Homework 9

# Homework 9 - homework 09 BAUTISTA ALDO Due 4:00 am 1...

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homework 09 – BAUTISTA, ALDO – Due: Oct 30 2006, 4:00 am 1 Mechanics - Basic Physical Concepts Math: Circle: 2 π r , π r 2 ; Sphere: 4 π r 2 , (4 / 3) π r 3 Quadratic Eq.: a x 2 + b x + c = 0, x = - b ± b 2 - 4 a c 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: ~ A = ( A x , A y ) = A x ˆ ı + A y ˆ Resultant: ~ R = ~ A + ~ B = ( A x + B x , A y + B y ) Dot: ~ A · ~ B = A B cos θ = A x B x + A y B y + A z B z Cross product: ˆ ı × ˆ = ˆ k , ˆ × ˆ k = ˆ ı , ˆ k × ˆ ı = ˆ ~ C = ~ A × ~ B = fl fl fl fl fl fl ˆ ı ˆ ˆ k A x A y A z B x B y B z fl fl fl fl fl fl C = A B sin θ = A B = A B , use right hand rule Calculus: d dx x n = n x 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 = m a [ M ][ L ][ T ] - 2 , or F = m v 2 r [ M ][ L ][ T ] - 2 Summation: N i =1 ( a x i + b ) = a N i =1 x i + b N Motion One dimensional motion: v = d s dt , a = d v 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 + a t s ( t ) : s = ¯ v t = v 0 t + 1 2 a t 2 , ¯ v = v 0 + v 2 v ( s ) : v 2 = v 2 0 + 2 a s Nonuniform acceleration: x = x 0 + v 0 t + 1 2 a t 2 + 1 6 j t 3 + 1 24 s t 4 + 1 120 k t 5 + 1 720 p t 6 + . . . , (jerk, snap, . . . ) Projectile motion: t rise = t fall = t trip 2 = v 0 y g h = 1 2 g t 2 fall , R = v ox t trip Circular: a c = v 2 r , v = 2 π r T , f = 1 T (Hertz=s - 1 ) Curvilinear motion: a = q a 2 t + a 2 r Relative velocity: ~v = ~v 0 + ~u Law of Motion and applications Force: ~ F = m~a, F g = m g, ~ F 12 = - ~ F 21 Circular motion: a c = v 2 r , v = 2 π r T = 2 π r f Friction: F static μ s N F kinetic = μ k N Equilibrium (concurrent forces): i ~ F i = 0 Energy Work (for all F): Δ W = W AB = W B - W A F k s = Fs cos θ = ~ F · ~s R B A ~ F · d~s (in Joules) Effects due to work done: ~ F ext = m~a - ~ F 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 = R B A m~a · d~s , K = 1 2 m v 2 K (conservative ~ F ): U B - U A = - R B A ~ F · d~s U gravity = m g y , U spring = 1 2 k x 2 From U to ~ F : F x = - ∂ U ∂x , F y = - ∂ U ∂y , F z = - ∂ U ∂z F gravity = - ∂ U ∂y = - m g , F spring = - ∂ U ∂x = - k x Equilibrium: ∂ U ∂x = 0, 2 U ∂x 2 > 0 stable, < 0 unstable Power: P = d W dt = F v k = F v cos θ = ~ F · ~v (Watts) Collision Impulse: ~ I = Δ ~ p = ~ p f - ~ p i R t f t i ~ F dt Momentum: ~ p = m~v Two-body: x cm = m 1 x 1 + m 2 x 2 m 1 + m 2 p cm M v 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 = M a cm K 1 + K 2 = K * 1 + K * 2 + K cm Two-body collision: ~ p i = ~ p f = ( m 1 + m 2 ) ~v cm v * i = v i - v cm , v 0 i = v *0 i + v cm Elastic: v 1 - v 2 = - ( v 0 1 - v 0 2 ), v *0 i = - v * i , v 0 i = 2 v cm - v i Many body center of mass: ~ r cm = m i ~r i m i = R ~r dm R m i Force on cm: ~ F ext = d~p dt = M~a cm , ~ p = ~ 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 = R r 2 dm I disk = 1 2 M R 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 M R 2 , I spherical shell = 2 3 M R 2 I = M (Radius of gyration) 2 , I = I cm + M D 2 Kinetic energies: K rot = 1 2 I ω 2 , K = K

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