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 can be approximated by y pp y a circle of radius R (radius of curvature):

2

" ) y' (1 R

2 3

+ =

( )

ds y"

R

2 2

d d d d = ψ

x ψ

2 2

dy dx ds + =

x ψ

Tangential and normal components g p y Velocity:

v v vT = =

y

vN =

Tangent Normal Acceleration: R

dt dv aT =

x ψ

R v a

2 N =

x ψ

R

Polar coordinates

+ = ˆ ˆ i ˆ y ˆ sin x ˆ cos r ˆ θ θ θ θ θ

y

( )

⎞ ⎛ + − = sin cos

• r

y ˆ cos x ˆ sin θ θ θ θ θ

y

r ˆ θˆ

( ) (

)

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = : s coordinate

• f

form In cos sin sin cos y ˆ x ˆ ˆ r ˆ θ θ θ θ θ

r

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

−

y x cos sin sin cos r r : s coordinate

• f

form In

1 r

θ θ θ θ

θ

θ

⎠ ⎝ ⎠ ⎝ ⎠ ⎝ y cos sin r θ θ

θ

x Note that and ar ith time

ˆ θˆ

Note that and vary with time.

r θ

Position and velocity in polar coordinates y p y Position:

r ˆ r r =

y

r ˆ θˆ

r Velocity:

&

θ

θ

θ

r v r vr & = =

x

θ θ ˆ r r ˆ r v

• r

& & v + =

Acceleration in polar coordinates p y Acceleration: y

r ˆ θˆ

Centripetal r

θ & & & r

• r

a

2 r =

θ

θ θ

θ

& & & & r 2 r a + =

x Coriolis