PHYSICS 231 Lecture 24: Walking on water & other ‘magic’ Remco Zegers Walk-in hour: Thursday 11:30-13:30 am Helproom PHY 231

h: distance between liquid surface and the point where you measure P h Pressure at depth h P = P0+ fluidgh h: distance between liquid surface and the point where you measure P h P Buoyant force for submerged object B = fluidVobjectg = Mfluidg = wfluid The buoyant force equals the weight of the amount of water that can be put in the volume taken by the object. If object is not moving: B=wobject object= fluid h B w Buoyant force for floating object The buoyant force equals the weight of the amount of water that can be put in the part of the volume of the object that is under water. objectVobject= waterVdisplaced h= objectVobject/(waterA) PHY 231

Bernoulli’s equation P1+½v12+gy1= P2+½v22+gy2 P+½v2+gy=constant The sum of the pressure (P), the kinetic energy per unit volume (½v2) and the potential energy per unit volume (gy) is constant at all points along a path of flow. Note that for an incompressible fluid: A1v1=A2v2 This is called the equation of continuity. PHY 231

Applications of Bernoulli’s law The examples shown are with air, not with fluid. Remember that we derived this law for an incompressible fluid. Air is not incompressible, so the situation is typically more complicated… But easier to show! PHY 231

Applications of Bernoulli’s law: moving a cart No spin, no movement Vair Spin and movement P1 V1=Vair+v V2=Vair-v P2 Near the surface of the rotating cylinder: V1>V2 P1+½v12= P2+½v22 P1-P2= ½(v22- v12) P1-P2= ½[(vair-v)2-(vair+ v) 2] P2>P1 so move to the left PHY 231

Applications of Bernoulli’s law: the golf ball Neglecting the small change in height between the top and bottom of the golf ball: P1+½v12= P2+½v22 P1-P2= ½(v22- v12) P1 P1-P2= ½(v22- v12)=0 v2=v1 No pressure difference, no lift P2 P1 P1-P2= ½(v2-v)2-(v1+ v) 2=0 P2>P1 so: Upward force: the ball goes higher and thus travels faster P2 PHY 231

Not the whole story: the dimples in the golf ball reduce the drag The drag is the force you feel when you are biking. The pressure in front of you is higher than behind you, so you feel a force against the direction of your motion. A B P1 P2 P1 P3 P3>P2 : there is less drag in case B PHY 231

Demo A floating globe PHY 231

Two liquid molecules like to sit close to each other (energy Surface tension Two liquid molecules like to sit close to each other (energy is gained) -Emin R R 2 liquid molecules PHY 231

A bunch of liquid molecules Inside the liquid 1 2 3 4 Near the surface of the liquid The molecule near the surface only gains 4 times Emin of energy. The summed energy is only reduced by 4Emin. 6 1 5 4 2 3 The molecule in the center gains 6 times Emin of energy. The summed energy is reduced by 6Emin It is energetically favorable to keep the surface of the liquid as small as possible PHY 231

Why does water make droplets on a surface and does not spread out? The liquid surface is smallest: energetically favorable. PHY 231

=mneedleg/(2Lsin) Surface tension Fs Fs Fg If you make the surface of the fluid larger, it tries to ‘push’ back. The force with which this is done: Fs=L where L is the length over which the force acts and  is the surface tension. The force works parallel to the water surface. Example: a needle on water Top view Fs Fs  L Fg Horizontal: Fscos-Fscos=0 Vertical: Fssin+Fssin-Fg=0 =mneedleg/(2Lsin) Units of : N/m=J/m2 Energy per unit surface PHY 231

Walking on water The insect uses surface tension! Surface tension depends on the type of liquid. PHY 231

Forces between molecules Cohesive forces: forces between like molecules Adhesive forces: forces between unlike molecules Adhesive Cohesive PHY 231

More on surfaces If cohesive forces are stronger than the adhesive ones like molecules in the drop try to stay together to reduce the total energy of the system; if adhesive forces are stronger the drop will spread out to reduce the total energy of the system. The spreading will stop when the surface tension becomes too strong. PHY 231

Same thing Adhesive > Cohesive The water wants to cover as much of the glass as its surface tension allows Adhesive < Cohesive The mercury wants to cover as little of the glass as its surface tension allows PHY 231

Capillary action Fsurface tesion=L= 2r Vertical (upwards) component: FSTvertical=2rcos The weight of the liquid in the tube: w=Mg=r2gh The liquid stops going up when: FSTvertical=w h=2cos/(gr) If r very large: h very small! PHY 231

Viscosity Viscosity: stickiness of a fluid One layer of fluid feels a large resistive force when sliding along another one or along a surface of for example a tube. PHY 231

Viscosity Contact surface A moving F=Av/d =coefficient of viscosity unit: Ns/m2 or poise=0.1 Ns/m2 fixed PHY 231

Poiseuille’s Law How fast does a fluid flow through a tube? R4(P1-P2) (unit: m3/s) Rate of flow Q= v/t= 8L PHY 231

Example R=[8QL/((P1-P2))]1/4=0.05 m Flow rate Q=0.5 m3/s Tube length: 3 m =1500E-03 Ns/m2 PP=106 Pa P=105 Pa What should the radius of the tube be? Rate of flow Q= R4(P1-P2) 8L R=[8QL/((P1-P2))]1/4=0.05 m PHY 231