Section 12.8 Planar Motion: Radial and Transverse Components RADIAL AND TRANSVERSE COMPONENTS (page 1) 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 ui? =1 r ?, i?, =1 ?, i?, = 0
Section 12.8 Planar Motion: Radial and Transverse Components RADIAL AND TRANSVERSE COMPONENTS (page 2) POSITION, VELOCITY, ACCELERATION · Position: i =r? r · Velocity: where Speed: V= VVÝ + V? · Acceleration: ? =a ú +a, ú, where a = " ! r " 2 and a, =r! +2 !!! v = v? + v, ?, rr V=1 r and V, =r! The expression for a, may be written alternatively as a, =1 d (rv r dt 1 d 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 ?, ="sim! i + cos! j · Time derivatives of unit vectors Notation: ? = df · Position: · Velocity: v="= d(ru.)=du +rd, = du +r/ u, · Acceleration: ! dt ! r ! u ! r F =r? r dt à = v= "(ru +r! u, ) E " =#u +Hd + (r! + #! )u, + r.u, = (#! ! r !"2)u + (r " + 2x+) u, 1 d (rv, ) == (r2 !) = - (r2 !! + 2rt! ) =r! + 2t! = a, r dt r r