User:Egm6322.s12.team2.steele.m2/Mtg22+23

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

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Notes, Sec 22 Plan: Model assignment table/contribution table: Team 1 R3 	Exact Nn-ODE’s (Non-linear nth order) Case n=1: N1-ODE 2nd exactness condition p. 22-6 cont’d .		Method 1 Method 2 Case n=2: N2-ODE 2nd exactness condition Method 1 Method 2 Particle moving with air resistance: p.14-1 cont’d WolframAlpha: Symbolic computation, and more A beautiful game: Soccer (Magnus force, Bernoulli’s principle)

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=Case n=1: N1-ODE p.21-8 cont’d= Equivalence of two forms of 2nd exactness condition, i.e., [[media:Pea1.f11.mtg21.djvu|Eqn(3) p.21-8]] and [[media:Pea1.f11.mtg16.djvu|Eqns(2)-(3) p.9-3]]. (2)	P.21-8 and [[media:Pea1.f11.mtg21.djvu|Eqn(3) p.21-7]]:

(3)	[[media:Pea1.f11.mtg21.djvu|Eqns(1)-(2) p.21-8,]] $$\phi_{xy} = \phi_{yx}$$	[[media:Pea1.f11.mtg9.djvu|Eqn(2) p.9-3]] (end Method 1) ///

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Method 2: Work with coefficients in 1st exactness condition.


 * 1) (3) p.21-8 and (3) p.21-7:

[[media:Pea1.f11.mtg21.djvu|Eqn(3) p.21-8]], (2) and (4): $$M_y = N_x$$		(3) p.9-3 (end Method 2) ///

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=Case n=2: N2-ODE= Equivalence of two forms of 2nd exactness condition: Form 2: (1)-(2) p.21-8:

Form 1: [[media:Pea1.f11.mtg17.djvu|Eqn(1) p.17-3]]: $$\phi_{xy} = \phi_{yx}, \phi_{yp} = \phi_{py}, \phi_{px} = \phi_{xp}$$		(1) p.17-3 Or equivalently [[media:Pea1.f11.mtg16.djvu|Eqns(1)-(2) p.16-5]]: $$f_{xx} + 2pf_{xy} + p^2f_{yy} = g_{xp} + pg_{yp}-g_y$$				(1) p.16-5 $$f_{xp} + pf_{yp} + 2f_y = g_{pp}$$					(2) p.16-5

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Method 1: Equivalence to symmetry of mixed 2nd partial derivatives of first integral $$\phi$$, i.e., (1) p.17-3. R*14.6: Show this equivalence; see p.22-1. Hints: Don’t expand the derivatives too far toward the detail (bottom) level; stay at the more general (top) level as much as possible. Specifically, first keep in mind that the first integral $$\phi$$ is a function of (x,y,y’) which are considered as 3 independent variables. Second, show that $$g_0 := \frac{\partial}{\partial y}\left(\frac{d \phi}{dx} \right)$$		(3) p.21-7

(end Hints) /// (end Method 1) ///

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=Method 2:= Work with coefficients in 1st exactness condition (1)	P.22-3 is equivalent to the 2 relations (1)-(2) p.16-5 in the 2nd exactness condition for N2-ODEs: See F09 Mtgs 13-14 for the details on the homework problem statement. Below is a detailed explanation. In addition to p(x) := y’(x)					(2) p.73
 * 1) (3) p.21-7:

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R*5.1: 	(1) p.22-3:

Define: (2)

Since 1 and q are linearly independent (q = y’’ is in general not a constant), we must have $$\bar g = 0 \Leftrightarrow$$(1) p.16-5 $$\bar f = 0 \Leftrightarrow$$(2) p.16-5 (end Method 2)///

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=Particle moving with air resistance: p.14-1 cont’d= A beautiful game: Soccer (Magnus force, Bernoulli’s principle) Soccer physics: Sport projectiles, must account for air resistance; see Mtg 14 (See illustration for $$F_D$$, D and U with vector diagram.)

Variable $$\rho$$ = air mass density			U = velocity magnitude A = cross section area		$$C_D$$ = drag coefficient D = ball diameter		$$\nu$$ = kinematic viscosity $$\mu$$ = dynamic / kinetic viscosity

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Goff, Power and spin in the beautiful game, Physics Today, Jul 2010. See graph of drag crisis with $$\nu = \frac{\mu}{\rho}$$ and Re = $$\frac{UD}{\nu}$$ See graph of $$\omega x U$$ and $$F_M$$ = Magnus force

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See graph of midfield free kick, soccer field with top view, and corner kick.

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See graph of Bernoulli’s principle, increased velocity, and decreased pressure.

See graph of life, wing, presser exerted by faster moving air, and flight.