Transcription
Chapter 11A – Angular MotionAA PowerPointPowerPoint PresentationPresentation bybyPaulPaul E.E. Tippens,Tippens, ProfessorProfessor ofof PhysicsPhysicsSouthernSouthern PolytechnicPolytechnic StateState UniversityUniversity 2007
WIND TURBINES suchas these can generatesignificant energy in away that is environmentally friendly andrenewable. Theconcepts of rotationalacceleration, angularvelocity, angulardisplacement, rotationalinertia, and other topicsdiscussed in thischapter are useful indescribing the operationof wind turbines.
Objectives: After completing thismodule, you should be able to: Define and apply concepts of angulardisplacement, velocity, and acceleration. Draw analogies relating rotational-motionparameters ( , , ) to linear (x, v, a)and solve rotational problems. Write and apply relationships betweenlinear and angular parameters.
Objectives: (Continued) Define moment of inertia and apply it forseveral regular objects in rotation. Apply the following concepts to rotation:1. Rotational work, energy, and power2. Rotational kinetic energy andmomentum3. Conservation of angular momentum
Rotational Displacement, Consider a disk that rotates from A to B:B Angular displacement :AMeasured in revolutions,degrees, or radians.1 rev 360 0 2 radTheThe bestbest measuremeasure forfor rotationrotation ofofrigid.rigid bodiesbodies isis thethe radianradian.
Definition of the RadianOne radian is the angle subtended atthe center of a circle by an arc length sequal to the radius R of the circle.ss R1 rad RR 57.30
Example 1: A rope is wrapped many timesaround a drum of radius 50 cm. How manyrevolutions of the drum are required toraise a bucket to a height of 20 m?s20 m R 0.50 m 40 radRNow, 1 rev 2 rad 1 rev 40 rad 2 rad 6.376.37 revrevh 20 m
Example 2: A bicycle tire has a radius of25 cm. If the wheel makes 400 rev, howfar will the bike have traveled? 2 rad 400 rev 1 rev 2513 rads R 2513 rad (0.25 m)ss 628628 mm
Angular VelocityAngular velocity, is the rate of change inangular displacement. (radians per second.) tAngular velocity in rad/s.Angular velocity can also be given as thefrequency of revolution, f (rev/s or rpm): ff AngularAngular frequencyfrequency ff (rev/s).(rev/s).
Example 3: A rope is wrapped many timesaround a drum of radius 20 cm. What isthe angular velocity of the drum if it lifts thebucket to 10 m in 5 s?s10 m R 0.20 m 50 radR 50 rad t5sh 10 m 10.010.0 rad/srad/s
Example 4: In the previous example, whatis the frequency of revolution for the drum?Recall that 10.0 rad/s. 2 f or f 2 10.0 rad/s 1.59 rev/sf 2 rad/revROr, since 60 s 1 min:rev 60 s revf 1.59 95.5s 1 min minff 95.595.5 rpmrpmh 10 m
Angular AccelerationAngular acceleration is the rate of change inangular velocity. (Radians per sec per sec.) tAngular acceleration (rad/s 2 )The angular acceleration can also be foundfrom the change in frequency, as follows:2 ( f ) tSince 2 f
Example 5: The block is lifted from restuntil the angular velocity of the drum is16 rad/s after a time of 4 s. What is theaverage angular acceleration? f ot0or fRt16 rad/srad 4.00 24ssh 20 m22 4.00rad/s 4.00 rad/s
Angular and Linear SpeedFrom the definition of angular displacement:s R Linear vs. angular displacement s R v t t t R v RLinearLinear speedspeed angularangular speedspeed xx radiusradius
Angular and Linear Acceleration:From the velocity relationship we have:v R Linear vs. angular velocity v v R v v R t t t a RLinearLinear accel.accel. angularangular accel.accel. xx radiusradius
Examples:R1Consider flat rotating disk:B 0; f 20 rad/st 4sWhat is final linear speedat points A and B?AR2R1 20 cmR2 40 cmvAf Af R1 (20 rad/s)(0.2 m);vAf 4 m/svAf Bf R1 (20 rad/s)(0.4 m);vBf 8 m/s
Acceleration ExampleConsider flat rotating disk:R1AB 0; f 20 rad/st 4sWhat is the average angularand linear acceleration at B? f 0t20 rad/s 4sa R (5 rad/s2)(0.4 m)R2R1 20 cmR2 40 cm22 5.00rad/s 5.00 rad/saa 2.002.00 m/sm/s22
Angular vs. Linear ParametersRecall the definition of linearacceleration a from kinematics.a v f v0tBut, a R and v R, so that we may write:a v f v0tbecomes R R f R 0t f 0Angular acceleration is the timerate of change in angular velocity. t
A Comparison: Linear vs. Angular v0 v fs vt 2 t 0 f t 2 v f vo at f o ts v0t at122s v f t at1222as v v2f20 t 0t t122 f t t22 20122f
Linear Example: A car traveling initiallyat 20 m/s comes to a stop in a distanceof 100 m. What was the acceleration?100 mSelect Equation:2as v 2f v02a 0 - v o22svo 20 m/s vf 0 m/s-(20 m/s)2 2(100 m)22aa -2.00m/s-2.00 m/s
Angular analogy: A disk (R 50 cm),rotating at 600 rev/min comes to a stopafter making 50 rev. What is theacceleration?Select Equation:222 f 0 o 600 rpmR f 0 rpm 50 revrev 2 rad 1 min 600 62.8 rad/smin 1 rev 60 s 0 - o22 -(62.8 rad/s)2 2(314 rad)50 rev 314 rad22 -6.29m/s-6.29 m/s
Problem Solving Strategy: Draw and label sketch of problem. Indicate direction of rotation. List givens and state what is to be found.Given: , , ( , , f, ,t)Find: , Select equation containing one and notthe other of the unknown quantities, andsolve for the unknown.
Example 6: A drum is rotating clockwiseinitially at 100 rpm and undergoes a constantcounterclockwise acceleration of 3 rad/s2 for2 s. What is the angular displacement?Given: o -100 rpm; t 2 s 2 rad/s2rev 1 min 2 rad 100 10.5 rad/smin 60 s 1 rev ot t ( 10.5)(2) (3)(2)122 -20.9 rad 6 rad12R2 -14.9-14.9 radradNet displacement is clockwise (-)
Summary of Formulas for Rotation v0 v fs vt 2 t 0 f t 2 v f vo at f o ts v0t at122s v f t at1222as v v2f20 t 0t t122 f t t22 20122f
CONCLUSION: Chapter 11AAngular Motion
until the angular velocity of the drum is 16 16 rad/s rad/s after a time of 4 s. What is the average angular acceleration? h 20 m R 4.00 rad/s 4.00 rad/s. 2. 2. 0. fo. or f tt 2. 16 rad/s rad 4.00 4 s s
