Formula Sheet - Vectors: Scalar Product (Dot Product)...

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Formulae: Vectors: A = A x i + A y j + A z k Scalar Product (“Dot” Product) A · B = A x B x + A y B y + A z B z = AB cos θ A , B Vector Product (“Cross” Product) direction: “ right-hand rule A×B = ( A y B z A z B y ) i + ( A z B x A x B z ) j + ( A x B y A y B x ) k | A×B | = AB sin θ A , B Kinematics: the “motion”: position r as function of time t r = r ( t ) (position) velocity v ; acceleration a : v d r / dt (speed | v | ); a d v / dt Linear motion with constant a : v = v 0 + a t, r = r 0 + v 0 t + ½ a t 2 ; eliminating t : v 2 = v 0 2 + 2 a ·( r r 0 ) rotation angle θ (radians; rotation radius R ): s (=arc length) /R = ( t ) (angular position) angular velocity ω ; angular acceleration α : v tan /R ; a tan /R ( tan = tangential) Circular motion (radius R ) with constant α : = 0 + t; = 0 + 0 t + ½ t 2 ; eliminating t : 2 = 0 2 + 2 ( 0 ) circular motion – radial acceleration a rad : a rad = a c = v T 2 / R (radially inwards) Center-of-Mass position of system (mass M ) r cm Σ i m i r i / Σ i m i = Σ i m i r i / M Moment of Inertia I : I Σ m i r i 2 = r 2 dm ; r i ( r ) = distance between rotation axis and m i ( dm ) thin uniform rod ( M , L ), axis rod through cm: I = ML 2 /12 hollow cylinder ( M , R in , R out ), axis is cylinder axis: I = ½ M ( R in 2 + R out 2 ) uniform solid sphere ( M , R ), axis through center: I = 2 MR 2 /5 Parallel-Axis Theorem I = I //, cm + Md 2 ( d =distance between the parallel axes) Perpendicular-Axis Theorem for a Planar body in the x-y plane: I z = I x + I y Momentum p p Σ i m i v i = M v cm Angular Momentum L L Σ i R i × m i v i (= I ω for rotation around symmetry axis) Moving axis: K tot ½ mv cm
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This note was uploaded on 09/26/2008 for the course PHY 131 taught by Professor Rijssenbeek during the Fall '03 term at SUNY Stony Brook.

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Formula Sheet - Vectors: Scalar Product (Dot Product)...

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