23_KINE426_Angular_Kinematics

1 Andrew Nordin, PhD Assistant Professor, Division of Kinesiology KINE 426 Exercise Biomechanics Angular Kinematics Chapter 9 Angle conventions • Direction of rotation (polarity) – Right hand rule • Thumb represents axis of rotation (vertex) • Fingers represent direction of rotation – Clockwise vs. counterclockwise • Thumb indicates positive rotation • Clockwise (- rotation) • Counterclockwise (+ rotation) r axis Δθ 12 3 6 9

2 Angle conventions Positive + Negative - Do not confuse with linear translation direction along the axis of rotation Right hand rule r Δθ 1 axis Angular kinematics Scalar Vector Distance Speed Acceleration Displacement Velocity Magnitude Magnitude & direction ∆ ! = ! ! − ! ! ! = ! ! − ! ! ! ! − ! ! ! = ! ! − ! ! ! ! − ! ! Δθ 2 Δθ 3 Angular distance: ∆ ! = ! ! + ! ! Angular displacement: ∆ ! = ! ! θ : theta ω : omega α : alpha Δθ Angular kinematics θ f θ i Δθ

4 Angular kinematics Angular kinematics Scalar Vector Distance Speed Acceleration Displacement Velocity Magnitude Magnitude & direction ∆ ! = ! ! − ! ! ! = ! ! − ! ! ! ! − ! ! ! = ! ! − ! ! ! ! − ! ! θ : theta ω : omega α : alpha 12 3 6 9 Direction / polarity from Right Hand Rule degrees or rad deg/s or rad/s deg/s 2 or rad/s 2 Acceleration • Linear acceleration & direction – Acceleration sign (+/-) alone does not directly indicate direction !""#$#%!&'( ! = !"#$%&'() !"#$% or !"#$% !"# = !"#$%&'()*)+, !"#$% or ! ! ! ! ! ! ! − ! ! ! ! − ! ! ! ! − ! ! d = 0m t = 0s d = 1m t = 1s d = 2m t = 2s d = 0 d = 1m d = 2m t = 2s t = 1s t = 0 ↑+ v ↓+ v + a - a ↓- v + a ↑- v - a v = 0m/s v = 0m/s v = 0m/s v = 0m/s v = 1m/s v = -1m/s a = 1m/s 2 a = -1m/s 2 a = -1m/s 2 a = 1m/s 2

5 Δθ 1 Acceleration • Angular acceleration & polarity – Acceleration sign (+/-) alone does not directly indicate polarity ! = ! ! − ! ! ! ! − ! ! ! = ! ! − ! ! ! ! − ! ! ↑ +࠵? ↓ +࠵? 0 °/࠵? 0 °/࠵? Δθ 1 ↑ −࠵? ↓ −࠵? 0 °/࠵? 0 °/࠵? +࠵? −࠵? +࠵? −࠵? • Human movement – Rotation → linear translation Linear-angular relationships

7 • Linear vs. angular displacement – ࠵? = ࠵?࠵?࠵?࠵?࠵? – ࠵? = ࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?ℎ – ࠵? = ࠵?࠵?࠵?࠵?࠵?࠵? • Radians are dimensionless – Use radians to work between linear and angular values • Linear quantities – Arc length – Tangential velocity – Tangential acceleration ! = ! ! r θ s axis v T a T Linear-angular relationships ! ! = ! → ! !"# = ! • Tangential velocity – Arc length – Time derivative – Instantaneous – Perpendicular to the radius r ∆ θ ∆ s axis v T Linear-angular relationships r ∆ θ ∆ s axis v T ∆ s ∆t = ࠵? ! ∆࠵? = ∆θ r

8 • Linear vs. angular displacement – Dimensionless radians • Linear vs. angular velocity – Tangential velocity • Linear vs. angular acceleration – Tangential acceleration ! = ! ! r Δθ Δs axis ! = ! ! − ! ! ! ! − ! ! ! ! = !" ω α v T a T ! ! = ! ! ! = ! ! − ! ! ! ! − ! ! ! !"# ! ! = ! ! ! ! !"# ! = ! ! Linear-angular relationships ࠵? = ࠵?࠵? → ࠵?(࠵?࠵?࠵?) • Linear components of angular acceleration – Tangential acceleration – Centripetal acceleration (radial) • Center seeking – Resultant acceleration ! ! = ! ! ! + ! ! ! a T a C ! ! = ! ! ! + ! ! = !" ! ! = ! ! ! ! Linear-angular relationships r Δθ a T a C a R