PHYS 1443 – Section 501 Lecture #20AnnouncementsFundamentals of RotationRotational KinematicsExample for Rotational KinematicsRotational AccelerationsExample for Rotational MotionRolling Motion of a Rigid BodyMore Rolling Motion of a Rigid BodyTorqueMoment of InertiaTorque & Angular AccelerationExample for Torque and Angular AccelerationRotational Kinetic EnergyExample for Rigid Body Moment of InertiaExample for Moment of InertiaCalculation of Moments of InertiaParallel Axis TheoremSlide 19Example for Parallel Axis TheoremTorque and Vector ProductProperties of Vector ProductMore Properties of Vector ProductSimilarity Between Linear and Rotational MotionsMonday April 12, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt1PHYS 1443 – Section 501Lecture #20Monday Apr 12, 2004Dr. Andrew Brandt•Rotation Review•Torque•Moment of Inertia•Rotational Kinetic Energy•Torque and Vector ProductsMonday April 12, 2004 2 PHYS 1443-501, Spring 2004Dr. Andrew BrandtAnnouncements•HW#9 on Ch. 10 will be assigned tomorrow and is due 4/19. There will be two more homeworks after this. 4/26+5/3. It would help you to do them.•Updated grades will be posted tomorrowMonday April 12, 2004 3 PHYS 1443-501, Spring 2004Dr. Andrew BrandtFundamentals of RotationConsider a motion of a rigid body – an object that does not change its shape – rotating about the axis protruding out of the slide. One radian is the angle swept by an arc length equal to the radius of the arc.360Since the circumference of a circle is 2r,rPsOrs The arc length, or sergita, isrsTherefore the angle, , is , and the unit of the angle is radians.radrr /22Monday April 12, 2004 4 PHYS 1443-501, Spring 2004Dr. Andrew BrandtRotational KinematicsThe first type of motion we learned about in linear kinematics was under a constant acceleration. We will learn about the rotational motion under constant angular acceleration, because these are the simplest motions in both cases.fJust like the case in linear motion, one can obtainAngular Speed under constant angular acceleration:Angular displacement under constant angular acceleration:fOne can also obtain 2fti221ttii ifi 22rvtta ra=Monday April 12, 2004 5 PHYS 1443-501, Spring 2004Dr. Andrew BrandtExample for Rotational KinematicsA wheel rotates with a constant angular acceleration of 3.50 rad/s2. If the angular speed of the wheel is 2.00 rad/s at ti=0, a) through what angle does the wheel rotate in 2.00s?ifUsing the angular displacement formula in the previous slide, one getsWhat is the angular speed at t=2.00s?tifUsing the angular speed and acceleration relationshipFind the angle through which the wheel rotates between t=2.00 s and t=3.00 s.rad0.112221tt 200.250.32100.200.2 rad0.11.75.1.20.11revrev srad /00.900.250.300.2 rad8.2100.350.32100.300.2232rad8.10Monday April 12, 2004 6 PHYS 1443-501, Spring 2004Dr. Andrew BrandtRotational AccelerationsrvtrPOxyatTwoHow many different types of linear acceleration do you see in a circular motion and what are they?Total linear acceleration isSince the tangential speed vt isWhat does this relationship tell you?Although every particle in the object has the same angular acceleration, its tangential acceleration differs proportional to its distance from the axis of rotation.Tangential, at, and the radial acceleration, ar.artaThe magnitude of tangential acceleration at isThe radial or centripetal acceleration ar israaWhat does this tell you?The father away the particle from the rotation axis the more radial acceleration it receives. In other words, it receives more centripetal force.adtdvt rdtddtdrrrv2 rr22r22rtaa 222rr 42rMonday April 12, 2004 7 PHYS 1443-501, Spring 2004Dr. Andrew BrandtExample for Rotational MotionAudio information on compact discs are transmitted digitally through the readout system consisting of laser and lenses. The digital information on the disc are stored by the pits and flat areas on the track. Since the speed of readout system is constant, it reads out the same number of pits and flats in the same time interval. In other words, the linear speed is the same no matter which track is played. a) Assuming the linear speed is 1.3 m/s, find the angular speed of the disc in revolutions per minute when the inner most (r=23mm) and outer most tracks (r=58mm) are read.Using the relationship between angular and tangential speedb) The maximum playing time of a standard music CD is 74 minutes and 33 seconds. How many revolutions does the disk make during that time?c) What is the total length of the track past through the readout mechanism?ld) What is the angular acceleration of the CD over the 4473s time interval, assuming constant ?rvsradmmsm/5.5610233.123/3.13min/104.5/00.92revsrev sradmmsm/4.2210583.158/3.13min/101.22revrv mmr 58mmr 23 2fi min/3752min/210540revrevfti revssrev4108.24473/603750 tvtmssm3108.54473/3.1 tif 23/106.74473/5.564.22sradssradMonday April 12, 2004 8 PHYS 1443-501, Spring 2004Dr. Andrew BrandtRolling Motion of a Rigid BodyWhat is a rolling motion?To simplify the discussion, let’s make a few assumptionsLet’s consider a cylinder rolling without slipping on a flat surfaceA more generalized case of a motion where the rotational axis moves together with the objectUnder what condition does this “Pure Rolling” happen?The total linear distance the CM of the cylinder moved isThus the linear speed of the CM isA rotational motion about the moving axis1. Limit our discussion on very symmetric objects, such as cylinders, spheres, etc2. The object rolls on a flat surfaceR ss=RRs dtdsvCMCondition for “Pure Rolling”dtdRRMonday April 12, 2004 9 PHYS 1443-501, Spring 2004Dr. Andrew BrandtMore Rolling Motion of a Rigid BodyAs we learned in the rotational motion, all points in a rigid body moves at the same angular speed but at a different linear speed.At any given time the point that comes to P has 0 linear speed while the point at P’
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