Team of Brazil
Problem 02
Elastic Space
reporter:
Denise Sacramento Christovam
Team of Brazil
Problem 2:
##Elastic
Title Space
Problem 2
Elastic Space
The dynamics and apparent interactions of massive
balls rolling on a stretched horizontal membrane are
often used to illustrate gravitation. Investigate the
system further. Is it possible to define and measure
the apparent “gravitational constant” in such a
“world”?
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Team of Brazil
Problem 2:
##Elastic
Title Space
Contents
Introduction
• Elastic membranes and deformation
• Membrane solution and general model
• Relativity and proposed model
• Field dimension
• Newtonian model: Gravitational constant and approach movement
Experiments
• Spring constant
• Same balls, different membranes
• Different balls, same membrane
• System’s dimensions
• Circular motion
Conclusion
• Analysis
• Comparison
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Problem 2:
##Elastic
Title Space
Elastic Membrane - Deformation
• Vertical displacement of the membrane:
– Continuous isotropic membrane
– Discrete system
– Using Hooke’s Law:
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Team of Brazil
Problem 2:
##Elastic
Title Space
Elastic Membrane - Deformation
Curvature
.
.
.
Massic density
(constant)
Space (mm)
.
18
16
14
12
10
8
6
4
2
0
0
20
40
60
80
Space (mm)
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Team of Brazil
Problem 2:
##Elastic
Title Space
Elastic Membrane - Potential
• Membrane obeys wave equation:
• Stationary form:
Vertical displacement
Distance from the center to any other
position (radial coordinates)
Adjustment
constant
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Team of Brazil
Problem 2:
##Elastic
Title Space
Elastic Membrane - Potential
• Ball is constrained to ride on the membrane.
Mass of the particle
Force due to
membrane’s potential
Earth’s Gravity
As the force does not vary with
1/r², our system is not
gravitational.
We find, then, an analogue G.
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Team of Brazil
Problem 2:
##Elastic
Title Space
Elastic Membrane - C1 (and analog G) definition
Analog G
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Team of Brazil
Problem 2:
##Elastic
Title Space
Elastic Membrane – Motion Equation
y
x
•Gravity
•Masses
•Radii
•Elastic
constants
Damping constant
(dissipative forces)
The validity of this
trajectory will be
shown qualitatively in
experiment.
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Team of Brazil
Problem 2:
##Elastic
Title Space
Elastic Membrane – Motion Equation
• Rolling ball’s behavior:
– Close to the great mass (“singularity”)
– Far from singularity
– The closer to singularity, the greater deviation from parabola
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Problem 2:
##Elastic
Title Space
Approach Movement
• Disregarding vertical motion (one-dimensional movement),
the smaller ball’s acceleration towards the other is equivalent
to the gravitational pull
Fg
M
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Team of Brazil
Problem 2:
##Elastic
Title Space
Elastic Membranes
Different rolling
masses
Great masses
compared to the
central one
Both deform the
sheet
Greater than real
gravitational
constant
Deformations add
themselves
meniscus-like
Constant
dependent on
depth of
deformation
Model applicability
finds limitation
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Problem 2:
##Elastic
Title Space
Gravitational Field’s Dimension
• Space-time: continuous
• Membrane: limited
Limit= Borders
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Problem 2:
##Elastic
Title Space
Relativity and Membranes
• Basic concepts: space-time and gravity wells
• Fundamental differences:
–
–
–
–
Time
Friction
Kinetic energy
Gravitational potential
• Our model: Earth’s gravity causes deformation (great ball’s
weight) (Rubber-sheet model)
• It’s not a classical gravitation model, but it’s possible to make
an analogy.
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Problem 2:
##Elastic
Title Space
Material
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
•
•
Strips of rubber balloon and lycra
Half of rubber balloon and lycra (sheet)
Baste
Styrofoam balls with lead inside
Metal spheres, marbles and beads
Two metal brackets
Scale (±0.0001 g - measured)
Sinkers
Measuring tape (±0.05 cm)
2
Plastic shovel with wooden
block (velocity control)
Adhesive tape
Camera
1
6
4
2
3
5
1
8
9
2
10
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
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Taiwan, 24th – 31th July, 2013
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Problem 2:
##Elastic
Title Space
Experimental Description
• Experiment 1: Validation of theoretical trajectory.
• Experiment 2: Determination of rubbers’ spring constant.
• Experiment 3: Same balls interacting, variation of rubbersheet and dimensions.
• Experiment 4: Same membrane and center ball (great mass),
different approaching ball.
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Problem 2:
##Elastic
Title Space
Experiment 1: Trajectory
• Using the shovel, we set the same initial velocity for
all the repetitions.
Trajectory
Spring
Constant
Dimensions
Different
ball
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Team of Brazil
Problem 2:
##Elastic
Title Space
Experiment 1: Trajectory
Trajectory (experiment)
25.00
20.00
Trajectory
15.00
Dimensions
10.00
Position (cm)
Spring
Constant
-20.00
Different
ball
5.00
0.00
-10.00
0.00
10.00
20.00
30.00
-5.00
-10.00
The obtained
trajectory fits
our model,
thus, validating
the proposed
equations.
-15.00
-20.00
Position (cm)
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Problem 2:
##Elastic
Title Space
Experiment 2: Spring Constant
• Experimental Setup:
Trajectory
Spring
Constant
Dimensions
Different
ball
• 5 sinkers, added one at a time.
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Team of Brazil
Problem 2:
##Elastic
Title Space
Experiment 2: Spring Constant
Force x Displacement
4
3.5
Spring
Constant
Force (N)
Trajectory
y = 0.95x
R² = 1.00
3
y = 0.82x
R² = 1.00
2.5
2
1.5
Dimensions
1
1
Different
ball
1.5
2
2.5
3
3.5
4
4.5
Displacement (cm)
•Yellow: k=0.82 N/cm
• Orange: k=0.95 N/cm
The yellow sheet will be the most deformed: higher gravitational
constant.
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Problem 2:
##Elastic
Title Space
Experiment 3: Different rubber-sheets
• Experimental procedure:
Trajectory
Spring
Constant
Dimensions
Different
ball
• Five repetitions for every ball/sheet combination
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Team of Brazil
Problem 2:
##Elastic
Title Space
Experiment 3: Different rubber-sheets
Position x Time
370
Trajectory
Dimensions
Position (mm)
Spring
Constant
y = -0.66x2 - 3.00x + 366.91
350
330
310
y = -2.74x2 - 1.50x + 334.15
290
270
250
Different
ball
0
1
2
3
4
5
6
Time (s)
• GaO: 6,3.10-12 m3/kg.s²
• GaY: 1,18.10-11 m3/kg.s²
• M = 77,3 g
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Team of Brazil
Problem 2:
##Elastic
Title Space
Experiment 3: Different rubber-sheets
Gravitational Constant
Trajectory
Spring
Constant
Theoretical
Gy
Go
6,67E-11
6.29E-12
1.17E-11
Deviation
90.56
82.45
Dimensions
Different
ball
m≈6.9.10-6%M
6,67.10-11
m³kg-1s-2
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Analogue
gravitational
constant
Taiwan, 24th – 31th July, 2013
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Problem 2:
##Elastic
Title Space
Experiment 3:
G(r)
Trajectory
Spring
Constant
Dimensions
Analogue G (mm³/gs²)
28
23
18
Yellow
13
Orange
8
3
200
Different
ball
220
240
260
280
300
320
340
360
380
Radial distance (mm)
The “gravitational constant” is dependent on the distance from
the center, showing that, again, our system is not gravitational.
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Team of Brazil
Problem 2:
##Elastic
Title Space
Experiment 4: Different rolling ball
Position x Time
350
Spring
Constant
330
Position (mm)
Trajectory
y = -1.48x2 - 4.48x + 342.26
R² = 1.00
310
290
y = -1.93x2 - 3.17x + 321.91
R² = 1.00
270
250
Dimensions
230
0.00
1.00
2.00
3.00
4.00
5.00
6.00
Time (s)
Different
ball
• Blue: marble
• Red: bead
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Problem 2:
##Elastic
Title Space
Conclusion
• The proposed system is not a gravitational system;
• It is possible to determine analogue analogue gravitational
constant of the system (exp. 3);
• This Ga is dependent on geometric and elastic characteristics
of the system;
• Ga not equal to G, as the system’s not gravitational;
• The model’s approach is limited to very different masses
(exp.4). Otherwise, the model is not applicable.
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
27
Team of Brazil
Problem 2:
##Elastic
Title Space
References
• CARROLL ,Sean M. Lecture Notes on General Relativity. Institute for
Theoretical Physics, University of California, 12/1997.
• MITOpenCourseWare – Physics 1: Classical Mechanics. 10 – Hooke’s
Law, Simple Harmonic Oscillator. Prof. Walter Lewin, recorded on
10/01/99.
• HAWKING, Stephen, MLODINOW, Leonard. A Briefer History of
Time, 1st edition.
• HALLIDAY, David, RESNICK, Robert. Fundamentals of Physics, vol 2:
Gravitation, Waves and Thermodynamics. 9th edition.
• HAWKING, Stephen. The Universe in a Nutshell.
• POKORNY, Pavel. Geodesics Revisited. Prague Institute of Chemical
Technology, Prague, Czech Republic.
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
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Team of Brazil
Problem 2:
##Elastic
Title Space
Thank you!
Team of Brazil: Liara Guinsberg, Amanda Marciano, Denise Christovam, Gabriel Demetrius, Vitor Melo Rebelo
Reporter: Denise Christovam
Taiwan, 24th – 31th July, 2013
29