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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 Dg 2R 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 Dg 2R 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 2R 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 xt v0 c Video analysis and fit c 18 Fitting the drag coefficient Distance passed x [m] 0.06 0.05 0.04 1 e ct xt v0 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