Linear and Angular Kinematics PDF

Preview

Summary

Dr. Thomas...

Description

Linear and Angular Kinematics Definitions  Kinematics describes position, velocity, acceleration (disregards what causes motion Linear vs. Angular Motion  Linear (meters): every part of the object experiences equivalent displacement  If you (object) are walking in a straight line, every part of you will get from A to B (equivalent displacement)  Angular: all parts of the object do not experience the same displacement  Units:  Degree  Revolution  Radian: ratio of circumference of a circle to its radius  1 revolution = 360deg  1 rev = 2pi radians = 6.28 radians  Normal human movement encompasses linear and angular displacement  Walking: you move linearly but your joints go through angular motion  Throwing ball: arm moves angularly to project ball linearly  Cycling  Joint rotations create force on pedals  Rotates gears  wheels  Results in linear motion of rider and bike Linear Motion Considerations  Acceleration  Defined as change in velocity WRT time  Can have + or – value  When direction of motion is described as something other than + or –  + acceleration = speeding up  - acceleration = speeding up  If direction of motion is described as +/ Direction of motion dictates sign and negative acceleration may still be mean speeding up  + = up, - = down  May also have acceleration = 0 if moving at constant velocity  Velocity tells you if you are moving or not, acceleration tells if speeding up or slowing down  Average and Instantaneous Quantities  Dictated by selection of time interval when analyzing motion  Average is sometimes sufficient but may lose information depending on length of t  Instantaneous important when considering projectile motion  Velocity at release dictates how far an object will travel Angular Motion  Angular Distance and Displacement  Similar to linear motion  Shoulder flexion  Move from 00 to 900  Distance: 900  Displacement: 900  If we now return to 00  Distance: 1800 (90 + 90)  Displacement: 00  Sign convention: clockwise motion is (-), counterclockwise motions are (+)  Angular speed and Velocity  Angular speed: angular distance/change in time  Angular velocity: angular displacement/change in time

 What are common angular velocities experienced during sports?  Baseball pitching  Elbow: 2320 0/s of extension  Shoulder: 7240 0/s of IR  Tennis serve  Elbow: 1510 0/s of extension  Shoulder: 2420 0/s of IR  **Baseball faster than tennis, shoulder faster than elbow  Angular Acceleration  Change in angular velocity/change in time Relation Between Linear and Angular Motion  Linear and Angular Displacement  The greater the radius b/w a given point on a rotating body and its axis of rotation, the greater the linear distance covered by that point during an angular motion  S: curvilinear distance  R: radius of rotation  Phi: angular displacement  Curvilinear distance traveled by the point of interest is the product of the point’s radius of rotation and angular distance through which the rotating body moves  Unit: radians  Equation is only valid if  Linear distance and radius of rotation are measured in the same units  Angular distance must be expressed in radians  Linear and Angular Velocity  Consider baseball (or tennis, golf, etc)  The greater R with which bat hits ball = greater linear velocity imparted on ball  The greater radius of the bat, the greater the linear velocity at that point, the faster the ball leaves the bat, the farther that it can travel  R = radius of rotation  Omega = angular velocity  Angular velocity is constant  Linear and Angular Acceleration  Acceleration of a body with angular motion has 2 components  Tangential  Radial  Tangential Acceleration  Ball will follow curved path while in pitcher’s hand (because moving around elbow’s fulcrum)  Tangential component represents change in linear speed of ball (what is the linear speed of the ball at that one spot in the arc of motion)  Goal is to maximize tangential velocity to throw far or fast  Once ball is released, tangential acceleration = 0 b/c pitcher is no longer applying force  Association b/w linear and angular  Looks exactly like v= r x omega  Take tangential acceleration, set it equal to the radius times the angular acceleration  Radial Acceleration  Rate of change in direction of a body in angular motion  Always directed toward center of curvature (radius of rotation, center of circular motion)  Ball follows curved path (until release) b/c hand restrains it  Restraining force causes radial acceleration  Upon release radial acceleration = 0  Ball follows path of tangent to curve at instant release  Where release occurs dictates where ball ends up...