team of brazil problem 03 string of beads reporter: bárbara cruvinel santiago

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.




Contents

Introduction

• Acceleration• Initial energy• Detachment

Experiments

• Experimental data

• Theory validation

Conclusion

• Analysis• Comparison




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.




Acceleration and motion features

When H>L




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.




Initial energy to overcome inertia

Initial energy:

Referential for energy calculation.




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




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.




DetachmentCut diametrically the

beads

Larger angle

Larger detachment arc

There’s a maximum

arc of tension




Detachment

d1 < d2 α1 > α2Higher tension

Takes different shapes due to the arrangement inside the beaker.

Larger bead’s size

Larger angle

Higher tension




Material

• Beakers• Chains (beads and nylon)• Table• Camera• Measuring tape (precision of ±0.05 cm)• Precision scale (precision of ±0.1 g)




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.




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




Experiment 1: acceleration analysis

Source of errors: measuring tape (±0.5 mm) and conversion scale.


Detachment in time

Height of launch

Beaker dimensions


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)




Experiment 1: acceleration analysisAccelerationInitial energyDetachment

shapeDetachment

in timeHeight of

launchBeaker

dimensionsDistance b et w een b eads




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


Detachment in time

Height of launch

Beaker dimensions


Average:0.039 J

Theoretical:0.038 J

Chain mass: 270 gChain length: 2.7 mBeaker height: 14 cm




Experiment 3: detachment analysisAccelerationInitial energyDetachment

shapeDetachment

in timeHeight of

launchBeaker






shapeDetachment

in timeHeight of

launchBeaker






shapeDetachment

in timeHeight of

launchBeaker


Possible to see the relation between patterns




Experiment 4: evolution of the detachment with timeAccelerationInitial energyDetachment

shapeDetachment

in timeHeight of

launchBeaker





Experiment 4: evolution of the detachment with timeAccelerationInitial energyDetachment

shapeDetachment

in timeHeight of

launchBeaker


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.




Experiment 5: height of launch variationAccelerationInitial energyDetachment

shapeDetachment

in timeHeight of

launchBeaker


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.




Experiment 6: beaker dimensions variationAccelerationInitial energyDetachment

shapeDetachment

in timeHeight of

launchBeaker


It is varied.





shapeDetachment

in timeHeight of

launchBeaker


0 0.5 1 1.5 2 2.50123456789

10


400 mL600 mL1000 mL

Time (s)

Heig

ht o

f det

achm

ent (

cm)





shapeDetachment

in timeHeight of

launchBeaker


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




Experiment 7: distance between beadsAccelerationInitial energyDetachment

shapeDetachment

in timeHeight of

launchBeaker






shapeDetachment

in timeHeight of

launchBeaker


0 0.5 1 1.5 2 2.5 3 3.50

1

2

3

4

5

6


2 mm20 mm

Time (s)

Heig

ht o

f det

achm

ent (

cm)





shapeDetachment

in timeHeight of

launchBeaker


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




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).




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




Thank you!

team of brazil problem 03 string of beads reporter: bárbara cruvinel santiago

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