Class 5: Tangential and normal components Class 5: Tangential and - PowerPoint PPT Presentation

Class 5: Tangential and normal components Class 5: Tangential and normal components Polar coordinates

Radius of curvature Any point on the path y can be approximated by pp y a circle of radius R ( (radius of curvature): ) + 3 2 (1 y' ) 2 = R y" " R ds = ψ d ψ ψ x x = + 2 2 2 2 d ds d dx d dy

Tangential and normal components g p y Velocity: y = v T v v N = = v 0 0 Tangent Acceleration: Normal dv a T = R dt 2 v N = a ψ ψ R R x x

Polar coordinates = θ + θ ˆ ˆ ˆ r cos x sin y y y θ θ = − θ θ + θ θ ˆ ˆ ˆ ˆ ˆ sin i x cos y θ ˆ ˆ r or ( ( ) ) ( θ θ − θ θ ⎛ ⎛ ⎞ ⎞ cos cos sin sin ) θ = ⎜ ⎟ ˆ ˆ ˆ ˆ r x y ⎜ ⎟ θ θ ⎝ ⎠ sin cos r In In form form of of coordinate coordinate s s : : − θ − θ 1 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ r cos sin x θ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = r ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ θ θ θ θ ⎝ ⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎠ r r sin sin cos cos y y θ θ x θ ˆ θ Note that Note that and vary with time. ˆ and ar ith time r

Position and velocity in polar coordinates y p Position: y y θ ˆ r = ˆ ˆ r r r Velocity: r = & v r r θ & = θ v r θ x v & = + θ ˆ θ & ˆ or v r r r

Acceleration in polar coordinates p Acceleration: y y θ ˆ Centripetal ˆ r & r = θ & & 2 a r - r r θ & & & = θ + θ & a r 2 r θ x Coriolis