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final_holcomb

# final_holcomb - nal 01 HOLCOMB DAVID Due 2:00 pm 1...

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fnal 01 – HOLCOMB, DAVID – Due: Dec 17 2007, 2:00 pm 1 Mechanics - Basic Physical Concepts Math: Circle: 2 π r , 2 ; Sphere: 4 2 , (4 / 3) 3 Quadratic Eq.: ax 2 + bx + c = 0, x = - b ± b 2 - 4 ac 2 a Cartesian and polar coordinates: x = r cos θ, y = r sin θ , r 2 = x 2 + y 2 , tan θ = y x Trigonometry: cos α cos β + sin α sin β = cos( α - β ) sin α + sin β = 2 sin α + β 2 cos α - β 2 cos α + cos β = 2 cos α + β 2 cos α - β 2 sin 2 θ = 2 sin θ cos θ, cos 2 θ = cos 2 θ - sin 2 θ 1 - cos θ = 2 sin 2 θ 2 , 1 + cos θ = 2 cos 2 θ 2 Vector algebra: H A =( A x ,A y )= A x ˆ ı + A y ˆ Resultant: H R = H A + H B A x + B x y + B y ) Dot: H A · H B = AB cos θ = A x B x + A y B y + A z B z Cross product: ˆ ı × ˆ = ˆ k × ˆ k ı , ˆ k × ˆ ı H C = H A × H B = ± ± ± ± ± ± ˆ ı ˆ ˆ k A x A y A z B x B y B z ± ± ± ± ± ± C = sin θ = A B = , use right hand rule Calculus: d dx x n = nx n - 1 , d dx ln x = 1 x , d sin θ = cos θ , d cos θ = - sin θ , d dx const = 0 Measurements Dimensional analysis: e.g. , F = ma [ M ][ L ][ T ] - 2 , or F = m v 2 r [ M ][ L ][ T ] - 2 Summation: N i =1 ( i + b a N i =1 x i + bN Motion One dimensional motion: v = ds dt , a = dv dt Average values: ¯ v = s f - s i t f - t i a = v f - v i t f - t i One dimensional motion (constant acceleration): v ( t ): v = v 0 + at s ( t s vt = v 0 t + 1 2 2 v = v 0 + v 2 v ( s v 2 = v 2 0 +2 as Nonuniform acceleration: x = x 0 + v 0 t + 1 2 2 + 1 6 jt 3 + 1 24 st 4 + 1 120 kt 5 + 1 720 pt 6 + ... , (jerk, snap, ) Projectile motion: t rise = t fall = t trip 2 = v 0 y g h = 1 2 gt 2 fall ,R = v ox t trip Circular: a c = v 2 r , v = 2 T , f = 1 T (Hertz=s - 1 ) Curvilinear motion: a = ² a 2 t + a 2 r Relative velocity: Hv = ² + Hu Law of Motion and applications Force: H F = mHa, F g = mg, H F 12 = - H F 21 Circular motion: a c = v 2 r ,v = 2 T =2 π r f Friction: F static μ s NF kinetic = μ k N Equilibrium (concurrent forces): i H F i =0 Energy Work (for all F): Δ W = W AB = W B - W A F ³ s = Fs cos θ = H F · Hs ³ B A H F · dHs (in Joules) E±ects due to work done: H F ext = mHa - H F c - H f nc W ext | A B = K B - K A + U B - U A + W diss | A B Kinetic energy: K B - K A = ³ B A · dHs , K = 1 2 mv 2 K (conservative H F ): U B - U A = - ³ B A H F · dHs U gravity = mg y , U spring = 1 2 kx 2 From U to H F : F x = - ∂ U ∂x , F y = - ∂y , F z = - ∂z F gravity = - = - mg , F spring = - = - Equilibrium: = 0, 2 U 2 > 0 stable, < 0 unstable Power: P = dW dt = Fv ³ = cos θ = H F · (Watts) Collision Impulse: H I = Δ H p = H p f - H p i ³ t f t i H F dt Momentum: H p = mHv Two-body: x cm = m 1 x 1 + m 2 x 2 m 1 + m 2 p cm Mv cm = p 1 + p 2 = m 1 v 1 + m 2 v 2 F cm F 1 + F 2 = m 1 a 1 + m 2 a 2 = Ma cm K 1 + K 2 = K * 1 + K * 2 + K cm Two-body collision: H p i = H p f m 1 + m 2 ) cm v * i = v i - v cm , v ² i = v i + v cm Elastic: v 1 - v 2 = - ( v ² 1 - v ² 2 ), v i = - v * i , v ² i v cm - v i Many body center of mass: H r cm = m i >r i m i = ³ >r dm ³ m i Force on cm: H F ext = d>p dt = MHa cm , H p = H p i Rotation of Rigid-Body Kinematics: θ = s r , ω = v r , α = a t r Moment of inertia: I = m i r 2 i = ³ r 2 dm I disk = 1 2 MR 2 , I ring = 1 2 M ( R 2 1 + R 2 2 ) I rod = 1 12 M ± 2 , I rectangle = 1 12 M ( a 2 + b 2 ) I sphere = 2 5 2 , I spherical shell = 2 3 2 I = M (Radius of gyration) 2 , I = I cm + MD 2 Kinetic energies: K rot = 1 2 I ω 2 , K = K rot + K cm Angular momentum: L = r mv = r mω r = Torque: τ = dL dt = m dt r = Fr = I

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final_holcomb - nal 01 HOLCOMB DAVID Due 2:00 pm 1...

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