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 Law
Experimental Setup
The Problem of Heating and Cooling: Newton‘s Law
The Problem of Heating and Cooling: Newton‘s Law
 t   ?
or
environ   (t )  
?
The Problem of Heating and Cooling: Newton‘s Law

dU
d
 mc
 h  A  environ   
dt
dt
U
InternalT hermalEnergyof theBody
t
T ime
m
Mass of thecooled/heated Body
c
Specific Heat Capacity

Bodys T emperatur
e
h
Heat T ransferCoefficient
A
Surface Area of theBody
environ Environmental T emperature
The Problem of Heating and Cooling: Newton‘s Law
Solution
environ     environ  initial e
with
h A
k
mc
 k t
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
From a Differential to a FiniteDifferenceEquation
Δ
mc
 h  A  environ    
Δt
h A
Δ 
 environ     t
mc

k
The Problem of Heating and Cooling: Newton‘s Law
Calculation: Finite Difference Equation (Spreadsheet)
The Problem of Heating and Cooling: Newton‘s Law
Calculation: 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“)
Response- T ime
1

k
1

s
0.3
 3.3s
The Problem of Heating and Cooling: Newton‘s Law
Evaporation Heat of Pentane: Exponential Regression
C5H12
The Problem of Heating and Cooling: Newton‘s Law
Evaporation Heat of Pentane or Acetone: Experimental Setup
The Problem of Heating and Cooling: Newton‘s Law
Evaporation Heat of Pentane ( C5H12 ): Regression Graph
Harmonic Oscillations of a Pendulum
Calculations:
Newton s 2 law :
M  I  
nd
Momentof T orque:
M  m  g    sin 
For small Angles :
M  m  g   
Momentof Inertia:

or
Solution :
or
with Angular Frequency
or P eriod
I  m  2
m  g      m   2  
g      
   0  cosω  t 
y  y0  cosω  t 
g


T
2 

 2 

g
Harmonic Oscillations of a Pendulum
Construction
Workshop: Pendulum
Experimental 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/Cinematographic
Recording of the Contact Surface from below
Force Plate
Clinical Motion- and Gait-Analysis (Rehabilitation and
prothesis enhancement, Kistler)
Force Plate
Gait Parameters: Forces Fx , Fy and Fz vs. time (Kistler)
Force Plate
Motion and Gait Analysis (Kistler)
Force Plate
Take off Forces in Performace Diagnostics (Kistler)
Force Plate
Strain Gages instead of Piezoelectric Sensors (Vernier)
Force Plate
d
School Application: Newton‘s 3 Law
Force Plate
Take off Forces during a Jump from the force plate
(Vernier, Nspire)
Force Plate
Take off Forces: Calculation of Jumping height (Vernier)
t
LinearMomentum(Impulse):
Δp  m  Δv   F  m  g   dt
0
Conservation of Mechanical Energy:
Height of theJump :
1
 m  v 2  m  g  h
2
tF

    g   dt 
 
v 2  0  m
h

2 g
2 g
2
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 Capacitor
Charging and Discharging Circuit
Workshop: Discharging a Capacitor
Capacitor, Resistor and 9-Volt Battery
Workshop: Discharging a Capacitor
EasyLink Interface and Voltage Probe
Workshop: Discharging a Capacitor
Nspire Discharging Scatter-Plot and Regression-Graph
Workshop: Discharging a Capacitor
Charging and Discharging Circuit
Workshop: Evaporation Heat of Pentane
Preparation of the Temperature Probe
The Problem of Heating and Cooling: Newton‘s Law
Evaporation Heat of Pentane or Acetone: Experimental Setup
Workshop: Evaporation Heat of Pentane
Optical Fit
Workshop: Evaporation Heat of Pentane
Regression
Workshop: Pendulum
Motion and Acceleration Sensors
Workshop: Pendulum
Measurement Procedure
Workshop: Pendulum
Experimental Setup
Workshop: Pendulum
Measurement Scatter-Plot, Spreadsheet
Workshop: Oscillating Circuit
Starting and Oscillating Circuit
Workshop: Oscillating Circuit
Experimental Setup
Workshop: Oscillating Circuit
Nspire Voltage Measurement (EasyLink)
Workshop: Faraday’sLaw of Induction
dΦm
U ind  
with
dt
 
Φm  B  A  A  B  cos
Workshop: Law of Induction
Measurement: Voltage Probe and EasyLink
Workshop: Law of Induction
Measurement: 50 Hz R-C-Ripple Filter (if necessary)
Workshop: Law of Induction
Experimental Setup
Workshop: Law of Induction
Measurement: Voltage Surge (Coil 23’000 Turns)
t2
1
Φ2  Φ1     U ind  dt
n t1
Workshop: Law of Induction
Oscillating Magnet (turn by 90° for Measurement)
Workshop: Law of Induction
Oscillating Magnet: Experimental Setup
Workshop: Law of Induction
Oscillating 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
LinearMomentum(Impulse):
t
Δp  m  Δv   F  m  g   dt
0
Conservation of Mechanical Energy:
1
 m  v 2  m  g  h
2
Height of theJump :
 F

    g   dt 
 
v 2  0  m
h

2 g
2 g
t
2
Workshop: Force Plate
Calculations: Integral (Trapezoidal Rule)
Thank you for your attention