1 Class #3 Vectors and Dot products Retarding forces Stokes Law (viscous drag) Newton’s Law (inertial drag) Reynolds number Plausibility of Stokes law Projectile motions with viscous drag Plausibility of Newton’s Law Projectile motions with inertial drag Worked Problems Homework Review :10

2 Vectors and Central forces Vectors Many forces are of form Remove dependence of result on choice of origin Origin 1 Origin 2 :30

3 Vector relationships Vectors Allow ready representation of 3 (or more!) components at once.  Equations written in vector notation are more compact :33

4 Dot product is a “projection” operator :33 O m h Block on ramp with gravity Choose coordinates consistent with “constraints”

5 Defining Viscosity Two planes of Area “A” separated by gap Top plane moves at relative velocity defines viscosity (“eta”) MKS Units of are Pascal-seconds Only CGS units (poise) are actually used 1 poise=0.1 =0.1 Pouseille y x A :30

6 Viscosity of Common Substances SubstanceViscosity (CentiPoise) Air (STP)0.018 Water Water ~ Antifreeze~20 Olive Oil~84 30 Weight motor oil~200 Honey (20 C)~10,000 Peanut Butter~250,000 Window Glass1E15

7 Viscous Drag I An object moved through a fluid is surrounded by a “flow-field” (red). Fluid at the surface of the object moves along with the object. Fluid a large distance away does not move at all. We say there is a “velocity gradient” or “shear field” near the object. Molecules (particularly long molecules) intertwine with their faster moving neighbors and hold them back. This “intermolecular friction” causes viscosity. A :35

8 Viscous Drag II “k” is a “form-factor” which depends on the shape of the object and how that affects the gradient field of the fluid. “D” is a “characteristic length” of the object The higher the velocity of the object, the larger the velocity gradient around it. Thus drag is proportional to velocity D :40

9 Viscous Drag III – Stokes Law Form-factor k becomes “D” is diameter of sphere Viscous drag on walls of sphere is responsible for retarding force. George Stokes [ ]  (Navier-Stokes equations/ Stokes’ theorem) D :45

10 Falling raindrops I Problems: A small raindrop falls through a cloud. At time t=0 its velocity is purely horizontal. Describe it’s velocity vs. time. Raindrop is 10  m diameter, density is 1 g/cc, viscosity of air is 180  Poise Work the same problem with a 100  m drop. z x :55

11 Falling raindrops II 1) Newton 2) On z-axis 3) Rewrite in terms of v 4) Variable substitution 5) Solve by inspection z x :60

12 Falling raindrops III 1) Our solution 2) Substitute original variable 3) Apply boundary conditions (v0=0) 4) Expand “b” 5) Define v terminal :05

13 Velocity Dependent Force  Forces are generally dependent on velocity and time as well as position  Fluid drag force can be approximated with a linear and a quadratic term = Linear drag factor (Stokes Law, Viscous or “skin” drag) = Quadratic drag factor ( Newton’s Law, Inertial or “form” drag) :15

14 The Reynolds Number R < 10 – Linear drag 1000< R < 300,000 – Quadratic R > 300,000 – Turbulent D v :20

15 Reynolds Number Regimes R < 10 – Linear drag 1000< R < 300,000 – Quadratic R > 300,000 – Turbulent

16 The Reynolds Number II :20 D “D”= “characteristic” length

17 The Reynolds Number III R < 10 – Linear drag 1000< R < 300,000 – Quadratic R > 300,000 – Turbulent D v

18 Vector Relationships -- Problem #3-1 “The dot-product trick” Given vectors A and B which correspond to symmetry axes of a crystal: Calculate: Where theta is angle between A and B A B :38

19 Falling raindrops L3-2 A small raindrop falls through a cloud. It has a 10  m radius. The density of water is 1 g/cc. The viscosity of air is 180  Poise. a) Quantify the force on the drop for a velocity of 10 mm/sec. b) What should be the terminal velocity of the raindrop? c) What is the Reynolds number of this raindrop? Work the same problem with a 100  m drop. :50

20 Pool Ball L3-3 A pool ball 6 cm in diameter falls through a graduated cylinder. The density of the pool ball is 1.57 g/cc. The viscosity of air is 180  Poise. a) Quantify the force on the ball for a velocity of 100 mm/sec. b) What should be the terminal velocity of the ball? c) Quantify the force if we assume quadratic drag :50

21 Stokes Dynamics :10