Hayner Final

# Hayner Final - nal 01 RUDE DAVID Due 10:00 pm 1 Mechanics...

This preview shows pages 1–2. Sign up to view the full content.

fnal 01 – RUDE, DAVID – Due: Dec 15 2007, 10:00 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: C A =( A x ,A y )= A x ˆ ı + A y ˆ Resultant: C R = C A + C B A x + B x y + B y ) Dot: C A · C 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 C C = C A × C 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: Cv = ² + Cu Law of Motion and applications Force: C F = mCa, F g = mg, C F 12 = - C 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 C F i =0 Energy Work (for all F): Δ W = W AB = W B - W A F ³ s = Fs cos θ = C F · Cs ³ B A C F · dCs (in Joules) E±ects due to work done: C F ext = mCa - C F c - C 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 · dCs , K = 1 2 mv 2 K (conservative C F ): U B - U A = - ³ B A C F · dCs U gravity = mg y , U spring = 1 2 kx 2 From U to C 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 = C F · (Watts) Collision Impulse: C I = Δ C p = C p f - C p i ³ t f t i C F dt Momentum: C p = mCv 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: C p i = C 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: C r cm = m i ;r i m i = ³ ;r dm ³ m i Force on cm: C F ext = d;p dt = MCa cm , C p = C 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 11/03/2010 for the course PHYSICS 303 taught by Professor Shih during the Spring '10 term at University of Texas.

### Page1 / 18

Hayner Final - nal 01 RUDE DAVID Due 10:00 pm 1 Mechanics...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online