Chapter 7 Rotational Motion Angles Angular Velocity and Angular Acceleration Universal Law of Gravitation Kepler s Laws Angular Displacement 1 2 3 Circular motion about AXIS Three different measures of angles Degrees Revolutions 1 rev 360 deg Radians 2 rad s 360 deg Angular Displacement cont Change in distance of a point s 2 r N N counts revolutions r is in radians Example An automobile wheel has a radius of 42 cm If a car drives 10 km through what angle has the wheel rotated a In revolutions b In radians c In degrees Solution Note distance car moves distance outside of wheel moves a Find N Known s 10 000 m r 0 42 m Basic formula s 2 rN r s N 3 789 2 r b Find in radians Known N 2 radians revolution N 2 38 x 104 rad c Find in degrees Known N 360 degrees revolution N 1 36 x 106 deg Angular Speed Can be given in Revolutions s Radians s Called Degrees s f i t in radians Linear Speed at r f i in revolutions v 2 r t 2 r f i in degrees 360 t 2 r f i in radians r 2 t Example A race car engine can turn at a maximum rate of 12 000 rpm revolutions per minute a What is the angular velocity in radians per second b If helipcopter blades were attached to the crankshaft while it turns with this angular velocity what is the maximum radius of a blade such that the speed of the blade tips stays below the speed of sound DATA The speed of sound is 343 m s Solution a Convert rpm to radians per second rev 12 000 min 2 rad 1256 radians s rev sec 60 min b Known v 343 m s 1256 rad s Find r Basic formula v r v r 27 m Angular Acceleration Denoted by f i t must be in radians per sec Units of angular acceleration are rad s Every portion of the object has same angular speed and same angular acceleration Analogies Between Linear and Rotational Motion Rotational Motion i f 2 t Linear Motion v v x t i f 2 1 2 i t t 2 1 2 x v i t at 2 i t v v i at 2 2 i 2 v 2 v i2 2a x Linear movement of a rotating point Distance s r Speed Acceleration v r Different points on the same object have different linear motions a r Only works when and are in radians Example A pottery wheel is accelerated uniformly from rest to a rate of 10 rpm in 30 seconds a What was the angular acceleration in rad s2 b How many revolutions did the wheel undergo during that time Solution First find the final angular velocity in radians s 1 rev rad rad f 10 2 1 047 min 60 sec min rev sec a Find angular acceleration Basic formula f i t f i 0 0349 rad s2 t b Find number of revolutions Known i 0 f 1 047 and t 30 First find in radians Basic formula i f 2 f t 15 7 rad 2 t rad N 2 5 rev 2 rad rev Solution b Find number of revolutions Known i 0 f 1 047 and t 30 First find in radians Basic formula i f t 2 f t 15 7 rad 2 rad N 2 5 rev 2 rad rev Example A coin of radius 1 5 cm is initially rolling with a rotational speed of 3 0 radians per second and comes to a rest after experiencing a slowing down of 0 05 rad s2 a Over what angle in radians did the coin rotate b What linear distance did the coin move Solution a Find Given i 3 0 rad s f 0 0 05 rad s2 Basic formula 2 2 f i 2 i2 2 90 radians 90 2 revolutions b Find s the distance the coin rolled Given r 1 5 cm and 90 rad Basic formula s r s r is in rad s 135 cm Centripetal Acceleration Moving in circle at constant SPEED does not mean constant VELOCITY Centripetal acceleration results from CHANGING DIRECTION of the velocity Centripetal Acceleration cont Acceleration is directed toward the center of the circle of motion v a t Basic formula Derivation a 2r v2 r From the geometry of the Figure v 2v sin 2 v for small From the definition of angular velocity t v v t 2 v v 2 a v r t r Forces Causing Centripetal Acceleration Newton s Second Law F ma Radial acceleration requires radial force Examples of forces Spinning ball on a string Gravity Electric forces e g atoms Example A space station is constructed like a barbell with two 1000 kg compartments separated by 50 meters that spin in a circle r 25 m The compartments spins once every 10 seconds a What is the acceleration at the extreme end of the compartment Give answer in terms of g s b If the two compartments are held together by a cable what is the tension in the cable Solution a Find acceleration a Given T 10 s r 25 m Basic formula rad rev 2 N rev s Basic formula 2 v 2 a r r First find in rad s 2 0 1 Then find acceleration a 2 r 9 87 m s2 1 006 g Solution b Find the tension Given m 1000 kg a 1 006 g Basic formula F ma T ma 9870 N Example A race car speeds around a circular track a If the coefficient of friction with the tires is 1 1 what is the maximum centripetal acceleration in g s that the race car can experience b What is the minimum circumference of the track that would permit the race car to travel at 300 km hr Solution a Find the maximum centripetal acceleration Known 1 1 Remember only consider forces towards center Basic formula f n F ma f mg ma mg a g Maximum a 1 1 g Solution b Find the minumum circumference Known v 300 km hr 83 33 m s a 1 1 g First find radius Basic formula 2 v 2 a r r v2 r a Then find circumference L 2 r 4 043 m In the real world tracks are banked Example AAyo yo is spun in a circle as shown If the length of the string is L 35 cm and the circular path is repeated 1 5 times per second at what angle with respect to the vertical does the string bend Solution F ma Basic formula Apply F ma for both the horizontal and vertical components ma y 0 Fy T cos mg T cos mg Basic formula 2 a r ma x m 2 r m 2 L sin F2 x T sin m L T r Lsin Solution We want to find given 2 1 5 L 0 35 2 2 ma x m r m …
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