Physics 1A Lecture 19 Rotation and Circular Motion Centripetal Acceleration November 14 2019 Describing Rotation Rotation Velocity Centripetal Acceleration Tangential Acceleration Motivation Moving in Circles In this module we focus on three closely related motions 1 Particle moving in circles 2 Particle moving in curved trajectories 3 Object rotating insert slide 1 23 Describing Rotation Rotation Velocity Centripetal Acceleration Tangential Acceleration Describing Angular Position Radians For linear motion position is given by x y z in some coordinate system relative to some origin For rotational motion angle is given by around some axis relative to some reference line Q position angular displacement given in radians Not Degrees 0 0 Angle de nition arc length radius s r is unitless length length 2 is a full rotation Note 2 4 all describe same angular position 2 23 Describing Rotation Rotation Velocity Centripetal Acceleration Tangential Acceleration Angular Displacement Angular displacement f i All points on a purely rotating rigid object have same angular displacement 3 23 Describing Rotation Rotation Velocity Centripetal Acceleration Tangential Acceleration Angular Displacement vs Linear Distance Angular displacement angle traveled Beware as an object rotates the angular position keeps coming back to 0 However the angular displacement is measured cumulatively e g Spinning around twice gives 4 Linear distance arc length traveled s Relating angular displacement and linear distance s r s r distance from axis 4 23 Describing Rotation Rotation Velocity Centripetal Acceleration Tangential Acceleration Concept Check Angular Displacement A rotating carousel has a horse near the outer edge and a lion half way out from the center The carousel rotates by a quarter turn Which animal has the bigger angular displacement The bigger linear distance traveled Clicker question a angular Horse linear Horse b angular Lion linear Horse c angular Equal linear Horse d angular Horse linear Equal e angular Equal linear Equal 5 23 Describing Rotation Rotation Velocity Centripetal Acceleration Tangential Acceleration insert correct Slide Angular Velocity Angular velocity measures how quickly an object rotates De nition of Angular Velocity Average angular velocity avg angular displacement t d dt Instantaneous angular velocity Sign convention in 2D positive counterclockwise negative clockwise 6 23 Describing Rotation Rotation Velocity Centripetal Acceleration Tangential Acceleration More on Angular Velocity Other descriptions of angular speed Frequency and Period Frequency how many full revolutions per unit time units s 1 Hz angular velocity rad s rev s f 2 rad rev Period how long does a full revolution take units sec 2T radlrev Tada Shen T 2 1 f q frequency 7 23 insert hole Describing Rotation Rotation Velocity Centripetal Acceleration Tangential Acceleration Linear Velocity in Circular Motion For a particle moving in a circle Velocity vector is tangent to the circle at each point Magnitude of velocity v speed 8 23 view fromtop Describing Rotation Rotation Velocity Centripetal Acceleration Tangential Acceleration Putting it Together Circular and Projectile Motion David is riding on a ying carousel which swings his seat in a horizontal circle The carousel rotates counterclockwise When he is on the south side of the carousel David s hat falls o Clicker question Where does the hat land a South of where it fell o b West of where it fell o c East of where it fell o d Southwest of where it fell o e Southeast of where it fell o south F 9 23 NSEWview from top Describing Rotation Rotation Velocity Centripetal Acceleration Tangential Acceleration Relating Angular and Linear Velocity Points further from axis cover more distance for the same angular rotation linear Velocity v lim t 0 dist traveled t lim t 0 r t r Linear Velocity for Circular Motion Linear velocity of a point on a rotating object is always tangent to the circle of rotation and has magnitude v r Objects further from axis move faster for the same angular velocity e linear velocity of green red angular velocity cid 15482 me for them 10 23 Describing Rotation Rotation Velocity Centripetal Acceleration Tangential Acceleration Example Rotation of ATP synthase F1Fo ATP synthase molecular motor converts H gradient to ATP Fo Subunit C brown radius 2nm Rotates 0 5 revolution every second frequency What is the linear speed of an atom on the outer edge of the subunit Stewart group 2016 angular speed t 3 14rad s linear speed 0 5 2 1s v r 6 3nm s 11 23 f O 5 Ms O 5 H Z appt 2hm radial Apd W fztrrwad o s ref 3 rev z nm 3 14 rays V 6 28 hm IS approach O sgtef futrxefnm J 6 no n m s Describing Rotation Rotation Velocity Centripetal Acceleration Tangential Acceleration Simple Model Uniform Circular Motion Simplest situation with circular motion assumes Single point particle moves in circular arc of radius r Constant speed during the motion 1 p speed v are constant Uniform circular motion describes a point particle moving in a perfect circle at constant speed Good model for particle following curved trajectory without speeding up or slowing down EON 12 23 Describing Rotation Rotation Velocity Centripetal Acceleration Tangential Acceleration Acceleration While Turning For a particle in uniform circular motion Speed is constant v r Velocity vector is not constant changes direction Particle moving in a circle is always accelerating even if speed constant your still aveteran ng Acceleration always points towards circle center 13 23 Describing Rotation Rotation Velocity Centripetal Acceleration Tangential Acceleration Centripetal Acceleration Centripetal pointing towards the center Acceleration in Uniform Circular Motion An object moving in a circle of radius r with constant speed v has an acceleration that always points towards the center of the circle of magnitude a v 2 r i tf 2x speed acceleration will quadruple see extra slides for derivation Iowa Ohigha if 2X radius a will be 42 as big 14 23 Describing Rotation Rotation Velocity Centripetal Acceleration Tangential Acceleration Application Centrifuge Centrifuges separate out components by rapid spinning A coagulation test requires separating plasma from blood cells by centrifuging a blood sample at an acceleration of 2000g If you have a centrifuge with radius 15 cm at what frequency in rpm revolutions per minute should you spin the sample a 2000g 2000 9 8 2 104 m s2 a v par 54m s f v 2 r 58Hz 3500rpm v 2 r 2 15 23 9 84