Team of Slovakia International Young Physicists’ Tournament, Seoul 2007 4. Spring thread.
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Transcript of Team of Slovakia International Young Physicists’ Tournament, Seoul 2007 4. Spring thread.
Team of Slovakia
International Young Physicists’ Tournament, Seoul 2007
4. Spring thread
No. 4. Spring ThreadPresented by Tomas Bzdusek
Task
Pull a thread through the button holes as shown in the picture. The button can be put into rotating
motion by pulling the thread. One can feel some elasticity of the thread. Explain the elastic
properties of such a system.
No. 4. Spring ThreadPresented by Tomas Bzdusek
Outline
• Theory– What is elasticity?– Deriving the motion equation
• Experiments
• Conlusion
No. 4. Spring ThreadPresented by Tomas Bzdusek
What is elasticity?
• Elastic material acts according to Hooke‘s law
• In other notation
• Force is directly proportional to relative extension.
• Hooke‘s law can be used also for compressing an elastic material. Then is relative contraction.
E
fF
No. 4. Spring ThreadPresented by Tomas Bzdusek
Our system
• Two extremal views of the situation:
1.) Fixed distance of ends of the thread.
- When rotating the button, the thread has to extend, so it acts due to Hooke‘s law
2.) The distance changes, we assume no extension in real lenght of the thread – It changes only due to entangling of the thread.
No. 4. Spring ThreadPresented by Tomas Bzdusek
Where is the reality?
• Somewhere between the extremes.– The distance changes significantly.– There are some small changes in real lenght of
the thread.
No. 4. Spring ThreadPresented by Tomas Bzdusek
Used threads
• We used three different types of thread– Thin thread (green, orange, yellow); r = 0.12 mm– Thick thread (white); r = 0.29 mm– Silon; r = 0.65 mm
No. 4. Spring ThreadPresented by Tomas Bzdusek
Measuring the Young modulus
No. 4. Spring ThreadPresented by Tomas Bzdusek
Young modulus of used threads
• Thin thread
{ } = 0.003 F
00.0050.01
0.0150.02
0.0250.03
0.0350.04
0.045
0 1 2 3 4 5 6 7 8F/[N]
No. 4. Spring ThreadPresented by Tomas Bzdusek
Young modulus of used threads
• Thick thread
{ } = 0.001 F + 0.0015
00.0020.0040.0060.008
0.010.0120.0140.0160.018
0.02
0 2 4 6 8 10 12 14 16
F/[N]
No. 4. Spring ThreadPresented by Tomas Bzdusek
Young modulus of used threads
• Silon
{ } = 0.0045 F - 0.0032
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 2 4 6 8 10 12 14 16
F/[N]
No. 4. Spring ThreadPresented by Tomas Bzdusek
Our model
• Extensibilities due to Hooke’s law are small – we will neglect them.
• The system is only being shortened due to convolution
No. 4. Spring ThreadPresented by Tomas Bzdusek
Is this system elastic?
• Force F can be arbitrary, whatever the relative contraction of the system is. – Against Hooke’s law
• Therefore THE SYSTEM IS NOT ELASTIC.
• However, we can feel some “elasticity”.
,,fF
No. 4. Spring ThreadPresented by Tomas Bzdusek
Parameters
• Half-lenght of the system in steady state (not shortened) – L0
• Actual half-lenght – L• Radius of the thread – r• Angle of rotation of the button (in comparison with
steady state) – • Angle of convolution of the thread -
No. 4. Spring ThreadPresented by Tomas Bzdusek
Parameters shown in a picture
L
r
No. 4. Spring ThreadPresented by Tomas Bzdusek
Basic equtions
• The thread is homogeneously rolled on a cyllindrical surface with radius r.
L
L0 r
rolled threadL
r tan
No. 4. Spring ThreadPresented by Tomas Bzdusek
Length of the system
• According to Phytagorean theorem:
• For shortening of the system:
– N is number of windings.
• For small number of windings:
220 rLL
222200
2200 422 rNLLrLLS
2
0
224N
L
rS
No. 4. Spring ThreadPresented by Tomas Bzdusek
Measuring the shortening
buttonfixed end free end
No. 4. Spring ThreadPresented by Tomas Bzdusek
Shortening of used threads
• Thin thread
0
50
100
150
200
250
300
0 100 200 300 400 500 600
Number of windings N
Sh
ort
en
ing
S/[
mm
]
measuerd
predicted
No. 4. Spring ThreadPresented by Tomas Bzdusek
Shortening of used threads
• Thick thread
0
10
20
30
40
50
60
70
80
0 5 10 15 20 25 30
Number of windings N
sh
ort
en
ing
S/[
mm
]
measuerd
predicted
No. 4. Spring ThreadPresented by Tomas Bzdusek
Shortening of used threads
• Silon
0
10
20
30
40
50
60
0 20 40 60 80 100
Number of windings N
sh
ort
en
ing
S/[
mm
]
measured
predicted
No. 4. Spring ThreadPresented by Tomas Bzdusek
Further equations
• In the thread there will be a strain T:
cos2TF
2
F2
F
tan2
F
No. 4. Spring ThreadPresented by Tomas Bzdusek
Motion equation
• For torque we can obtain:
• For small number of windings– Direct proportion:
0LL
L
Frr
FM
22tan
24
0
22
L
FrM
No. 4. Spring ThreadPresented by Tomas Bzdusek
Prooving the direct proportion
weight of mass F/gchangeable wieght
No. 4. Spring ThreadPresented by Tomas Bzdusek
Prooving the direct proportion
No. 4. Spring ThreadPresented by Tomas Bzdusek
Measuring the direct proportion
• In equilibrum:
• Theoretically:
– d = radius of a button– M = mass suspended at free end of the system– m = mass of changeable weight
mgdL
Fr
22
22 22 Mr
Ldm
F
g
r
Ldm
No. 4. Spring ThreadPresented by Tomas Bzdusek
Results
• Thin threadM = 1kg ; d = 14 mm ; L0= 0.2 m ; m = 0.5 g
0
50
100
150
200
250
0 10 20 30 40 50
number of weights n
nu
mb
er
of
win
din
gs
N
theoretically
measured
No. 4. Spring ThreadPresented by Tomas Bzdusek
Results
• Thick threadM = 1kg ; d = 14 mm ; L0= 0.6 m ; m = 0.5 g
0
5
10
15
20
25
30
35
40
0 10 20 30 40 50
number of weights n
nu
mb
er
of
win
din
gs
N
theoretically
measured
No. 4. Spring ThreadPresented by Tomas Bzdusek
Results
• SilonM = 1kg ; d = 14 mm ; L0= 0.6 m ; m = 0.5 g
020406080
100120140160180200
0 10 20 30 40 50 60
number of weights n
nu
mb
er
of
win
din
gs
N
theoretically
measured
No. 4. Spring ThreadPresented by Tomas Bzdusek
Motion equations
• We obtained
• Therefore
• Linear harmonic oscillator with period
0
22
L
FrM
LI
Fr
I
M 22
222
Fr
LIT
No. 4. Spring ThreadPresented by Tomas Bzdusek
Motion equations
• If we assume some damping b:
(where )
• Well-known solution of this equation (for 0=0) is:
0 Kb
LI
FrK
22
Ct
C
bCte
tb
sin2
cos20
4
2bKC
No. 4. Spring ThreadPresented by Tomas Bzdusek
Damped oscillations
-250
-200
-150
-100
-50
0
50
100
150
200
250
0 2 4 6 8 10time / [s]
nu
mb
er
of
win
din
gs
No. 4. Spring ThreadPresented by Tomas Bzdusek
Spring thread as a toy
• When playing with the toy, we act – with larger force when the system is expanding– with smaller force when the system is shortening
• In our model, we suppose the forces to be F and F/2.
No. 4. Spring ThreadPresented by Tomas Bzdusek
Simulation• We can see, that the system begins to rezonate
Spring Thread Resonation
-150
-100
-50
0
50
100
150
0 0,5 1 1,5 2 2,5 3 3,5 4
t/[s]
fi/[
rad
]
No. 4. Spring ThreadPresented by Tomas Bzdusek
Conclusion
• Elasticity of the system is a dynamic property.– We can feel it only when the button is rotating.
• Spring thread is rezonating torsional oscillator - THIS IS THE ELASTICITY we can feel.
xm
kx
k
m
LI
Fr 22
No. 4. Spring ThreadPresented by Tomas Bzdusek
Thank you for your attention
No. 4. Spring ThreadPresented by Tomas Bzdusek18. Appendix
The motion equation (1)
We suppose a solution . When substitued into the differential equation, we obtain:
We suppose that
tie 0
0
02
0002
Kib
eKeibe tititi
42
2
2;1
bK
bi
4/2bK
No. 4. Spring ThreadPresented by Tomas Bzdusek19. Appendix
The motion equation (2)
Substitution:
We have:
For 0:
Since 0 is real, and
Cb
K 4
2
iCtBiCtAe
tiCb
BtiCb
A
tb
expexp
2exp
2exp
2
BA0*AB RA20
No. 4. Spring ThreadPresented by Tomas Bzdusek20. Appendix
The motion equation (3)
For angular velocity:
In zero time:
iCtiCttb
BiCb
AiCb
e
tiCb
BiCb
tiCb
AiCb
t
exp2
exp2
2exp
22exp
2d
d
2
BAiCBAb
20
No. 4. Spring ThreadPresented by Tomas Bzdusek21. Appendix
The motion equation (4)
We have and
Therefore we can simplify:
Finally we have:
We will use:
IiAiCb
BAiCBAb
222 00
IR iAAA IR iAAA *
20
RAC
b
AI 22 00
xiee
xeeixix
ixix
sin2
cos2
No. 4. Spring ThreadPresented by Tomas Bzdusek22. Appendix
The motion equation (5)
By putting together and simplifying we obtain:
Under special condition :
Ct
C
b
Ctetb
sin2cos00
02
00
Ct
C
bCte
tb
sin2
cos20