ROTATIONAL KINEMATICS
Angular Velocity Vector
● Consider a rotation which evolves over time: R t R t R t xx xy xz x ' t , y ' t , z ' t or R t = R t R t R t yx yy yz Rzx t Rzy t Rzz t – Both axis and rate of rotation may change over time
● Over infinitesimal time interval between t and t+dt: – Rotation about an “instantaneous axis” by small angle dθ: – = R d R d R d R d A x x y y z z (in any order) – Can write rate of rotation as a vector – xyz treated equally
d d d (Points ● Angular velocity vector: ≡ x i y j z k along axis dt dt dt of rotation)
Rotational Kinematics
● Taylor Series for angular velocity: 2 d 1 d 2 = ∣ t − t ∣ t − t ... 0 0 2 0 dt t 2 dt 0 t 0
“angular acceleration” ( ) – may or may not point along – Similar to translational kinematics, with no “position vector”
● For rotations about a constant axis: – Rotations do commute → can assign an “angular position” θ – Taylor Series for rotation angle (about a constant axis only):
2 d 1 d 2 = ∣ t − t ∣ t − t ... 1-D kinematics 0 0 2 0 dt t 2 dt equations for x,v ,a 0 t 0 x x can now be applied Ω to θ, Ω, α 0 0 Time Derivative in Rotating Frames
● R matrix transforms components of a vector: u ' = R u – Time derivatives in rotating frames must take into account: – 1) time dependence of the actual vector – 2) changing direction of coordinate axes→time dependence of R
● Example: Frame S' rotating with angular velocity – d u ' = d u d R Using the chain rule: R u dt dt dt − − dR = R t dt R t = R dt 1 ● R t By definition: dt dt dt
− − 1 z dt y dt 0 = dR z y − = − Plug in R dt dt 1 dt z x z 0 x R dt − dt 1 dt − y x y x 0 Time Derivative in Rotating Frames − 0 z y ● − Examine the effect of matrix z 0 x on a vector − y x 0 − − 0 b x z b y y bz z y − = − = − × The cross product can be 0 b z x b y x bz z b x written as an operator in − y x 0 − matrix form! bz y bx x b y
● Time derivative of a vector in a rotating frame: d u ' = d u − [ × ] Vectors which are defined by a time R R u derivative (e.g. velocity) pick up an dt dt extra term when they are transformed
● [ ] Examples: Velocity → v ' = R v − × R r d v ' d v dR d r dR = R v − [ × R r ] ● Acceleration dt dt dt dt dt
[ ] [ ] a ' = R a − 2 × R v × × R r Coriolis and Centrifugal Acceleration [ ] [ ] a ' = R a − 2 × R v × × R r ● More useful to relate a' to v' and r' (instead of v and r) – This way, all measurements can be made in rotating frame Plug in R r = r ' [ ] and R v = v ' × r '
[ ] [ ] a ' = R a − 2 × v ' × r ' × × r '
[ ] [ ] a ' = R a − 2 × v ' − × × r '
Inertial acceleration Centrifugal acceleration Coriolis acceleration
acentrifugal (Index Notation)
= − [ × × ] = − × × acentrifugal r i ei j e j r k ek = − × = − acentrifugal i ei j rk jkl el i j rk jkl ilm e m = − = − acentrifugal jkl mil i j r k em ji km jm ki i j rk e m = − acentrifugal i i r k ek i j ri e j 2 a = r − ⋅r centrifugal r ⊥ 2 a = r − r cos centrifugal r ∥ = 2 − = 2 r acentrifugal r r ∥ r ⊥
● acentrifugal always points directly away from rotation axis
Coriolis / Centrifugal Examples ● Bead on spoke of bicycle wheel rotating at Ω – = − Slides outward such that r r0 k 0 (ignore gravity) – Calculate r'(t), v'(t), Coriolis and centrifugal accelerations
● Spacecraft approaches S.B. from above (34º latitude) – At some instant: – Earth rotates at angular velocity Ω, spacecraft is at r = 2RE – Falling straight down at speed v (in inertial space) – Calculate (North/South, East/West, Up/Down) components of v' – Calculate components of Coriolis and centrifugal accelerations Fictitious vs. Real Forces
● Normal, Friction, Tension, Compression – Exerted by particles on other particles nearby – Newtonian Physics does not attempt to describe further
● Does a particle exert Coriolis and centrifugal forces? – No! They are a property of the rotating space itself
● Which type of force is gravity? Has aspects of both... – Gravitational force is directly exerted by space – But has particles as its source – Einstein's work shows gravity acts more like a fictitious force – Examples: 1) Does gravity affect light? 2) orbit of Mercury