# 1. Deformation of water surface

```7.
Drawing pins
Michal Hled&iacute;k
7. Drawing pins
A drawing pin (thumbtack) floating on the
surface of water near another floating object
is subject to an attractive force. Investigate
and explain the phenomenon.
Is it possible to achieve a repulsive force by
a similar mechanism?
2
• Video of attracting
• +picture of pins
3
Content
1. Deformation of water surface
2. Attraction of pins
•
•
•
Mechanism
Calculating the motion
Theory vs. experiments
3. Repulsion of objects
• Mechanism
• Attracting/repelling boundary
4
1. Deformation
of water surface
5
Forces analysis
FBU
Gravity
FST
Buoyant force
Surface tension
G
Force equilibrium
6
Force equilibrium
G  FBU  FST
D

R
Mg  R Dg  2R sin 
2
Only unknown quantities
Eq. (1)
7
Water displacement
Pressure – hydrostatic
Absent water – compensated by surface
tension
8
Water displacement
r
z r 
Resulting
function:
z (r )   tan ` 
 g 

K 0  r`




   tan `  f r` 
g  g 

K1  R




Dominic Vella, L. Mahadevan, The ‘‘Cheerios effect,’’9 (2005)
Finding contact angle
R
z R 
D
T
z ( R)  T  D  tan   f R   T
Eq. (1):
Mg  R Dg  2R sin 
Predicted angle:
2
  29.4  2
10
Contact angle
measurement
Analyzing size
Distant
light source
Pin on water
11
Measuring the contact angle
Applying Snell’s law, fitting contact angle
Contact angle:
  33,3  3,1
12
Shape of water surface
0
0
5
10
15
r
20
25
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
z (r )
[mm]
[mm]
13
2. Attraction of
pins
14
Why are they attracting?
2 pins on water –
inclined to each
other
G  FB
Mass of a pin &gt; mass of water
displaced
Potential energy of water and
pin – decreases as pin
descends
15
Determining the acceleration
EP
F
 G  FB
y
FD
a
 FST  2R sin 
β
Horizontal motion:
F
FD
a  sin  cos  
cos 
m
m
F
16
Slope of one pin
Given by the deformation of
water surface by the other pin

Our approximation:
z ( far end )  z (near end )
tan(  ) 
diameter
17
Drag force
FD
Assuming FD  v
v
2
d x
dx
 c
2
dt
dt
x…position
1  e  ct
xt   v0
c
Video
analysis and
fit c
18
Fitting the drag coefficient
Distance passed x [m]
0.06
0.05
0.04
1  e  ct
xt   v0
c
0.03
0.02
c  0.53s
0.01
1
Time [s]
0
0
1
2
3
19
4
Acceleration  distance in time
v
v
Dependence of
acceleration on distance
and velocity
r
a(r , v)
Numerical solution
20
Attracting – experiment
21
Theory vs. experiment
Distance of the pins [mm]
30
25
20
15
10
5
0
0
5
10
Time [s]
15
22
3. Repelling
objects
23
Repulsive force
FBU
Object wetted by water
 FST acts downwards
G  FB
 object floats up
FST
G
24
Repelling objects
Plastic caps from pins
– float upwards
Behavior depends on weight
There is a critical mass – does not repel or attract
25
Both caps wetted by water
26
+ A little weight on the yellow cap
27
Empty cap and a cap with a weight
Distance between the caps [cm]
7
6
0,204g
5
0,162g
4
Greater mass
 stronger repulsion
3
0,062g
2
0,041g
0,027g
1
0
0
2
4
6
8
10
Time [s]12
28
Conclusion
We explained the mechanism of
‒ floating, attraction, repulsion
Determined the deformation of water
surface
Described the motion quantitatively
‒ theory correlates with experiments
Found the border between
attraction/repulsion
29
APPENDICES
7. Drawing pins
Drawing pin “dipole” – attracts different
objects on different sides
31
Water displacement
d z 1 dz
g

z
2
dr
r dr

• Boundary conditions:z ' ( R)  tg
r    z (r )  0
2
• Solution:
 g 

K 0  r`





z (r )  tg
g 

g

K1  R




Dominic Vella, L. Mahadevan, The ‘‘Cheerios effect,’’ (2005)
32
Critical mass
• Water is not deformed
• Surface tension resultant force = 0
G  FBuoyant
D
M  R D
2
33
Critical mass
•
•
•
•
Pin caps – bent edges
Mass theoretically: M T  0,16 g
Depth: H  1,3mm
Mass experimentally:M E  0,12 g
34
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