Chapter 10 Notes
Chapter 10 Notes Physics 125
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This 4 page Class Notes was uploaded by Nora Salmon on Friday February 27, 2015. The Class Notes belongs to Physics 125 at University of Alabama - Tuscaloosa taught by Prof. Andreas Piepke in Fall2015. Since its upload, it has received 209 views. For similar materials see Physics 1 w/Calculus in Physics 2 at University of Alabama - Tuscaloosa.
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Date Created: 02/27/15
Chapter 10 o 101 What is Physics 0 So far we have discussed translational motion in which an object moves along a straight or curved line 0 In this chapter we will examine rotational motion in which an object turns about an axis 0 102 The Rotational Variables o A rigid body is a body that can rotate with all its parts locked together and without any change in its shape A xed axis means that the rotation occurs about an axis that does not move 0 A xed axis can be called the quotaxis of rotationquot or quotrotation axisquot 0 In pure rotation every point of the body moves in a circle whose center lies on the axis of rotation o This means that every point of the body moves at a xed radius away from the axis of rotation o Angular Position When examining rotational motion we use a reference line a line xed in the rigid body perpendicular to the rotation axis and rotating with the body The angular position of this line is the angle of this line relative to a xed direction 0 This xed direction is the zero angular position 0 We have an equation for the angular position 0 0 where sis arc length extending from the zero angular position to the reference line and ris the radius of the circle We see then that theta is measured in radians o Angular Displacement If a body rotates and changes its angular position the body undergoes an angular displacement This angular displacement holds not only for the rigid body as a whole but also for every particle in the body Counterclockwise angular displacement is positive and clockwise is negative 0 Angular Velocity 0 Chapter 10 Average angular velocity is the angular displacement divided by the time interval tft Average angular velocity is o 001im At gt0 t The magnitude of an angular velocity is called the angular speed Angular Acceleration The instantaneous angular acceleration of an object is the time derivative of its angular velocity A00 0 x 11m At gt0 t o 103 Are Angular Quantities Vectors O 0 Yes and we can treat them as vectors by using the right hand rule A vector de nes an axis of rotation not a direction in which something moves nonetheless the vector also de nes the rotational motion For rotations around a xed axis we don t need to use vectors because there are only two directions clockwise negative and counterclockwise positive Anguar displacements cannot be treated as vectors because you cannot add angular displacements commutatively o 104 Rotation with Constant Angular Acceleration 0 Rotation with constant angular acceleration has a set of equations that parallels the kinematic equations angular velocity replaces linear velocity angular acceleration replaces linear acceleration and angular displacement replaces linear displacement o 105 Relating the Linear and Angular Variables O In a rigid body all the particles make one revolution in the same amount of time they all have the same angular speed However the farther a particle is from the axis the larger the circumference it must travel at the same time as particles closer to the axis of rotation They must have a larger linear speed The equation for the period of revolution T relates circumference linear speed v and angular velocity 2Hr2H v o The linear acceleration has two components 0 T Chapter 10 Tangential component change in magnitude of linear velocity 0 atO r Radial component change in direction of linear velocity 2 V 2 arZ Zw r r o 106 Kinetic Energy of Rotation o How do we treat a rigid body to nd its kinetic energy We treat it as a bunch of particles then add up the kinetic energies of the particles to nd the kinetic energy of the whole body 1 Zam wrlf where is the 1th particle o The rotational inertia of an object tells us how the mass of the rotating body is distributed about its axis of rotation 39 2 miri2 0 Now kinetic energy has an equation that relates rotational inertia and angular velocity I Z llwz 2 o 107 Calculating the Rotational Inertia 0 Since rigid bodies are usually composed of many particles we use an integral to de ne the rotational inertia of the body and write fr2dm 0 Table 102 on p255 lists nine common rotational inertia formulas based on nine body shapes 0 ParallelAxis Theorem If we want to nd the rotational inertia of a body of mass Mabout an axis we can use a shortcut if we know the rotational inertia of the body about a parallel axis stretching through that body s center of mass If h is the perpendicular distance between the two axes then we have paralle Mh2 o This is called the parallelaxis theorem 0 For a mathematical proof of this theorem see p 25455 0 108 Torque 0 Think about opening a door it doesn t only matter that you apply the force but also where you apply the force the knob Chapter 10 This causes the door to rotate A force applied parallel to the door would not cause the door to rotate 0 So we have a quantity torque that equals the product of the magnitude of the force applied and the distance from the origin perpendicular to the axis 39 TrsinF This perpendicular distance is called the moment arm of F So what is torque Comes from Latin word quotto twistquot and can be de ned as the turning or twisting of F 0 SI Unit Nm Torque is either or based on the rotation CCW is positive CW is negative When several torques act on a body the net resultant torque is the sum of these torques 109 Newton s 2nCI Law for Rotation 0 Using the 2ncl law we can relate the net torque on a body to its angular acceleration about a rotation axis Net force becomes net torque mass becomes rotational inertia I and acceleration becomes angular This gives TICX 1010 Work and Rotational KE 0 O O This section is basically just equations that he ll give us in class When a torque accelerates a body around a xed axis rotating it the torque does work on the body This means that the rotational KE can change 0 We can use the workkinetic energy theorem for rotational KE by replacing KE with KErot Also we can calculate the work as the de nite integral of torque from the body s initial to nal angular positions with respect to those positions measured in radians IDS
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