# 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*

- what is meant by "uniform" circular motion
- 1.1.2. Newton's laws and uniform circular motion
*R4*- relationship between net force and acceleration
*R4*

- relationship between net force and acceleration
- 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*

- what is meant by "non-uniform" circular motion
- 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*

- importance of finding circles that match the curvature of the path
- 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*

- only 2 new "big ideas" in circular motion
- 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*

- table of ideas and principles needed to solve circular motion problems

- types of situations covered by
- 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 v
_{x}and v_{y}vs. time to find a_{x}and a_{y}*R12*

- what is meant by "simple" projectile motion
- 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*

- using a graph to write an expression for horizontal position vs. time
- 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*

- defining symbols for the vectors
- 1.2.4. Free-fall acceleration
*R14*- difference between
*g*and*a*_{g}*R14* - why we use the symbol
*a*to denote free-fall acceleration_{g}*R14*

- difference between
- 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*

- what is meant by the term
- 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*

- seeing patterns in how the speed and velocity of a projectile change
- 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*

- recognizing that time
- 2 examples
*R19,20* - how to interpret a negative root
*R20*

- 4 relationships needed to solve problems in simple projectile motion
- 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*

- 4 relationships needed to solve problems in 2-dimensional motion

- what is meant by
- 1.3. RELATIVE MOTION
*R23-35*- situations covered by
*relative motion**R23*- some goals of studying relative motion
*R23*

- some goals of studying relative motion
- 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*

- 4 people at the airport on or near a moving walkway
- 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*

- what is meant by
- 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*

- labeling frames as "primed" and "unprimed"
- 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*

- Jamal throws a ball into the air while riding a skateboard
- 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*

- a boat is crossing a river, while Sue is running along the shore
- 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*

- science experiments on a train moving with constant velocity relative to the ground
- 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*

- throwing a ball from the ground and from a moving train
- 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*

- the
- sometimes, a situation is easier to analyze in one frame than another
*R32,33*

- only 3 new ideas
- 1.3.9. Solving problems with relative motion ideas
*R33-35*- many common problems involve navigation
*R33,34* - definition of the term
*heading**R35*

- many common problems involve navigation

- situations covered by

- 3 independent sections: circular motion, projectile motion & relative motion
- 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*

- why we need to introduce a new set of kinematic quantities
- 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*

- description of linear motion
- 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*

- why the radian is different from other units of measure
- 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*

- angular velocity and linear velocity are very different quantities
- 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*

- relationship between angular speed and angular velocity

- what is meant by
- 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*

- what is meant by
- 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*

- 4 factors affecting the torque
- 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*

- 3 factors affecting the moment of inertia
- 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*

- mathematical description of Newton's 2nd law for rotations about a fixed axis
- 2.2.5. Angular vs. linear dynamics
*R48*- table comparing linear and angular dynamics
*R48*

- table comparing linear and angular dynamics
- 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*

- for static situations, every axis is a fixed axis of rotation
- 2.2.7. Solving problems in angular dynamics
*R51*- an example
*R51* - relationship between angular acceleration and linear acceleration
*R51*

- an example

- situations covered by
- 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*

- rewriting the kinetic energy using rotational quantities
- 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*

- how energy can be stored in a rotational system
- 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*

- table comparing energy for linear and rotational motion
- 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 examples
- 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*

- how conservation of energy and the Work-Kinetic Energy Theorem are applied

- 2.3.1. Kinetic energy of rotating objects
- 2.4. SOLVING PROBLEMS IN ROTATIONAL MOTION
*R56*- general guidelines for solving problems in rotational motion
*R56*

- general guidelines for solving problems in rotational motion

- situations covered by

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