An Introduction to Angular Kinematics

1. An Introduction to Angular Kinematics Angular Kinematics

2. Aim • The aim of these slides is to introduce the variables used in angular kinematics • These slides include: • An introduction to conventions used to measure angles and vectors used to represent angular quantities • Definitions and calculations of angular displacement, velocity and acceleration • Application and interpretation of angular displacement, velocity and acceleration in simple motion

3. Angular Kinematic Analysis • Angular Kinematics • Description of the circular motion or rotation of a body • Motion described in terms of (variables): • Angular position and displacement • Angular velocity • Angular acceleration • Rotation of body segments • e.g. Flexion of forearm about transverse axis through elbow joint centre • Rotation of whole body • e.g. Rotation of body around centre of mass (CM) during somersaulting

4. Absolute and Relative Angles • Absolute angles • Angle of a single body segment, relative to (normally) a right horizontal line (e.g. trunk, head, thigh) • Relative Angles • Angle of one segment relative to another (e.g. knee, elbow, ankle)

5. arc (d) θ radius (r) Units of Measurement • Angles are expressed in one of the following units: • Revolutions (Rev) • Normally used to quantify body rotations in diving, gymnastics etc. • 1 rev = 360º or 2 π radians • Degrees (º) • Normally used to quantify angular position, distance and displacement • Radians (rad) • Normally used to quantify angular velocity and acceleration • Convert degrees to radians by dividing by 57.3

6. Right-hand thumb rule Fingers of right hand curled in direction of rotation Direction of extended thumb coincides with the direction of the angular motion vector Counter clockwise rotations are positive Clockwise rotations are negative Angular Motion Vectors

7. 1 2 θ1 = 90º θ2 = 110º Angular Distance and Displacement • Angular distance: • The sum of all the angular changes of a rotating body between its initial and final positions • Denoted by ∆θ • Angular displacement • The difference between the final and initial positions of a rotating body • Calculated by θ2 - θ1 • Denoted by ∆θ Angular Displacement: ∆θ = 110º - 90º = 20º

8. Angular Velocity • Angular velocity (ω) is equal to the angular displacement (∆θ)divided by change in time (∆t) • Units: º/s or º·s-1 rad/s or rad·s-1 Example: ∆θ = 45º∆t = 0.6 s

9. Angular Acceleration • Angular acceleration (α) is equal to change in angular velocity (∆ω)divided by change in time (∆t) • Units: º/s/s or º·s-2 rad/s/s or rad·s-2 Example: ∆ω= 1.31 rad·s-2∆t = 0.5 s

10. 1 3 2 θ = 170º t1 = 0 s θ = 91º t2 = 0.5 s θ = 185º t3 = 0.8 s Standing Vertical Jump – Displacement • Angular displacement (∆θ) • During countermovement = θ2 - θ1 or ∆θ = 91 - 170 = -79º • During upward movement = θ2 - θ1 or θ = 185 - 91 = 94º N.B. Flexion = -ve displacement Extension = +ve displacement

11. 1 3 2 θ = 170º t1 = 0 s θ = 91º t2 = 0.5 s θ = 185º t3 = 0.8 s Standing Vertical Jump - Velocity • Angular velocity (ω) • During countermovement ω= -158º·s-1or -2.76 rad·s-1 • During upward movement ω= 313º·s-1or 5.47 rad·s-1 N.B. Flexion = -ve velocity Extension = +ve velocity

12. Standing Vertical Jump - Acceleration • Angular acceleration (α) • Between countermovement and upward movement α = 27.4 rad·s-2 N.B. Positive angular acceleration decreases negative angular velocity to zero at bottom of countermovement and increases positive angular velocity from bottom of countermovement

13. Summary • Angular distance is the angle between two bodies • Angular displacement is the angle through which a body has been rotated • Average angular velocity is the angular displacement divided by the change in time • Average angular acceleration is the change in angular velocity divided by the change in time

14. Recommended Reading • Enoka, R.M. (2002). Neuromechanics of Human Movement (3rd edition). Champaign, IL.: Human Kinetics. Pages 3-10 & 27-33. • Grimshaw, P., Lees, A., Fowler, N. & Burden, A. (2006). Sport and Exercise Biomechanics. New York: Taylor & Francis. Pages 22-29. • Hamill, J. & Knutzen, K.M. (2003). Biomechanical Basis of Human Movement (2nd edition). Philadelphia: Lippincott Williams & Wilkins. Pages 309-336. • McGinnis, P.M. (2005). Biomechanics of Sport and Exercise (2nd edition). Champaign, IL.: Human Kinetics. Pages 147-158.