B i o L a b - B i o m e c h a n i c s T e a c h i n g & L e a r n i n g T o o l B o xB i o L a b - B i o m e c h a n i c s T e a c h i n g & L e a r n i n g T o o l B o x

Angular KinematicsAngular Kinematics

An Introduction to Angular Kinematics

An Introduction to Angular Kinematics

AimAim

• 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

• 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

Angular Kinematic AnalysisAngular Kinematic Analysis

Absolute and Relative AnglesAbsolute 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)

Units of MeasurementUnits 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

θ

radius (r)

arc (d)

d = =

rθ 1 radian

Angular Motion VectorsAngular Motion Vectors

• 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 Distance and DisplacementAngular 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

11

θ1 = 90º

22

θ2 = 110º

Angular Displacement:

∆θ = 110º - 90º = 20º

Angular Displacement:

∆θ = 110º - 90º = 20º

Angular VelocityAngular Velocity

• Angular velocity (ω) is equal to the angular displacement (∆θ) divided by change in time (∆t)

t =

t=2 1ω

- θ θ

θ

• Units:

º/s or º·s-1

rad/s or rad·s-1

Example:

∆θ = 45º ∆t = 0.6 s

-1 -1ω45

= = 75 º·s 1.31 rad= 6

·s0.

Angular AccelerationAngular Acceleration

• Angular acceleration (α) is equal to change in angular velocity (∆ω) divided by change in time (∆t)

t =

t=2 1α

- ω ω

ω

• 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

-21.31= = α 2.18 rad

6·s

0.

11

θ = 170º

t1 = 0 s

22

θ = 91º

t2 = 0.5 s

θ = 185º

t3 = 0.8 s

33

Standing Vertical Jump – DisplacementStanding 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

θ = 170º

t1 = 0 s

22

θ = 91º

t2 = 0.5 s

• Angular velocity (ω)

– During countermovement

ω = -158º·s-1 or -2.76 rad·s-1

– During upward movement

ω = 313º·s-1 or 5.47 rad·s-1

N.B.

Flexion = -ve velocity

Extension = +ve velocity

= t

θ -79ω =

0.5

= t

θ 94ω =

0.3

Standing Vertical Jump - VelocityStanding Vertical Jump - Velocity

θ = 185º

t3 = 0.8 s

33

Standing Vertical Jump - AccelerationStanding 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

= t

ωα

= (5.47 - (-2.76)) 8.22

α = 0.3 0.3

SummarySummary

• 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

• 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.

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