Note for PH 101 at UA-General Physics I (8)
Note for PH 101 at UA-General Physics I (8)
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Date Created: 02/06/15
GIANCOLI Lecture PowerPoints Chapter 8 Physics Principles with Applications 6 edition Giancoli Chapter 8 Copyright 2005 Pearson Prentice Hall Inc Units of Chapter 8 Angular Quantities Constant Angular Acceleration Rolling Motion Without Slipping Torque Rotational Dynamics Torque and Rotational Inertia Solving 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 All points on a straight line drawn through the axis move through the same angle in the same time The angle 9 in radians is defined 9 A 81 a I where l is the arc length 81 Angular Quantities Angular displacement A6 62 91 The average angular velocity is defined as the total angular displacement divided by time 5 E 82a At The instantaneous angular velocity A6 a 11m 82b At gt0 Copyright 2005 Pearson Prentice Hall Inc 81 Angular Quantities The angular acceleration is the rate at which the angular velocity changes with time 02 11 ACO a 83a Ar Ar The instantaneous acceleration S 1 Am a A3930 A Mb 81 Angular Quantities Every point on a rotating body has an angular velocity w and a linear velocity v They are related 84 Copyright 2005 Pearson Prentice Hall Inc 81 Angular Quantities Therefore objects farther from the axis of rotation will move faster Copyright 2005 Pearson Prentice Hall Inc Copyright 2005 Pears 81 Angular Quantities a tan on Prentice Hall Inc If the angular velocity of a rotating object changes it has a tangential acceleration Even if the angular velocity is constant each point on the object has a centripetal acceleration 3902 r002 2 aR a r 86 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 r0 0 velocity u v no atan acceleration a atan ra Copyright 2005 Pearson Prentice Hall Inc 81 Angular Quantities The frequency is the number of complete revolutions per second 0 f 2 4739 Frequencies are measured in hertz 1 HZ 1 8 1 The period is the time one revolution takes T 1 88 f 82 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 ww0ozt U v0 l 6 wot a12 x not at2 02 mg 209 02 223 24513 a 10 0 l 00 Si N 3 ll 2 39 2 83 Rolling Motion Without Slipping p o In a a wheel is rolling without C f V slipping The point P touching 4 the ground is instantaneously at rest and the center moves with velocity v ln 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 rm Copyright 2005 Pearson Prentice Hall Inc 84 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 84 Torque Alonger lever arm is very helpful in rotating objects 393 Copyright 2005 Pearson Prentice Hall Inc 84 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 Axis of rotation 84 Torque Point of application of force b Copyright 2005 Pearson Prentice Hall Inc The torque is defined as 739 11F 810a Or 39lI 17 i take the perpendicular component of the force 85 Rotational Dynamics Torque ancl Rotational Inertia Knowing that Fma we see that 7 mrzor 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 Emrzh 812 85 Rotational Dynamics Torque ancl Rotational Inertia The quantity I Emr2 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 Incl a b C d 6 f g h Location Moment of Objecl of axis inertia Axis Thin hoop Through radius R center MR2 Axis Thin hoop Through radius R central I I I width W diameter EMRZ t EMWZ Axis Solid cylinder Through l radius R center EMRZ Axis Hollow cylinder Through I 2 2 inner radius R center 3 Ell RI R2 outer radius R Uniform sphere Through radius R center MR Axis Long uniform rod Through i MLZ length L center L I Axis Long uniform rod Through 1 M L2 length L end 7 LiI 3 Axis Rectangular Through LMU W3 thin plate center V g 12 L length L width W Copyright 2005 Pearson Prentice Hall Inc 85 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 Worked Example Moment of Inertia Two small blocks are placed on a light rod as shown in the diagram below Calculate the moment of inertia of this rod a when rotated about an axis midway between the masses b when rotated about an axis 05m to the left of the first I I I L 40 m l I I 50 kg I Axis a 4 gt 40m gt l I I 50kg 70kg b Newton s 2nd Law for Rotation The analog of Fma for rotation is IIa Recall the linearangular analogs force F gt torque C Acceleration a gt angular acceleration or Mass m gt Moment of inertia I 86 Solving Problems in Rotational Dynamics 1 Draw a 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 86 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 Worked Example Rotational Dynamics A force F is used to move a pulley radius R and mass M Assuming the pulley is a solid cylinder find the resulting angular acceleration of the pulley 87 Rotational Kinetic Energy The kinetic energy of a rotating object is given by KE 26 mv39z 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 2 1 KB MvCM ICMQ2 8 6 Worked Example Sphere rolling down incline A solid sphere of mass M and radius R rolls without slipping down a inclined plane If it starts at height H what will be its linear speed v at the bottom 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 Solid cylinder DCell q Sphere marble Box sliding Copyright 2005 Pearson Prentice Hall Inc Answer to Conceptual Question he faster the object rotates also the faster it moves So the slowest are the ones with the most resistance to rotation ie have the largest rotational inertia Hence the order of arrival is Box which doesn t roll at all followed by Sphere l 25 MR2 Solid Cylinder l 12 MR2 Hollow Cylinder l 12 M R12 R22 and finally Hoop l M R2 87 Rotational Kinetic Energy The torque does work as it moves the wheel through an angle 9 W we 817 17 Copyright 2005 Pearson Prentice Hall Inc 88 Angular Momentum and Its Conservation In analogy with linear momentum we can define angular momentum L L Im 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 IQ 10aD 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 1 Copyright 2005 Pearson Prentice Hall Inc Copyright 2005 Pearson Prentice HallI Incl Worked Example Angular Momentum Conservation A small mass m is attached to a string and revolves in a circle on a smooth horizontal table The string is passed through a hole in the center of the table Initially the mass revolves with a speed v1 in a circle of radius r1 The string is then pulled slowly through the hole until the radius of revolution is 7 1 reduced to r2 K What is the new I m vi speed of the revolving mass H 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 Im If the net torque on an object is zero its angular momentum does not change Backup 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 Lperson E platform b Copyright 2005 Pearson Prentice Hall Inc
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