# Relationships Between Linear And Angular Motion

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Relationships between linearand angular motion Body segment rotationscombine to producelinear motion of thewhole body or of aspecific point on a bodysegment or implement– Joint rotations createforces on the pedals.– Forces on pedals rotatecrank which rotates gearswhich rotate wheels.– Rotation of wheels resultin linear motion of thebicyclist and his bike.Examples Running– Coordinate joint rotations to createtranslation of the entire body. Softball pitch– Rotate body to achieve desired linearvelocity of the ball at release. Golf– Rotate body to rotate club to strike the ballfor intended distance and accuracy. Example specific to your interests:1

Key concept:– the motion of any point on a rotating body(e.g., a bicycle wheel) can be described inlinear terms Key information:axisradius– axis of rotation– radius of rotation: distance from axis topoint of interest Linear and angular displacementd θxr***WARNING***θ must be expressed in the units ofradians for this expression to be validNOTE: radians are expressed by a “unit-less” unit. That is, the units ofradians seem to be invisible in each of the equations which relatedlinear and angular motion.2

Exampler Bicycle odometers measure linear distance traveledper wheel rotation for a point on the outer edge ofthe tire You describe bicycle wheel radius (r 0.33 m) Device counts rotations (θ 1 rev 2 π rad) Question: How many times did a Tour de France’scyclist’s wheel rotate (d 3427.5 km)?––––know:need:use:answer: Linear and angular velocityvT ω x r***WARNING***ω must be expressed in the units ofradians/s for this expression to be valid Although vT may appear to be a newterm, it is simply the linear or tangentialvelocity of the point of interest.3

Example: Hockey wrist shot A hockey player isrotating his stick at 1700deg/s at the instant ofcontact. If the blade ofthe stick is located 1.2 mfrom the axis of rotation,what is the linear speed ofthe blade at impact?––––know:need:use:answer:Follow up questions What would happen to blade velocity if thestick was rotated two times faster? What would happen to blade velocity if thestick (radius of rotation) was 25% shorter?4

What does the vT ωr relationship tellus about performance?– In many tasks, it is important to maximize thelinear velocity (vT) of a projectile or of a particularendpoint (usually distal) club head speed in golf ball velocity in throwing– Theoretically: vT can be increased in two ways: increasing r increasing ω– Problem: it is more difficult to rotate an objectwhen its mass is distributed farther from the axisof rotation.– What are some examples of this tradeoff? Linear and angular acceleration– Newton’s 1st law of motion states that anobject must be forced to follow a curvedpath.– A change of direction represents achange in velocity (a vector quantity).– Therefore, even if the magnitude of avelocity vector remains constant (10m/s), a change in direction of the velocityvector results in acceleration.5

Radial acceleration Radial acceleration (aR) - the linearacceleration that serves to describe thechange in direction of an objectfollowing a curved path.– Radial acceleration is a linear quantity– It is always directed inward, toward thecenter of a curved path.Example – Radial acceleration Skaters or skiers on a curvemust force themselves tochange directions. Changes of direction result inchanges in velocity - even if thespeed remains constant (why?) Changes of velocity, bydefinition, result in accelerations(aR). This radial acceleration iscaused by the component of theground reaction force (GRF)that is directed toward thecenter of the turn.6

aR vT2/r (ωr)2/r ω2r This relationship demonstrates:– for a given r, higher vT is related to a higher aR;which means a higher force is needed to produceaR (i.e., to maintain curved path).– for a given r, higher w is also related to a higheraR; which means a higher force is needed toproduce aR (i.e., to maintain curved path).– for a given vT, lower r (i.e., a tighter “turningradius”) results in a higher aR (and the need for agreater force to maintain a curved path)Example scenarios Two bicyclists are racing on a rainy day and both enter a slippery corner at 25m/s. If the one cyclist takes a tighter turning radius than the other, whichcyclist experiences the greatest radial acceleration?– Who is at greater risk for slipping or skidding?– What strategies can cyclists take to reduce the risk of skidding?– Which strategy is theoretically more effective?7

Other examples– A baseball pitcher delivers two pitches with exactly the sametechnique. However, the first pitch is thrown two times fasterthan the second (e.g. fastball vs very slow change up). During which pitch does the athlete experience greater radialaccelerations? In which direction(s) are the radial accelerations experienced? How these accelerations relate to injury (e.g., rotator cuffdamage)?– In preparation for his high-bar dismount, a gymnastincreases his rate of rotation by a factor of three. His radiusof rotation remains the same. By what factor does his radial acceleration change during thistime? Tangential acceleration (aT) - the linearacceleration that serves to describe therate of change in magnitude oftangential velocity.aT (vTf – vTi)/t Although aT may appear to be a newterm, it is simply the change in linear ortangential velocity of the point ofinterest.8

Resultant Acceleration Vector Rotational and curvilinearmotions will always result inradial acceleration because thedirection of the velocity vector isalways changing. If the magnitude of the velocityvector also changes, tangentialacceleration will also bepresent. Therefore, during all rotationaland curvilinear motions theresultant acceleration iscomposed of the radial andtangential accelerations.aresultantatangentialaradial9

Linear and angular acceleration – Newton’s 1st law of motion states that an object must be forced to follow a curved path. – A change of direction represents a change in velocity (a vector quantity). – Therefore, even if the magnitude of a velocity vector remains constant (10 m/s), a change in direction of the velocity vector results in acceleration. 6 Radial acceleration .