Test3.sol - midterm 03 RAHMAN TARIQUE Due Apr 3 2008 1:00...

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midterm 03 – RAHMAN, TARIQUE – Due: Apr 3 2008, 1:00 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 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 sin2 θ = 2 sin θ cos θ, cos2 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: vector A = ( A x , A y ) = A x ˆ ı + A y ˆ Resultant: vector R = vector A + vector B = ( A x + B x , A y + B y ) Dot: vector A · vector 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 vector C = vector A × vector B = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ˆ ı ˆ ˆ k A x A y A z B x B y B z vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 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 = 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 + 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 = radicalBig a 2 t + a 2 r Relative velocity: vectorv = vectorv + vectoru Law of Motion and applications Force: vector F = mvectora, F g = m g, vector F 12 = vector 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 vector F i = 0 Energy Work (for all F): Δ W = W AB = W B W A F bardbl s = Fs cos θ = vector F · vectors integraltext B A vector F · dvectors (in Joules) Effects due to work done: vector F ext = mvectora vector F c vector f nc W ext | A B = K B K A + U B U A + W diss | A B Kinetic energy: K B K A = integraltext B A mvectora · dvectors , K = 1 2 m v 2 K (conservative vector F ): U B U A = integraltext B A vector F · dvectors U gravity = m g y , U spring = 1 2 k x 2 From U to vector 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 = dW dt = F v bardbl = F v cos θ = vector F · vectorv (Watts) Collision Impulse: vector I = Δ vector p = vector p f vector p i integraltext t f t i vector F dt Momentum: vector p = mvectorv 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
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