physics Experiments for Mathematical Education

Hans Kammer, Berner Maturitätsschule für Erwachsene, CH-3012 Berne, Switzerland

Motivation for dealing with empiric Data in mathematical Education

• real world experiments engage students • application in a practical field stimulates more than „grey theory“ alone• well selected applications show beginners the enormous practical importance of mathematics• managing great data volumes is a modern everyday application of mathematics (=>meteorology)• working with practical experiments from physics, biology, sports etc. shows the interdisciplinar character of mathematics• and first of all: calculator methods are simple!

Evaluation of experimental Data: Procedure

• measuring physical data with sensors and calculator

• guessing the mathematical function describing these data• fitting optically the measured data with this function varying its parameters with sliders or

• fitting by regression with a (limited) set of built in standard functions.

What should you learn for your instruction?

• Handling of at least one sensor e.g. Temperature• Data Acquisition with the TI-NspireCAS Handheld Calculator and Computer Software: Spreadsheet and Scatterplot in Graphs-Application• Enter a Function and fitting it to the measured data by settings ist parameters with sliders• automatic Data Regression with built in standard function• Expanding the Regression Technique with Parameters/Sliders

Subjects

• The Problem of Heating and Cooling: Newton‘s Law

• Harmonic Oscillations of a Pendulum

• Force Platform

• and more in the workshops …

The Problem of Heating and Cooling: Newton‘s Law

The Problem of Heating and Cooling: Newton‘s LawExperimental Setup

The Problem of Heating and Cooling: Newton‘s Law

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The Problem of Heating and Cooling: Newton‘s Law

The Problem of Heating and Cooling: Newton‘s Law

The Problem of Heating and Cooling: Newton‘s Law

The Problem of Heating and Cooling: Newton‘s Law

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The Problem of Heating and Cooling: Newton‘s Law

Calculation: Finite Difference Equation (Spreadsheet)

The Problem of Heating and Cooling: Newton‘s LawCalculation: Finite Difference Equation (Graph)

The Problem of Heating and Cooling: Newton‘s Law

Measurement: Response Time of an EasyTemp Sensor

The Problem of Heating and Cooling: Newton‘s Law

Measurement: Response Time („Optical Fitting“)

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The Problem of Heating and Cooling: Newton‘s Law

Evaporation Heat of Pentane: Exponential Regression

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The Problem of Heating and Cooling: Newton‘s LawEvaporation Heat of Pentane or Acetone: Experimental Setup

The Problem of Heating and Cooling: Newton‘s Law

Evaporation Heat of Pentane ( ): Regression Graph125HC

Calculations:Harmonic Oscillations of a Pendulum

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Harmonic Oscillations of a Pendulum

Construction

Workshop: PendulumExperimental Setup

Harmonic Oscillations of a Pendulum

Sensors: Distance (ulrasonic, contactless), Acceleration

Harmonic Oscillations of a Pendulum

Measurement: Distance, Graph

Harmonic Oscillations of a Pendulum

Measurement: Distance, Spreadsheet, Regression

Harmonic Oscillations of a Pendulum

Measurement: Velocity, Calculated from distance data

Harmonic Oscillations of a Pendulum

Measurement: Acceleration, Calculated from velocity

Force Plate (Clinic, Biomechanical Research, Sports)

3-d Professional Force Platform for Gait and Balance Analysis (Kistler 9285BA, piezoelectric)

Force Plate (Kistler 9285 BA)Force Measurements and Photograhic/CinematographicRecording of the Contact Surface from below

Clinical Motion- and Gait-Analysis (Rehabilitation and prothesis enhancement, Kistler)

Force Plate

Gait Parameters: Forces vs. time (Kistler)zyx and , FFF

Force Plate

Force PlateMotion and Gait Analysis (Kistler)

Force Plate

Take off Forces in Performace Diagnostics (Kistler)

Force PlateStrain Gages instead of Piezoelectric Sensors (Vernier)

Force Plate

School Application: Newton‘s Lawd

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Force Plate

Take off Forces during a Jump from the force plate (Vernier, Nspire)

Force Plate

Take off Forces: Calculation of Jumping height (Vernier)

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Force Plate

Jump from the Force Plate: Data Analysis (Vernier, Nspire)

And many other applications from natural and environmental sciences, sports, medicine …

Thank you for your attention

Physics Experiments for Mathematical Education:Workshop

Hans Kammer, Berner Maturitätsschule für Erwachsene, CH-3012 Berne, Switzerland

Subjects

• Discharging a Capacior with a Resistance• Evaporation Heat of Pentane• Pendulum: Distance and Acceleration• R-L-C-Oscillator (Oscillating Circuit)• Faradays Law of Induction • Electric Characteristics (Diode, Bulb)• Force Platform

Workshop: Discharging a CapacitorCharging and Discharging Circuit

Workshop: Discharging a CapacitorCapacitor, Resistor and 9-Volt Battery

Workshop: Discharging a CapacitorEasyLink Interface and Voltage Probe

Workshop: Discharging a CapacitorNspire Discharging Scatter-Plot and Regression-Graph

Workshop: Discharging a CapacitorCharging and Discharging Circuit

Workshop: Evaporation Heat of PentanePreparation of the Temperature Probe

The Problem of Heating and Cooling: Newton‘s LawEvaporation Heat of Pentane or Acetone: Experimental Setup

Workshop: Evaporation Heat of PentaneOptical Fit

Workshop: Evaporation Heat of PentaneRegression

Workshop: PendulumMotion and Acceleration Sensors

Workshop: PendulumMeasurement Procedure

Workshop: PendulumExperimental Setup

Workshop: PendulumMeasurement Scatter-Plot, Spreadsheet

Workshop: Oscillating CircuitStarting and Oscillating Circuit

Workshop: Oscillating CircuitExperimental Setup

Workshop: Oscillating CircuitNspire Voltage Measurement (EasyLink)

Workshop: Faraday’sLaw of Induction

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Workshop: Law of InductionMeasurement: Voltage Probe and EasyLink

Workshop: Law of InductionMeasurement: 50 Hz R-C-Ripple Filter (if necessary)

Workshop: Law of InductionExperimental Setup

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Workshop: Law of InductionMeasurement: Voltage Surge (Coil 23’000 Turns)

Workshop: Law of InductionOscillating Magnet (turn by 90° for Measurement)

Workshop: Law of InductionOscillating Magnet: Experimental Setup

Workshop: Law of InductionOscillating Magnet (Coil with 23’000 turns)

2 Oscillating Modes

Workshop: Characteristics (Bulb. Diode LED)Measuring Circuit: With 2 Nspires and 2 Voltage Probes

Workshop: Characteristics Measuring: Characteristics of an ideal and a real Bulb

Workshop: Characteristics Measuring: Characteristics of a LED (GaAS/GaP)

Workshop: Force Plate Measuring: Vernier Force Platfrm (with Strain Gauges)

Workshop: Force Plate Measuring: Nspire F vs. t Graph

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Workshop: Force Plate Calculations: Integral (Trapezoidal Rule)

Thank you for your attention