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# 449 Class Note for PHYS 211 at PSU

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This 31 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 31 views.

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Date Created: 02/06/15

g Rotation of a Rigid Body Radian Measure Rotational Kinematics Kinetic Energy Moment of Inertia and Parallel Axis Theorem Torque Rotational Kinematics Extended objects Allows consideration of the whole object Allows separation of motion into Motion of the center of mass Motion rotation about the center of mass Definition of Radian Measure Measuring the angle in radians Definition of Kinematic quantities A6 6149 wavg At a Angular velocity a 2 Ad dd Angular a avg At dt acceleration Example 1 What is the angular speed of the daily rotation of the earth Determining the direction g of rotation Counterclockwise looking down Right hand rule Angular Displacement Refe rence 11quot a nquot f 39i Rotation axis 1h Follow the point P which undergoes circular motion a distance r from the axis of rotation Velocity and Angular V 39Velocity Accelerati on and Angular acceleration trdt r rw v dd 61 ldv a a r rat a dt dt rdt r t Kinematic equations for i constant angular acceleration A6 wwot 2 0 00 at 6 60 wot at2 2a6 60a2 a Comparison of kinematic equations 7 Rotational Motion About a Fixed Axis quotwith a Constant Linear Motion with a Constant Variables 639 and 0 Variables x and v ww0cxt vv0at 9Bo w0wlt xxa 1zv vjt B 90 not cx 2 x x0 not E15am w an 200163 60 v2 1232 2011 x0 Example 10 1 9 r rud o maq L Rom no ll n uc V 772cm 1 anguur gym PIilXn Y 3 negative 1 In a I Example 2 What is the angular displacement and angular acceleration of a flywheel that accelerates uniformly from rest to 100 rpm in 20 s Rotational Kinetic Energy Begin with the kinetic energy of a system of rotating particles n n 1 2 1 2 9 K jmivz jmi i1 i1 n n 1 2 2 1 2 2 K Emlr a 3 E mlr a 1 11 K 1102 Moment of inertia For a system of particles quot n b I E miri2 A J i 1 For a continGous distribution of mass M Example 3 Three particles are connected by rigid rods of negligible mass lying along the y axis If the system rotates about the x axis with and angular speed of 200 rads what is the moment of inertia about the x axis and the total rotational kinetic energy 2 F x F Units are N m 239 rFsin 9 Newton s Second Law for Rotation The net torque acting on an object is proportional to the angular acceleration proportionality is moment of inertia Il Tnet 2 2391 2 Ia i1 i Moments of Inertia 1 Hoop or cylindrical shell ICM MR2 Solid cylinder or disk 1 CM m2 Fla110w cyhndcr I I m MR12 1222 39 J Rectangular plate 18M 1i2 Mm 172 amp Moments of Inertia 2 Long thin rod with rotation axis through center 1 CM ML2 12 50 id sphe re Icm MR2 e L l Long min rod with rotation axis through and i 2 I 3ML Thin spherical shell 9 ICM 3 MR2 Ca 1 L Cl Example 4 What is the kinetic energy of a 100 g 10 cm long rod rotating about its center of mass with a constant angular speed of 10 rpm Parallel Axis Theorem The Moment of Inertia about any axis of rotation is equal to the Moment of Inertia about the center of mass plus the mass times the distance from the center mass to the new axis squared Con Ill n I it 39 1h gh manner or mass Example 5 What is the moment of inertia for a sphere of mass M and radius R rotating about an axis 0733 from its center of mass 2 F x F Units are N m 239 rFsin 9 rFsin6 r Sin6F 11F I loop about central axis 1 MR2 2 Solid cylinder or disk about ceuml diameter L IMR2 2nIL2 4 Axis r TIl in I spherical shell about any 21 diameter t I MR2 g 1121141 Axis R I MRE Annular cylinder ol ring about central axis 0 Thin rod about axis Lhrough ccxxtcr pcrpendicular lo length 8 Hoop aboul any dialncxcr h I MR2 I lgMugz M Solid cylinder or disk about ccutml axis c Solid sphere aboxu any dialn me 21 I f Slab about pcrpendiculnr axis m rough celner aquot I lExample 10 8 and 710 10 a I lExample 1011

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