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
? )(
or
?
environ
t
t
The Problem of Heating and Cooling: Newton‘s Law
eTemperatur talEnvironmen
Body theof Area Surface
tCoefficienTransfer Heat
eTemperatur Bodys
Capacity Heat Specific
Body tedcooled/hea theof Mass
Time
Body theofEnergy Thermal Internal
environ
environ
A
h
c
m
t
U
Ahdt
dcm
dt
dU
The Problem of Heating and Cooling: Newton‘s Law
cm
Ahk
e tk
with
Solution
initialenvironenviron
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
t
Equation Difference Finite a toalDifferenti a From
environ
environ
k
cm
AhΔ
AhΔt
Δcm
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“)
s 3.3
s3.0
1
1
Time-Response
k
The Problem of Heating and Cooling: Newton‘s Law
Evaporation Heat of Pentane: Exponential Regression
125HC
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
g
g
tωyy
tω
g
mgm
mI
gmM
gmM
IM
22
T Periodor
Frequency Angular with
cos or
cos :Solution
or
:Inertia ofMoment
: Angles smallFor
sin :Torque ofMoment
:law 2 snNewto
0
0
2
2
nd
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
3
Force Plate
Take off Forces during a Jump from the force plate (Vernier, Nspire)
Force Plate
Take off Forces: Calculation of Jumping height (Vernier)
g
dtgmF
g
vh
hgmvm
dtgmFΔvmΔp
t
t
22 :Jump theofHeight
2
1 :Energy Mechanical ofon Conservati
:(Impulse) MomentumLinear
2
02
2
0
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
cos
th wi
BAABΦ
dt
dΦU
m
mind
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
2
1
ind12
1t
t
dtUn
ΦΦ
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
hgmvm
dtgmFΔvmΔpt
2
0
2
1
:Energy Mechanical ofon Conservati
:(Impulse) MomentumLinear
g
dtgmF
g
vh
t
22
:Jump theofHeight 2
02
Workshop: Force Plate Calculations: Integral (Trapezoidal Rule)
Thank you for your attention
Top Related