Section 12.8 Planar Motion: Radial and Transverse Components RADIAL AND TRANSVERSE COMPONENTS (page 1) 0 1 r Ur ? 0 0 Position (a) fig12_30a.jpg Copyright @ 2010 Pearson Prentice Hall, Inc. · Unit vectors u = cos! i + sin! j u, = "sin ! i + cos! j ?i? =1 ?, i?, =1 ?, i ù, = 0
Section 12.8 Planar Motion: Radial and Transverse Components RADIAL AND TRANSVERSE COMPONENTS (page 2) POSITION, VELOCITY, ACCELERATION · Position: · Velocity: where V=1' ? and v, =r !! Speed: V = V2 + V · Acceleration: å = a ú + a, ú, where a = # ! r " 2 and a, =r!+ 2+! i =r ù r v = v?, + v, ?, rr The expression for a, may be written alternatively as a, = = 1 d (rv , ) r dt This latter form for a, will be used in Chapter 15 in deriving the equation expressing Conservation of Angular Momentum.
Section 12.8 Planar Motion: Radial and Transverse Components RADIAL AND TRANSVERSE COMPONENTS (page 3) DERIVATION OF EQUATIONS FOR VELOCITY AND ACCELERATION · Unit vectors u = cos! i + sin! j ?, = "sin! i + cos! j · Time derivatives of unit vectors Notation: } = df · Position: · Velocity: v ="'= "(ru) =du +ru = tu +r/u, · Acceleration: dt ! ! r ?, = "?ú r i =r ù dt E dt å = v= "(ru + r! ?, ) E " = Hu +Hd + (r !! + w! )u, + r. u, = (#! ! r !"2)u + (r "! + 2}}) u, 1 d r (rv , )= - (r2 }) = 1(r2 !! + 2rt!) = r !! + 2x! = a, r dt r