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# 354 Class Note for PHYS 250 at PSU

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This 34 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 21 views.

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Date Created: 02/06/15

Chapter 8 Copyright 2005 Pearson Prentice Hall Inc Units of Chapter 8 Anguar Quantities Constant Angular Acceleration Torque Rotationa Dynamics Torque and Rotational Inertia Soving Problems in Rotational Dynamics Units of Chapter 8 Rotationa Kinetic Energy Anguar Momentum and Its Conservation Vector Nature of Angular Quantities 81 Angular Quantities Copyright 2005 Pearson Prentice Hall Inc In purely rotational motion all points on the object move in circles around the axis of rotation O The radius of the circle is r l is the arc length 81 a Angular Quantities Angular displacement A6 62 61 The average angular velocity displacement divided by time 5 amp 82a At The instantaneous angular velocit y A6 o 2 11m 82b Alf gt0 Ar Copyright 2005 Pearson Prentice Hall lnc Angular Quantities The angular acceleration is the rate at which the angular velocity changes with time 12 01 Am 83 a At At a The instantaneous acceleration 1 Aw 83b CEMew ltgt 81 Angular Quantities Every point on a rotating body has an angular velocity w and a linear velocity v They are related Copyright 2005 Pearson Prentice Hall Incl 81 Angular Quantities Therefore objects farther from the axis of rotation will move faster Copyright 2005 Pearson Prentice Hall Inc Angular Quantities If angular velocity of a rotating object changes has a 3tan tangential acceleration atan Fa Even if the angular velocity is constant each point on the object has a centripetal acceleration 2 2 7 no mgr 6 Copyright 2005 Pearson Prentice Hall Inc r r 81 Angular Quantities Here is the correspondence between linear and rotational quantities TABLE 8 1 Linear and Rotational Quantities Linear Type Rotational Relation x displacement 6 x r6 1 velocity co 1 no atan acceleration a atan ra Copyright 2005 Pearson Prentice Hall Inc Constant Angular Acceleration The equations of motion for constant angular acceleration are the same as those for linear motion with the substitution of the angular quantities for the linear ones Angular Linear ww0at vv0at 6 wot arg x 3900 aif2 12 mg 206 v2 1 Zax a l 600 J U 3900 8 ll cl ll 2 2 Angular Quantities The frequency is the number of complete revolutions per second f t 2339 Frequencies are measured in hertz 1 Hz 1 8 1 The period is the time one revolution takes T 1 88 f Find ang Vel Linear vel Tangential acc Centripital acc Total acc Rolling Motion Without Slipping In a a wheel is rolling without slipping The point P touching the ground is instantaneously at rest and the center moves with velocity v In b the same wheel is seen from a reference frame where C is at rest Now point P is moving with velocity v The linear speed of the wheel is related to its angular speed b v no Copyright 2005 Pearson Prentice Hall Inc Torque To make an object start rotating a force is needed the position and direction of the force matter as well The perpendicular distance from the axis of rotation to the line along which the force acts is called the lever arm Copyright 2005 Pearson Prentice Hall Inc Tmzawg arm g very hampfu in mta39it ng bjgctgn gt Copyright 2005 Pearson Prentice Hall Inc Torque Here the lever arm for FA is the distance from the knob to the hinge the lever arm for FD is zero and the lever arm for FC is as shown Copyright 2005 Pearson Prentice Hall Inc 84 Torque Point of AXIS of application rotation of force The torque is defined as Ilt r I 1 riF 810a a b Copyright 2005 Pearson Prentice Hall Inc Rotational Dynamics Torque and Rotational Inertia Knowing that Fma we see that 1 mr2a 811 This is for a single point mass what about an extended object As the angular acceleration is the same for the whole object we can write 2739 Emr2oz 812 85 Rotational Dynamics Torque and Rotational Inertia The quantity I 2 er is called the rotational inertia of an object The distribution of mass matters here these two objects have the same mass but the one on the left has a greater rotational inertia as so much of its mass is far from the axis of rotation Copyright 2005 Pearson Prentice Hall Inc a b c d e f g h Location Moment of Object of axis inertia Axis Thin hoop Through 7 radius R center MR Axis Thin hoop Through radius R central l v I 7 width W diameter tnquot EMR39 EMW Solid cylinder Through l radius R Center MR2 Hollow cylinder Through l 2 l inner radius R center MRl R2 outer radius R2 Uniform sphere Through radius R center M R Long uniform rod Through l MLQ length L center 12 L f Axis ong uni orm rod Through l 2 length L end lt L gtl 3ML Axis Rectangular Through LML2 W2 thin plate cemer 2 NW L length L width W Copyright 2005 Pearson Prentice Hall Inc Rotational Dynamics Torque and Rotational Inertia The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation compare f and g for example Solving Problems in Rotational Dynamics 1 Draw a Free Body diagram 2 Decide what the system comprises 3 Draw a freebody diagram for each object under consideration including all the forces acting on it and where they act 4 Find the axis of rotation calculate the torques around it Solving Problems in Rotational Dynamics 5 Apply Newton s second law for rotation If the rotational inertia is not provided you need to find it before proceeding with this step 6 Apply Newton s second law for translation and other laws and principles as needed 7 Solve 8 Check your answer for units and correct order of magnitude Copyright 2005 Pearson Prentice Hall Inc 87 Rotational Kinetic Energy The kinetic energy of a rotating object is given by KE 2 mvz By substituting the rotational quantities we find that the rotational kinetic energy can be written rotational KE Iw2 815 A object that has both translational and rotational motion also has both translational and rotational kinetic energy 1 KB M v2 M ICM02 8 6 87 Rotational Kinetic Energy When using conservation of energy both rotational and translational kinetic energy must be taken into account All these objects have the same potential energy at the top but the time it takes them to get down the incline depends on how much rotational HOOP inertia they have Ernpty can Splid cylinder Dcell lt Sphere marble Copyright 2005 Pearson Prentice Hall Inc 87 Rotational Kinetic Energy The torque does work as it moves the wheel through an angle 9 W me 817 Copyright 2005 Pearson Prentice Hall Inc 88 Angular Momentum and Its Conservation In analogy with linear momentum we can define angular momentum L L I a 818 We can then write the total torque as being the rate of change of angular momentum If the net torque on an object is zero the total angular momentum is constant 1m Iowo constant 88 Angular Momentum and Its Conservation Therefore systems that can change their rotational inertia through internal forces will also change their rate of rotation I large I small a small a large a b Copyright 2005 Pearson Prentice Hall Incl Cooyrighl 2005 Pearson Prentice Hall Inc J39 g 89 Vector Nature of Angular Quantities The angular velocity vector points along the axis of rotation its direction is found using a right hand rule Copyright 2005 Pearson Prentice Hall Inc 89 Vector Nature of Angular Quantities Axis Angular acceleration and angular momentum vectors also point along the axis of rotation t L person i platform b Copyright 2005 Pearson Prentice Hall Inc Summary of Chapter 8 Angles are measured in radians a whole circle is 211 radians Angular velocity is the rate of change of angular position Angular acceleration is the rate of change of angular velocity The angular velocity and acceleration can be related to the linear velocity and acceleration The frequency is the number of full revolutions per second the period is the inverse of the frequency Summary of Chapter 8 cont The equations for rotational motion with constant angular acceleration have the same form as those for linear motion with constant acceleration Torque is the product of force and lever arm The rotational inertia depends not only on the mass of an object but also on the way its mass is distributed around the axis of rotation The angular acceleration is proportional to the torque and inversely proportional to the rotational inertia Summary of Chapter 8 cont An object that is rotating has rotational kinetic energy If it is translating as well the translational kinetic energy must be added to the rotational to find the total kinetic energy Angular momentum is L Ia If the net torque on an object is zero its angular momentum does not change

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