7.
Drawing pins
Michal Hledík
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
of shade of the pin
Distant
light source
Pin on water
11
Measuring the contact angle
Applying Snell’s law, fitting contact angle
(size of the shade)
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 > 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
aD  cv
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
Thank you for your attention
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