1Relationships between linear and angular motion• Body segment rotations combine to produce linear motion of the whole body or of a specific point on a body segment or implement– Joint rotations create forces on the pedals.– Forces on pedals rotate crank which rotates gears which rotate wheels.– Rotation of wheels result in linear motion of the bicyclist and his bike.Examples• Running– Coordinate joint rotations to create translation of the entire body.• Softball pitch– Rotate body to achieve desired linear velocity of the ball at release.•Golf– Rotate body to rotate club to strike the ball for intended distance and accuracy.• Example specific to your interests:2• Key concept:– the motion of any point on a rotating body (e.g., a bicycle wheel) can be described in linear terms• Key information:– axis of rotation– radius of rotation: distance from axis to point of interest axisradius• Linear and angular displacementd = θ xr***WARNING***θ must be expressed in the units of radiansfor this expression to be validNOTE: radians are expressed by a “unit-less” unit. That is, the units of radians seem to be invisible in each of the equations which related linear and angular motion.3Example• Bicycle odometers measure linear distance traveled per wheel rotation for a point on the outer edge of the tire…r• 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’s cyclist’s wheel rotate (d = 3427.5 km)?– know:– need:–use:– answer:• Linear and angular velocityvT= ω xr***WARNING***ω must be expressed in the units of radians/sfor this expression to be valid• Although vTmay appear to be a new term, it is simply the linear or tangential velocity of the point of interest.4Example: Hockey wrist shot• A hockey player is rotating his stick at 1700 deg/s at the instant of contact. If the blade of the stick is located 1.2 m from the axis of rotation, what is the linear speed of the blade at impact?– know:– need:–use:– answer:Follow up questions• What would happen to blade velocity if the stick was rotated two times faster?• What would happen to blade velocity if the stick (radius of rotation) was 25% shorter?5• What does the vT= ωr relationship tell us about performance?– In many tasks, it is important to maximize the linear velocity (vT) of a projectile or of a particular endpoint (usually distal)• club head speed in golf• ball velocity in throwing– Theoretically: vTcan be increased in two ways:• increasing r• increasing ω–Problem: it is more difficult to rotate an object when its mass is distributed farther from the axis of rotation.– What are some examples of this tradeoff? • 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.6Radial acceleration• Radial acceleration (aR) - the linear acceleration that serves to describe the change in direction of an object following a curved path.– Radial acceleration is a linear quantity– It is always directed inward, toward the center of a curved path. Example – Radial acceleration• Skaters or skiers on a curve must force themselves to change directions. • Changes of direction result in changes in velocity - even if the speed remains constant (why?) • Changes of velocity, by definition, result in accelerations (aR). • This radial acceleration is caused by the component of the ground reaction force (GRF) that is directed toward the center of the turn.7aR= vT2/r = (ωr)2/r = ω2r• This relationship demonstrates:– for a given r, higher vTis related to a higher aR; which means a higher force is needed to produce aR(i.e., to maintain curved path).– for a given r, higher w is also related to a higher aR; which means a higher force is needed to produce aR(i.e., to maintain curved path). – for a given vT, lower r (i.e., a tighter “turning radius”) results in a higher aR(and the need for a greater force to maintain a curved path)Example scenarios• Two bicyclists are racing on a rainy day and both enter a slippery corner at 25 m/s. If the one cyclist takes a tighter turning radius than the other, which cyclist 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?8Other examples– A baseball pitcher delivers two pitches with exactly the same technique. However, the first pitch is thrown two times faster than the second (e.g. fastball vs very slow change up). • During which pitch does the athlete experience greater radial accelerations? • In which direction(s) are the radial accelerations experienced?• How these accelerations relate to injury (e.g., rotator cuff damage)?– In preparation for his high-bar dismount, a gymnast increases his rate of rotation by a factor of three. His radiusof rotation remains the same. • By what factor does his radial acceleration change during this time?• Tangential acceleration (aT) - the linear acceleration that serves to describe the rate of change in magnitude of tangential velocity. aT= (vTf–vTi)/t• Although aTmay appear to be a new term, it is simply the change in linear or tangential velocity of the point of interest.9Resultant Acceleration Vector• Rotational and curvilinear motions will always result in radial acceleration because the direction of the velocity vector is always changing.• If the magnitude of the velocity vector also changes, tangential acceleration will also be present.• Therefore, during all rotational and curvilinear motions the resultant acceleration is composed of the radial and tangential