84.5 The circular motion 6The relation of v(t), a(andr(t) v(t)=o(t)×r(t) v(t=ro(t) at=@(k @Centripetal acceleration and the angular speed of uniform circular motion r(t=rose(t)i+rsin g(t)j a(0= d2r(t)d dr(o dt dt de de sin g(t)Ji +[.cose(t)Ij 84.5 The circular motion d de a2(1)= rsin g(t)Ji+lr--cos g(t) dt d 8 dicos e(0)i+/r sin 0(01) l ar=@(Ok () dv(t) =|o(t)xr() dt dt a(t) dr(t) Q(rxv(t) Direction: opposite to Magnitude: a=ra 1515 §4.5 The circular motion r r θ x y z t z t k ˆ ω( )=ω ( ) r v r 5The relation of v(t), (t) and r(t) r r r ω v(t) (t) r(t) r r r =ω × v(t) = rω(t) 6Centripetal acceleration and the angular speed of uniform circular motion r t r t i r t j ˆ sin ( ) ˆ ( ) = cosθ ( ) + θ r ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = − + = = t j t t i r t r t t r t t t r t a t c ˆ cos ( )] d d [ ˆ sin ( )] d d [ d d ] d d ( ) [ d d d d ( ) ( ) 2 2 θ θ θ θ r r r §4.5 The circular motion { } ( ) ˆ [ sin ( )] ˆ ) [ cos ( )] d d ( ˆ cos ( )] d d [ ˆ sin ( )] d d [ d d ( ) 2 2 r t r t i r t j t t j t t i r t r t a t z c r r ω θ θ θ θ θ θ θ = − = − + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = − + Magnitude: 2 c z a = rω Direction: opposite to r r ( ) ( ) d d ( ) ( ) [ ( ) ( )] d d d d ( ) ( ) t v t t r t t t r t t t v t a t c r r r r r r r r = × = × = = × ω ω ω r θr x y z t z t k ˆ ω( )=ω ( ) r v r a r