Wednesday, July 14, 2004 PHYS 1441-501, Summer 2004Dr. Jaehoon Yu1PHYS 1441 – Section 501Lecture #13Wednesday, July 14, 2004Dr. Jaehoon Yu• Rolling Motion• Torque• Moment of Inertia• Rotational Kinetic Energy• Angular Momentum and Its Conservation• Conditions for Mechanical EquilibriumToday’s homework is #6 due 7pm, Friday, July 23!!Remember the second term exam, Monday, July 19!!Wednesday, July 14, 2004 PHYS 1441-501, Summer 2004Dr. Jaehoon Yu2Using what we have learned in the previous slide, how would you define the angular displacement?=∆θAngular Displacement, Velocity, and AccelerationHow about the average angular speed?≡ωAnd the instantaneous angular speed?≡ωBy the same token, the average angular acceleration≡αAnd the instantaneous angular acceleration?≡αWhen rotating about a fixed axis, every particle on a rigid object rotates through the same angle and has the same angular speed and angular acceleration.θiθfifθθ−=−−ififttθθt∆∆θ=∆∆→∆ttθlim0dtdθ=−−ififttωωt∆∆ω=∆∆→∆ttωlim0dtdωUnit? rad/sUnit? rad/sUnit? rad/s2Unit? rad/s2Wednesday, July 14, 2004 PHYS 1441-501, Summer 2004Dr. Jaehoon Yu3How about the acceleration?vrω=TwoHow many different linear accelerations do you see in a circular motion and what are they?Total linear acceleration isSince the tangential speed v 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.taThe magnitude of tangential acceleration atisThe radial or centripetal acceleration arisraWhat does this tell you?The father away the particle is from the rotation axis, the more radial acceleration it receives. In other words, it receives more centripetal force.avt∆=∆()rtω∆=∆rtω∆=∆αr=rv2=()rr2ω=2ωr=22rtaa +=()()222ωαrr +=42ωα+= rWednesday, July 14, 2004 PHYS 1441-501, Summer 2004Dr. Jaehoon Yu4Rolling 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 surfaceRθss=RθθRs=dtdsvCM =Condition for “Pure Rolling”dtdRθ=ωR=Wednesday, July 14, 2004 PHYS 1441-501, Summer 2004Dr. Jaehoon Yu5More 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’ has twice the speed of CMThe magnitude of the linear acceleration of the CM isA rolling motion can be interpreted as the sum of Translation and RotationCMaWhy??PP’CMvCM2vCMCM is moving at the same speed at all times.PP’CMvCMvCMvCM+PP’CMv=Rωv=0v=Rω=PP’CM2vCMvCMCMvt∆=∆Rtω∆=∆αR=Wednesday, July 14, 2004 PHYS 1441-501, Summer 2004Dr. Jaehoon Yu6TorqueTorque is the tendency of a force to rotate an object about an axis. Torque, τ, is a vector quantity.≡τMagnitude of torque is defined as the product of the force exerted on the object to rotate it and the moment arm.FφdLine of ActionConsider an object pivoting about the point Pby the force F being exerted at a distance r. PrMoment armThe line that extends out of the tail of the force vector is called the line of action.The perpendicular distance from the pivoting point P to the line of action is called Moment arm.When there are more than one force being exerted on certain points of the object, one can sum up the torque generated by each force vectorially. The convention for sign of the torque is positive if rotation is in counter-clockwise and negative if clockwise. d2F221τττ+=∑2211dFdF−==φsinrF FdWednesday, July 14, 2004 PHYS 1441-501, Summer 2004Dr. Jaehoon Yu7R1Example for TorqueA one piece cylinder is shaped as in the figure with core section protruding from the larger drum. The cylinder is free to rotate around the central axis shown in the picture. A rope wrapped around the drum whose radius is R1exerts force F1to the right on the cylinder, and another force exerts F2on the core whose radius is R2downward on the cylinder. A) What is the net torque acting on the cylinder about the rotation axis?The torque due to F1111FR−=τSuppose F1=5.0 N, R1=1.0 m, F2= 15.0 N, and R2=0.50 m. What is the net torque about the rotation axis and which way does the cylinder rotate from the rest?R2F1F2and due to F2222FR=τUsing the above result=+=∑21τττSo the total torque acting on the system by the forces is2211FRFR +−=∑τThe cylinder rotates in counter-clockwise.2211FRFR+−mN •=×+×−= 5.250.00.150.10.5Wednesday, July 14, 2004 PHYS 1441-501, Summer 2004Dr. Jaehoon Yu8Moment of Inertia Rotational Inertia:What are the dimension and unit of Moment of Inertia?∑≡iiirmI22mkg⋅[]2MLMeasure of resistance of an object to changes in its rotational motion. Equivalent to mass in linear motion.Determining Moment of Inertia is extremely important for computing equilibrium of a rigid body, such as a building.dmrI∫≡2For a group of particlesFor a rigid bodyWednesday, July 14, 2004 PHYS 1441-501, Summer 2004Dr. Jaehoon Yu9Torque & Angular AccelerationLet’s consider a point object with mass m rotating on a circle.What does this mean?The tangential force Ftand radial force FrThe tangential force FtisWhat do you see from the above relationship?mrFtFrWhat forces do you see in this motion?tFThe torque due to tangential force FtisrFt=τατI=Torque acting on a particle is proportional to the angular acceleration.What law do you see from this relationship?Analogs to Newton’s 2ndlaw of motion in rotation.αmr=rmat=α2mr=αI=tma=Wednesday, July 14, 2004 PHYS 1441-501, Summer 2004Dr. Jaehoon Yu10Rotational Kinetic EnergyWhat do you think the kinetic energy of a rigid object that is undergoing a circular motion is? Since a rigid body is a collection of masslets, the total kinetic energy of the rigid object isSince moment of Inertia, I, is defined asKinetic
View Full Document
Unlocking...