Team of Brazil
Problem 03String of beads
reporter:
Bárbara Cruvinel Santiago
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 2
Team of BrazilProblem 3: String of beads
Problem 03String of beads
A long string of beads is released from a beaker by pulling a sufficiently long part of the chain over the edge of the beaker. Due to gravity the speed of the string increases. At a certain moment the string no longer touches the edge of the beaker (see picture). Investigate and explain the phenomenon.
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 3
Team of BrazilProblem 3: String of beads
Contents
Introduction
• Acceleration• Initial energy• Detachment
Experiments
• Experimental data
• Theory validation
Conclusion
• Analysis• Comparison
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 4
Team of BrazilProblem 3: String of beads
Introduction
• “sufficiently long part of the chain”: there is a minimum energy to overcome inertia.
• “the speed of the string increases”: two cases– Initial height > String’s length: 2 phases for acceleration behavior– Initial height < String’s length: 3 phases for acceleration behavior
• “the string no longer touches the edge of the beaker”: there is a mechanism which makes the chain detach from the beaker.
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 5
Team of BrazilProblem 3: String of beads
Acceleration and motion features
When H>L
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 6
Team of BrazilProblem 3: String of beads
Acceleration and motion features
When H<L• 1st part: till x = H
• 2nd part:
• 3rd part:
Total length that already left the beaker, including what already touched the floor.
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 7
Team of BrazilProblem 3: String of beads
Initial energy to overcome inertia
Initial energy:
Referential for energy calculation.
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 8
Team of BrazilProblem 3: String of beads
Detachment
• Cause: tension causes an impulse that pull the chain upwards.
Velocity increases
Tension increases
Detachment height increases
with time
*
*WANG, C. W.; YASUI, K.. Falling chains. University of California. 20 Feb 2006
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 9
Team of BrazilProblem 3: String of beads
Detachment
Beads touch the floor
Transference into gravitational potential energy
Detachment
Dissipation of energy
The height of the beaker influences the height of detachment above the beaker’s border.
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 10
Team of BrazilProblem 3: String of beads
DetachmentCut diametrically the
beads
Larger angle
Larger detachment arc
There’s a maximum
arc of tension
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 11
Team of BrazilProblem 3: String of beads
Detachment
d1 < d2 α1 > α2Higher tension
Takes different shapes due to the arrangement inside the beaker.
Larger bead’s size
Larger angle
Higher tension
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 12
Team of BrazilProblem 3: String of beads
Material
• Beakers• Chains (beads and nylon)• Table• Camera• Measuring tape (precision of ±0.05 cm)• Precision scale (precision of ±0.1 g)
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 13
Team of BrazilProblem 3: String of beads
Experimental description
• Experiment 1: acceleration analysis. • Experiment 2: initial energy analysis.• Experiment 3: detachment analysis.• Experiment 4: detachment evolution with time.• Experiment 5: height of launch variation.• Experiment 6: beaker dimensions variation.• Experiment 7: variation of the distance between beads.
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 14
Team of BrazilProblem 3: String of beads
0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40
1
2
3
4
5
6
7
8
f(x) = 0.199951 x⁵ − 1.4487 x⁴ + 4.45099 x³ − 6.19948 x² + 7.18254 x − 2.33827R² = 0.999999999812995
f(x) = 0R² = 0 Velocity in function of "x"
Part "x" hanging from the beaker (m)
Velo
city
(m/s
)
Experiment 1: acceleration analysis
Function is differentiable
AccelerationInitial energyDetachment shape
Detachment in time
Height of launch
Beaker dimensions
Distance b et w een b eads
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 15
Team of BrazilProblem 3: String of beads
Experiment 1: acceleration analysis
Source of errors: measuring tape (±0.5 mm) and conversion scale.
AccelerationInitial energyDetachment shape
Detachment in time
Height of launch
Beaker dimensions
Distance b et w een b eads
0 0.5 1 1.5 2 2.5 30
2
4
6
8
10
12
Acceleration in function of "x"
TheoryExperiment
Part "x" hanging from the beaker (m)
Acc
eler
atio
n (m
/s²)
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 16
Team of BrazilProblem 3: String of beads
Experiment 1: acceleration analysisAccelerationInitial energyDetachment
shapeDetachment
in timeHeight of
launchBeaker
dimensionsDistance b et w een b eads
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 17
Team of BrazilProblem 3: String of beads
Experiment 2: initial energy analysis
Minimum energy (J)0.0440.0380.0330.0380.0380.0440.038
Standard deviation: 0.004 J
Source of error: measuring tape (±0.5 mm) e of the scale (±0.1 g).
Experiment agrees with theory
AccelerationInitial energyDetachment shape
Detachment in time
Height of launch
Beaker dimensions
Distance b et w een b eads
Average:0.039 J
Theoretical:0.038 J
Chain mass: 270 gChain length: 2.7 mBeaker height: 14 cm
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 18
Team of BrazilProblem 3: String of beads
Experiment 3: detachment analysisAccelerationInitial energyDetachment
shapeDetachment
in timeHeight of
launchBeaker
dimensionsDistance b et w een b eads
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 19
Team of BrazilProblem 3: String of beads
Experiment 3: detachment analysisAccelerationInitial energyDetachment
shapeDetachment
in timeHeight of
launchBeaker
dimensionsDistance b et w een b eads
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 20
Team of BrazilProblem 3: String of beads
Experiment 3: detachment analysisAccelerationInitial energyDetachment
shapeDetachment
in timeHeight of
launchBeaker
dimensionsDistance b et w een b eads
Possible to see the relation between patterns
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 21
Team of BrazilProblem 3: String of beads
Experiment 4: evolution of the detachment with timeAccelerationInitial energyDetachment
shapeDetachment
in timeHeight of
launchBeaker
dimensionsDistance b et w een b eads
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 22
Team of BrazilProblem 3: String of beads
Experiment 4: evolution of the detachment with timeAccelerationInitial energyDetachment
shapeDetachment
in timeHeight of
launchBeaker
dimensionsDistance b et w een b eads
0 0.5 1 1.5 2 2.50
1
2
3
4
5
6
Height of detachment x Time
Time (s)
Heig
ht o
f det
achm
ent (
cm)
Tends to increase
Oscillates with time due to energy dissipation by collision.
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 23
Team of BrazilProblem 3: String of beads
Experiment 5: height of launch variationAccelerationInitial energyDetachment
shapeDetachment
in timeHeight of
launchBeaker
dimensionsDistance b et w een b eads
0.5 1 1.5 2 2.5 3 3.50123456789
10
Height of launch x Height of detachment
Height of launch (m)
Heig
ht o
f det
achm
ent (
cm)
Optimum point
Dissipation starts to overcome the transference into potential energy.
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 24
Team of BrazilProblem 3: String of beads
Experiment 6: beaker dimensions variationAccelerationInitial energyDetachment
shapeDetachment
in timeHeight of
launchBeaker
dimensionsDistance b et w een b eads
It is varied.
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 25
Team of BrazilProblem 3: String of beads
Experiment 6: beaker dimensions variationAccelerationInitial energyDetachment
shapeDetachment
in timeHeight of
launchBeaker
dimensionsDistance b et w een b eads
0 0.5 1 1.5 2 2.50123456789
10
Height of detachment x Time
400 mL600 mL1000 mL
Time (s)
Heig
ht o
f det
achm
ent (
cm)
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 26
Team of BrazilProblem 3: String of beads
Experiment 6: beaker dimensions variationAccelerationInitial energyDetachment
shapeDetachment
in timeHeight of
launchBeaker
dimensionsDistance b et w een b eads
0 0.5 1 1.5 2 2.50123456789
10
Height of detachment x Time (linear comparison)
Time (s)
Heig
ht o
f det
achm
ent (
cm)
Larger initial distance from
the border
Larger distance for the
detachment to overcome
Lower detachment
height
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 27
Team of BrazilProblem 3: String of beads
Experiment 7: distance between beadsAccelerationInitial energyDetachment
shapeDetachment
in timeHeight of
launchBeaker
dimensionsDistance b et w een b eads
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 28
Team of BrazilProblem 3: String of beads
Experiment 7: distance between beadsAccelerationInitial energyDetachment
shapeDetachment
in timeHeight of
launchBeaker
dimensionsDistance b et w een b eads
0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
4
5
6
Height of detachment x Time
2 mm20 mm
Time (s)
Heig
ht o
f det
achm
ent (
cm)
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 29
Team of BrazilProblem 3: String of beads
Experiment 7: distance between beadsAccelerationInitial energyDetachment
shapeDetachment
in timeHeight of
launchBeaker
dimensionsDistance b et w een b eads
0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
4
5
6
Height of detachment x Time (linear comparison)
Time (s)
Heig
ht o
f det
achm
ent (
cm)
Larger distance between beads
Lower tension
Lower detachment
height
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 30
Team of BrazilProblem 3: String of beads
Conclusion
• Initial energy: related to the position of the center of mass (exp. 3).
• Acceleration increases until g is reached in three phases (exp. 2).
• Detachment:– Impulse of the tension pulls the string upwards;– Different shapes due to chain’s arrangement (exp. 4);– Energy conversion (exp. 5);– Initial height (exp. 5);– Beaker’s dimensions (exp. 6);– Distance between beads (exp. 7).
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 31
Team of BrazilProblem 3: String of beads
References
• 1. GREWAL, Anoop; JOHNSON, Phillip; RUINA, Andy. A chain that accelerates rather than slows due to collisions: how compression can cause tension. Cornell University. 13 mar. 2011.
• 2. http://mathworld.wolfram.com/Catenary.html• 3. WANG, Chun Wa; YASUI, Kosuke. Falling chains. University
of California. 20 Feb 2006
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012Reporter: Bárbara Cruvinel Santiago 32
Team of BrazilProblem 3: String of beads
Thank you!