PHY 2311

PHYSICS 231Lecture 24: Walking on water & other

‘magic’

Remco ZegersWalk-in hour: Thursday 11:30-13:30 am

Helproom

PHY 2312

P = P0+ fluidghh: distance between liquid surface and the point where you measure P

P0

P

h

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

Pressure at depth h

Buoyant force for submerged object

Buoyant force for floating objecthB

w

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 2313

Bernoulli’s equation

P1+½v12+gy1=

P2+½v22+gy2

P+½v2+gy=constant

The sum of the pressure (P), the kinetic energy per unit volume (½v2) 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 ofcontinuity.

PHY 2314

Applications of Bernoulli’s law

The examples shown are with air, not with fluid.Remember that we derived this law for anincompressible fluid. Air is not incompressible,so the situation is typically more complicated…

But easier to show!

PHY 2315

Applications of Bernoulli’s law: moving a cart

No spin, no movementVair

Spin and movementP1 V1=Vair+v

V2=Vair-v P2

Near the surface of the rotating cylinder: V1>V2P1+½v1

2= P2+½v22

P1-P2= ½(v22- v1

2)P1-P2= ½[(vair-v)2-(vair+ v)

2]P2>P1 so move to the left

PHY 2316

Applications of Bernoulli’s law: the golf ball

P1

P2

Neglecting the small change in height between thetop and bottom of the golf ball:

P1+½v12= P2+½v2

2

P1-P2= ½(v22- v1

2)

P1

P2

P1-P2= ½(v22- v1

2)=0v2=v1

No pressure difference, no lift

P1-P2= ½(v2-v)2-(v1+ v) 2=0

P2>P1 so:Upward force: the ball goes higherand thus travels faster

PHY 2317

Not the whole story: the dimples in the golf ball reduce the drag

P1 P2 P1 P3

P3>P2 : there is less drag in case B

A B

The drag is the force you feel when you are biking. The pressure in front of you is higher than behind you, so youfeel a force against the direction of your motion.

PHY 2318

Demo

A floating globe

PHY 2319

Surface tension

R

Energy

0

R

2 liquid molecules

Two liquid molecules like tosit close to each other (energyis gained)

-Emin

PHY 23110

A bunch of liquid molecules

16

54

32

The molecule in the centergains 6 times Emin of energy.The summed energy is reduced by 6Emin

Inside the liquid

1

2 3

4

Near the surface of the liquid

The molecule near thesurface only gains 4 timesEmin of energy. The summedenergy is only reduced by4Emin.

It is energetically favorable to keepthe surface of the liquid as smallas possible

PHY 23111

Why does water make droplets on asurface and does not spread out?

The liquid surface is smallest:energetically favorable.

PHY 23112

Surface tensionIf you make the surface of the fluid larger, it tries to‘push’ back. The force with which this is done: Fs=Lwhere L is the length over which the force acts and is the surface tension. The force works parallel to thewater surface. Example: a needle on water

Top view

L

Fs Fs

FgHorizontal: Fscos-Fscos=0Vertical: Fssin+Fssin-Fg=0

=mneedleg/(2Lsin)

Units of : N/m=J/m2

Energy per unit surface

PHY 23113

Walking on water

The insect uses surface tension!

Surface tension depends on the type of liquid.

PHY 23114

Forces between molecules

Cohesive forces: forces between like moleculesAdhesive forces: forces between unlike molecules

Cohesive Adhesive

PHY 23115

More on surfaces

If cohesive forces are stronger than the adhesive oneslike 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 totalenergy of the system. The spreading will stop when thesurface tension becomes too strong.

PHY 23116

Same thing

Adhesive > CohesiveThe water wants tocover as much ofthe glass as its surface tension allows

Adhesive < CohesiveThe mercury wants tocover as little of the glassas its surface tension allows

PHY 23117

Capillary action

Fsurface tesion=L= 2rVertical (upwards) component:FSTvertical=2rcos

The weight of the liquid in the tube: w=Mg=r2gh

The liquid stops going up when:FSTvertical=wh=2cos/(gr)

If r very large: h very small!

PHY 23118

Viscosity

Viscosity: stickiness of a fluidOne layer of fluid feels a largeresistive force when slidingalong another one or along asurface of for example a tube.

PHY 23119

ViscosityContact surface A

fixed

movingF=Av/d

=coefficient of viscosityunit: Ns/m2

or poise=0.1 Ns/m2

PHY 23120

Poiseuille’s Law

How fast does a fluid flowthrough a tube?

Rate of flow Q= v/t=R4(P1-P2)

8L(unit: m3/s)

PHY 23121

Example

PP=106 Pa P=105 Pa

Flow rate Q=0.5 m3/sTube length: 3 m=1500E-03 Ns/m2

What should the radius of the tube be?

Rate of flow Q=R4(P1-P2)

8L

R=[8QL/((P1-P2))]1/4=0.05 m