Unit 2: Angular Kinematics Angular Kinematics Angular measures are similar to linear measures Angular measures have linear analogs 2 methods exist for assessing angles Relative angles Absolute angles Relative angles The angle between 2 adjacent segments Absolute Angles The angle between a segment and a reference plane Angular Kinematics Units Units for relative and absolute angles: Degrees and radians (1 rad = 57.3º) Angular Distance & Angular Displacement Angular Distance—sum of all angular changes undergone by a rotating body/segment Angular Displacement—angle formed between initial and final positions of a rotating body/segment and noting direction Angular Displacement (θ) Angular displacement (θ) is the angle formed between some initial and final angle and noting direction. Expressions of displacement: θ = θfinal – θinitial or θf – θi Angular Direction Linear & Angular Displacement Relationship Linear displacement (d) = radius of rotation(r) x angular displacement(θ) Or d = r θ Angular Speed & Angular Velocity Angular speed (scalar)—measures how quickly a given angular distance is covered = Δangular distance/ Δtime Angular Velocity (ω) (Vector)—measures how quickly a body changes angular position and the direction of movement ω= Δangular displacement/ Δtime Example: What is the angular velocity about the knee when moved from 88 to 175 deg in .15sec? (175-88deg)/(.15s-0) = 580 degrees/second Linear & Angular Velocity Relationship Linear velocity (v) = radius of rotation (r) * angular velocity (ω) v= r ω The greater the angular velocity of a baseball bat, the farther a struck ball will travel, other conditions being equal Linear & Angular Velocity Relationship Example Linear velocity (v) = radius of rotation (r) * angular velocity (ω) v= r ω 2 baseballs are consecutively hit by a bat. The first ball is hit .20 meters from the bat’s axis of rotation, and second ball is hit from .40 meters from the bat’s axis of rotation. If the angular velocity of the bat was 30 rad/s at the instant that both balls were contacted, what was the linear velocity of the bat at the 2 contact points? Ball 1 v=.20m*30rad/s = 6m/s Ball 2 v=.40m*30rad/s = 12 m/s Angular Acceleration Angular acceleration is a vector quantity and is the rate change in angular velocity. Angular acceleration (α) = 훥angular velocity/훥time α = ωf- ωi/ tf-ti α=ω/t Linear-Angular Relationships d = r θ v= r ω r is radius of rotation (distance between point of interest and axis of rotation. Angular units must be in radians! Increase r, increase d & v! To Summarize… Most human body movement involves the rotation of bones around the joint centers or axes of rotation. The angular kinematic quantities (distance and displacement, speed and velocity, and acceleration) possess the same interrelationships as their linear counterparts. Angular kinematic variables may be quantified for the relative angle formed by two body segments articulating at a joint, or for the absolute angular orientation of a single segment with respect to a fixed, horizontal or vertical, reference line. There is a direct relationship between the angular motion characteristics of a segment or body and the curvilinear displacement and velocity of any point on that body. When calculating angular motion from linear motion (or vice versa), the use of radian-based units is necessary..