Transcript Lecture5
Lecture #5 of 25 Moment of inertia 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 1 :10 Moment of inertia L5-1 R Given a solid quarter disk with uniform massdensity s and radius R: j O1 r Calculate I total Write r in polar coords Write out double integral, both r and phi components Solve integral Calculate 2 I CM r 2 dm r s (r )dA I O1 Given that CM is located at (2R/3, p/4) Calculate ICM 2 :10 Velocity Dependent Force F F (r , r , t ) Forces are generally dependent on velocity and time as well as position 2 Fr (r ) br cr Fluid drag force can be approximated with a linear and a quadratic term Ratio f quad f lin is important drag factor b = Linear (Stokes Law, Viscous or “skin” drag) c = Quadratic drag factor ( Newton’s Law, Inertial or “form” drag) 3 :15 The Reynolds Number density viscosity R R < 10 – Linear drag 1000< R < 300,000 – Quadratic R > 300,000 – Turbulent D vD R inertial (quad ) drag viscous (linear ) drag v 4 :20 vD R The Reynolds Number II D vD R R < 10 – Linear drag Fd 1 2 v 2 CD v 1000< R < 300,000 – Quadratic R > 300,000 – Turbulent Linear Regime kD v CD 1 2 v A 2 density viscosity 1 D 2 vD Re Quadratic Regime CD kA v 2 1 2 1 2 v A 2 k 5 :25 Defining Viscosity Fdrag y A u xˆ y Fdrag u A y x Two planes of Area “A” separated by gap y Top plane moves at relative velocity u xˆ u defines viscosity (“eta”) F A y 2 N s / m MKS Units of are Pascal-seconds Only CGS units (poise) are actually used 2 1 poise=0.1 N s / m 6 :30 Viscous Drag I Fdrag A u xˆ Fdrag du A xˆ dy 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. We are changing the momentum of the nearby fluid. This dp/dt creates a force which we call the viscous drag. 7 :35 Viscous Drag II Fdrag D u xˆ Fdrag k D u xˆ “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 8 :40 Viscous Drag III – Stokes Law Fdrag D u xˆ Fdrag b r Fdrag 3p D u xˆ Form-factor k becomes 3p “D” is diameter of sphere Viscous drag on walls of sphere is responsible for retarding force. George Stokes [1819-1903] (Navier-Stokes equations/ Stokes’ theorem) 9 :45 Falling raindrops L5-2 A small raindrop falls through a cloud. It has a 10 mm radius. The density of water is 1 g/cc. The viscosity of air is 180 mPoise. a) Draw the free-body diagram. b) Quantify the force on the drop for a velocity of 10 mm/sec. c) What is the Reynolds number of this raindrop d) What should be the terminal velocity of the raindrop? Work the same problem with a 100 mm drop. 10 :50 Falling raindrops I Fdrag , mg z x Problems: A small raindrop falls through a cloud. At time t=0 its velocity is purely horizontal. v0 3 xˆ m / s 2 Describe it’s velocity vs. time. Raindrop is 10 mm diameter, density is 1 g/cc, viscosity of air is 180 mPoise Work the same problem with a 100 mm drop. 11 :55 Falling raindrops II Fdrag mr mgzˆ br Assume vertical motion mz mg bz dvz b g vz dt m mg z x b b Define u g vz u vz m m b t b u u u u (0)e m m 1) Newton 2) On z-axis 3) Rewrite in terms of v 4) Variable substitution 5) Solve by inspection 12 :60 Falling raindrops III b t m u u (0)e b mg m u g vz vz u m b b b t mg m vz u (0)e m b b b mg t mg m vt vz 1 e 3pD b 3pD t mg 1 e m vz 3pD g t v vz v 1 e 1) Our solution 2) Substitute original variable 3) Apply boundary conditions 4) Expand “b” 5) Define vterminal 13 :05 Stokes Dynamics 14 :10 Lecture #5 Wind-up . R vD . g t . v vz v 1 e mg vt 3pD Read sections Taylor 2.1-2.4 Office hours today 3-5 Registration closes Friday 15 :72