HW11S - 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 , y = r sin , r 2 = x 2 + y 2 , tan = y x Trigonometry: cos cos + sin sin = cos( - ) sin + sin = 2 sin + 2 cos - 2 cos + cos = 2 cos + 2 cos - 2 sin2 = 2 sin cos , cos2 = cos 2 - sin 2 1- cos = 2 sin 2 2 , 1 + cos = 2 cos 2 2 Vector algebra: ~ A = ( A x , A y ) = A x + A y Resultant: ~ R = ~ A + ~ B = ( A x + B x , A y + B y ) Dot: ~ A ~ B = AB cos = A x B x + A y B y + A z B z Cross product: = k , k = , k = ~ C = ~ A ~ B = fl fl fl fl fl fl k A x A y A z B x B y B z fl fl fl fl fl fl C = AB sin = A B = AB , use right hand rule Calculus: d dx x n = nx n- 1 , d dx ln x = 1 x , d d sin = cos , d d cos =- sin , 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 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 r T = 2 r f Friction: F static s N F kinetic = k N Equilibrium (concurrent forces): i ~ F i = 0 Energy Work (for all F): W = W A B = W B- W A = F s k = F k s = Fs cos = ~ 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 = ~ F ~v (Watts) Collision Impulse: ~ I = ~ 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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HW11S - Mechanics - Basic Physical Concepts Mathematics...

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