# HW3S - Mechanics Basic Physical Concepts Mathematics...

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Unformatted text preview: Mechanics - Basic Physical Concepts Mathematics 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: ~ 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 = AB 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 = AB sin θ = A ⊥ B = AB ⊥ , use right hand rule Calculus: d dx x n = nx n- 1 , d dx ln x = 1 x , d dθ sin θ = cos θ , d dθ 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 ( 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 + a t s ( t ) : s = ¯ v t = v t + 1 2 a t 2 , ¯ v = v + v 2 v ( s ) : v 2 = v 2 + 2 a s Nonuniform acceleration: x = x + v 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 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 + ~u Law of Motion and applications Force: ~ F = m~a, F g = mg, ~ 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 A → B = W B- W A = F s k = F k s = Fs cos θ = ~ F · ~s → R B A ~ F · d~s (in Joules) Effects due to work done: F ext = ma + 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 mv 2 K (conservative ~ F ): U B- U A =- R B A ~ F · d~s U gravity = mg 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 =- mg , F spring =- ∂ U ∂x =- k x Equilibrium: ∂ U ∂x = 0, ∂ 2 U ∂x 2 > 0 stable, < 0 unstable Power: P = dW 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 =...
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HW3S - Mechanics Basic Physical Concepts Mathematics...

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