AT: Advanced Topics in Mechanics

Activities & Reader (ISBN 0-7872-5411-8, 172 pages)

How to Use this Book    xv

Acknowledgments    xvii

Activities

  • AT·1 - Exploring Ideas About Circular Motion    1
  • AT·2 - Finding Acceleration for Circular Motion    5
  • AT·3 - Finding Radial Acceleration for Circular Motion    9
  • AT·4 - Finding Tangential Acceleration for Circular Motion    13
  • AT·5 - Reasoning About Circular Motion    15
  • AT·6 - Solving Problems in Circular Motion    19
  • AT·7 - Exploring Ideas About Projectile Motion    23
  • AT·8 - Relating Kinematic Quantities for Two-Dimensional Motion    29
  • AT·9 - Reasoning About Projectile Motion    35
  • AT·10 - Solving Problems in Projectile Motion    39
  • AT·11 - Solving Problems in Two-Dimensional Motion    43
  • AT·12 - Exploring Ideas About Relative Motion    47
  • AT·13 - Exploring Relative Motion in Two Dimensions    51
  • AT·14 - Reasoning About Relative Motion    55
  • AT·15 - Solving Problems in Relative Motion    59
  • AT·16 - Graphing Rotational Motion    63
  • AT·17 - Introducing Rotational Kinematics    67
  • AT·18 - Solving Rotational Kinematics Problems    71
  • AT·19 - Introducing Rotational Dynamics    75
  • AT·20 - Solving Rotational Dynamics Problems    79
  • AT·21 - Identifying Energy in Rotational Systems    83
  • AT·22 - Solving Problems with Energy in Rotational Systems    87
  • AT·23 - Solving Problems in Rotational Motion    91

Reader: Advanced Topics in Mechanics

  • Chapter 1. Circular, Projectile & Relative Motion
    • 3 independent sections: circular motion, projectile motion & relative motion    R1
    • 1.1. CIRCULAR MOTION    R1-10
      • types of situations covered by circular motion    R1,2
      • 1.1.1. Uniform circular motion    R2-4
        • what is meant by "uniform" circular motion    R2
        • factors affecting acceleration: speed and radius of circle    R2
        • starting with the definition of acceleration    R2
        • diagram showing the change in velocity [delta]v for a small time period    R3
        • table showing the average acceleration for smaller and smaller time periods    R3
        • 1 effect of doubling the radius of the circular path    R3
        • 2 effects of doubling the speed of the ball    R3
        • magnitude of the acceleration for uniform circular motion    R4
        • direction of the acceleration for uniform circular motion    R4
      • 1.1.2. Newton's laws and uniform circular motion    R4
        • relationship between net force and acceleration    R4
      • 1.1.3. Non-uniform circular motion    R5,6
        • what is meant by "non-uniform" circular motion    R5
        • definition of the radial component of acceleration    R5
        • definition of the tangential component of acceleration    R5
        • magnitude of the radial component of acceleration for motion along any circle    R5
        • direction of the radial component of acceleration    R5
        • magnitude of the tangential component of acceleration for motion along any circle    R5
        • direction of the tangential component of acceleration    R5
        • finding the forces responsible for the radial and tangential accelerations    R5,6
      • 1.1.4. Motion along a curved path    R6,7
        • importance of finding circles that match the curvature of the path    R6
        • radial acceleration points toward the center of curvature    R6
        • radius of curvature is the radius of the matching circle    R7
        • magnitude of the radial component of acceleration for motion along any path    R7
        • direction of the radial component of acceleration    R7
      • 1.1.5. Reasoning with circular motion ideas    R7-9
        • only 2 new "big ideas" in circular motion    R7
        • integrating old ideas into new situations    R7
        • using a free-body diagram to analyze circular motion    R8
        • using energy ideas to analyze circular motion    R8,9
      • 1.1.6. Solving problems with circular motion ideas    R9,10
        • table of ideas and principles needed to solve circular motion problems    R9
        • example showing all the ideas that can impact a circular motion problem    R10
    • 1.2. PROJECTILE MOTION    R11-22
      • what is meant by projectile motion    R11
      • 1.2.1. Simple projectile motion    R11,12
        • what is meant by "simple" projectile motion    R11
        • an example using strobe diagram of a ball thrown into the air    R11,12
        • relationship of strobe diagram and plots to Newton's laws and force ideas    R12
        • using plots of vx and vy vs. time to find ax and ay    R12
      • 1.2.2. Algebraic representation of simple projectile motion    R12,13
        • using a graph to write an expression for horizontal position vs. time    R12
        • using a graph of velocity vs. time to derive expressions for vertical velocity vs. time and height vs. time    R12,13
      • 1.2.3. Algebraic representation of two-dimensional motion    R13
        • defining symbols for the vectors r, v, and a    R13
        • kinematic expressions for position and velocity as functions of time for constant acceleration    R13
      • 1.2.4. Free-fall acceleration    R14
        • difference between g and ag    R14
        • why we use the symbol ag to denote free-fall acceleration    R14
      • 1.2.5. Special features of simple projectile motion    R14
        • what is meant by the term trajectory    R14
        • 3 special features of a trajectory: time of flight, range, and maximum altitude    R14
        • labeled diagram of trajectory showing special features    R14
        • what the time of flight depends on    R14
        • what the range depends on    R14
        • what the maximum altitude depends on    R14
      • 1.2.6. Reasoning about simple projectile motion    R15-17
        • seeing patterns in how the speed and velocity of a projectile change    R15
        • comparing trajectories to understand projectile motion    R16
        • applying Newton's laws to projectile motion    R17
        • applying conservation of energy to projectile motion    R17
      • 1.2.7. Solving problems in simple projectile motion    R18-20
        • 4 relationships needed to solve problems in simple projectile motion    R18
        • 4 keys to solving projectile motion problems    R18,19
          • recognizing that time t is the same in all 4 relationships    R18
          • translating given information properly into equation form    R18
          • focusing on special features of trajectories    R18
          • realizing when you have enough equations to solve for the unknown    R18,19
        • 2 examples    R19,20
        • how to interpret a negative root    R20
      • 1.2.8. Solving problems in two-dimensional motion    R21,22
        • 4 relationships needed to solve problems in 2-dimensional motion    R21
        • 2 examples    R21,22
    • 1.3. RELATIVE MOTION    R23-35
      • situations covered by relative motion    R23
        • some goals of studying relative motion    R23
      • 1.3.1. Relative motion in one dimension    R23,24
        • 4 people at the airport on or near a moving walkway    R23
        • table of velocities as seen from 2 different perspectives    R24
      • 1.3.2. Reference frames    R24
        • what is meant by reference frame    R24
        • table of positions as measured in 2 different frames at <nobr>t = 0.0 s</nobr>    R24
        • why some positions change but other positions stay the same    R24
      • 1.3.3. Notation and language    R25
        • labeling frames as "primed" and "unprimed"    R25
        • labeling positions and velocities as "primed" and "unprimed"    R25
        • reasons someone's speed can be zero even though everyone agrees he is moving    R25
      • 1.3.4. Relative motion in two dimensions    R26
        • Jamal throws a ball into the air while riding a skateboard    R26
        • to Jamal, motion of the ball is 1-dimensional    R26
        • to Betty, motion of the ball is 2-dimensional    R26
      • 1.3.5. Position and velocity transformations    R26-29
        • a boat is crossing a river, while Sue is running along the shore    R26
        • in 2 dimensions, each reference frame has 2 coordinate axes    R26
        • graphical representation of position transformation    R26,27
        • numerical and symbolic representations of position transformation    R27
        • general expressions for transforming positions    R27
        • general expression for transforming velocity    R27
        • 3 representations of velocity transformation    R27
        • general expression for transforming acceleration    R28
        • 2 examples of velocity transformation    R28,29
      • 1.3.6. Newton's laws in different reference frames    R29,30
        • science experiments on a train moving with constant velocity relative to the ground    R29
        • laws of physics are the same in a frame moving with constant velocity    R29
        • science experiments on a train slowing down relative to the ground    R29,30
        • Newton's laws and empirical laws are different in an accelerating frame    R30
        • small accelerations have only small effects on Newton's laws    R30
        • definition of the phrase inertial frame    R30
      • 1.3.7. Conservation of energy in different reference frames    R30,31
        • throwing a ball from the ground and from a moving train    R30,31
        • change in kinetic energy depends on the frame of reference    R31
        • work done by a force depends on the frame of reference    R31
        • table showing how the scenarios look different in different frames    R31
      • 1.3.8. Reasoning with relative motion ideas    R32,33
        • only 3 new ideas    R32
          • the reference frame is the key to determining positions, velocities, and energy    R32
          • when the frames are inertial, forces, masses, and accelerations are the same in all frames    R32
          • there is no preferred reference frame    R32
        • sometimes, a situation is easier to analyze in one frame than another    R32,33
      • 1.3.9. Solving problems with relative motion ideas    R33-35
        • many common problems involve navigation    R33,34
        • definition of the term heading    R35
  • Chapter 2. Rotational Motion
    • situations covered by rotational motion    R36
    • how we are going to approach rotational motion    R36
    • why we are going to always use a fixed axis    R36
    • 3 main sections: angular kinematics, angular dynamics & energy in rotating systems    R36
    • 2.1. ANGULAR KINEMATICS    R37-42
      • what is meant by angular kinematics    R37
        • why we need to introduce a new set of kinematic quantities    R37
      • 2.1.1. Angular vs. linear kinematics    R37,38
        • description of linear motion    R37
        • description of angular motion    R37
        • what is meant by "CCW" and "CW"    R37
        • CCW rotations are positive    R37
        • table comparing linear motion and rotational motion (fixed axis)    R38
      • 2.1.2. The radian    R38,39
        • why the radian is different from other units of measure    R38
        • why the radian is the preferred unit for angles    R38
        • an example using arc length    R38,39
        • 2 examples applying the radian    R39
        • why certain relationships are not proper    R39
      • 2.1.3. Reasoning with angular kinematics ideas    R40,41
        • angular velocity and linear velocity are very different quantities    R40
        • linear velocity depends on your location on the spinning object    R41
        • the linear velocity can be zero even though the object is spinning    R41
      • 2.1.4. Solving problems in angular kinematics    R41,42
        • relationship between angular speed and angular velocity    R41
        • graphs can help organize information and help solve problems    R42
    • 2.2. ANGULAR DYNAMICS    R43-51
      • situations covered by angular dynamics    R43
      • 2.2.1. Pivots    R43
        • what is meant by pivot    R43
        • an example using a hinged door    R43
        • why we ignore forces parallel to the axis of rotation    R43
        • what is meant by "about the pivot" or "about the point p"    R43
      • 2.2.2. Torque    R44-46
        • 4 factors affecting the torque    R44
        • 2 definitions of torque for rotations about a fixed axis    R44
        • finding the direction of torque    R44
        • SI unit of torque (N·m)    R44
        • 2 examples    R45
        • definition of net torque for rotations about a fixed axis    R46
      • 2.2.3. Moment of inertia    R46,47
        • 3 factors affecting the moment of inertia    R46
        • definition of moment of inertia (point mass)    R46
        • definition of moment of inertia (composite object)    R46
        • 2 examples    R47
      • 2.2.4. Newton's 2nd law in rotational form    R48
        • mathematical description of Newton's 2nd law for rotations about a fixed axis    R48
      • 2.2.5. Angular vs. linear dynamics    R48
        • table comparing linear and angular dynamics    R48
      • 2.2.6. Reasoning with angular dynamics ideas    R48-50
        • for static situations, every axis is a fixed axis of rotation    R48,49
        • 3 examples    R49,50
        • the gravitational force acts "as though" through the center of gravity or balance point    R49
      • 2.2.7. Solving problems in angular dynamics    R51
        • an example    R51
        • relationship between angular acceleration and linear acceleration    R51
    • 2.3. ENERGY IN ROTATIONAL SYSTEMS    R52-56
      • 2.3.1. Kinetic energy of rotating objects    R52
        • rewriting the kinetic energy using rotational quantities    R52
      • 2.3.2. Potential energy in rotational systems    R52
        • how energy can be stored in a rotational system    R52
        • torque law for a torsional spring    R52
        • potential energy for a torsional spring    R52
      • 2.3.3. Energy for linear vs. rotational motion    R53
        • table comparing energy for linear and rotational motion    R53
        • why we do not refer to "angular energy"    R53
      • 2.3.4. Reasoning with energy ideas in rotational systems    R53,54
        • 2 examples    R53,54
        • importance of using the center of gravity in energy problems    R54
      • 2.3.5. Solving problems with energy ideas in rotational systems    R54-56
        • how conservation of energy and the Work-Kinetic Energy Theorem are applied    R54,55
        • why there is no such thing as "angular" energy    R55
        • 2 examples    R55,56
    • 2.4. SOLVING PROBLEMS IN ROTATIONAL MOTION    R56
      • general guidelines for solving problems in rotational motion    R56