Understanding The Physics Of Toys: Principles, Theory And Exercises

Contents
Preface
Part I: Introduction
1. Fundamental Concepts
1.1 Kinematic Quantities of Motion
1.2 Moment of Inertia
1.3 Dynamics: Equations of Motion and Conservation Laws
1.4 Euler Angles
1.5 Types of Motion
1.5.1 Linear Motion
1.5.2 Rotational Motion
1.5.3 Rolling Motion
1.5.4 Spin, Precession and Nutation
1.6 Friction
1.7 Lagrangian Dynamics
1.8 Equilibrium Points and Their Stability Analysis
1.8.1 Equilibrium Points
1.8.2 Stability Criteria
1.9 Bernoulli’s Principle
1.10 Laws of Electromagnetism
1.11 Conclusion
1.12 Bibliography
1.13 Exercises
Part II: Mechanical Toys
1. Roly-Poly
1.1 Type-I and II Balancing Toys
1.2 Centre of Gravity
1.3 Analysis of a Model of the Roly-Poly
1.4 Conclusion
1.5 Bibliography
1.6 Exercises
2. Seesaw
2.1 Machines
2.2 Some Basics of Seesaw
2.2.1 A Simple Seesaw Model
2.2.2 Seesaw as a Big Balance
2.2.3 Balance Equation
2.3 A Theoretical Model of Seesaw
2.4 Conclusion
2.5 Bibliography
2.6 Exercises
3. Newton’s Cradle
3.1 Arrangement of a Cradle
3.2 Working Principle of the Newton’s Cradle
3.3 Number of Balls Ejected From the Cradle
3.4 Duration of Collision
3.5 A Two-Body Model
3.6 Conclusion
3.7 Bibliography
3.8 Exercises
4. Falling Slinky
4.1 Mathematical Model of a Slinky
4.1.1 Equation of Motion
4.1.2 The Hanging Slinky
4.1.3 The Falling Slinky with Instant Collapse of Turns
4.1.4 What Does Happen After the Time tc?
4.2 Illustration of Falling of the Slinky
4.3 Falling Elastic Bars
4.4 Conclusion
4.5 Bibliography
4.6 Exercises
5. Walking Slinky
5.1 Walking Behavior of a Slinky
5.2 Rate of Walking Down the Stairs
5.3 Slinky Whistlers
5.4 Some Applications of a Slinky
5.5 Conclusion
5.6 Bibliography
5.7 Exercises
6. Woodpecker
6.1 A Theoretical Model
6.1.1 Oscillation
6.1.2 Oscillation with Sliding
6.1.3 Pure Sliding
6.1.4 Boundary Conditions
6.1.5 Numerical Calculation
6.1.6 Determination of Relevant Parameters
6.2 Conclusion
6.3 Bibliography
6.4 Exercises
7. Cartesian Diver
7.1 Gas Laws, Buoyant Force and Archimedes’ Principle
7.2 Methods of Constructing Cartesian Divers
7.3 Change in Pressure Necessary for Irreversible Sinking
7.4 Expression for the Critical Height hc
7.5 Conclusion
7.6 Bibliography
7.7 Exercises
Part III: Spinning Toys
1. Top
1.1 Playing with the Spinning Top
1.2 Rotation of a Spinning Top
1.3 Effects of Height of the CoM and Weight of the Top
1.4 Theoretical Analysis
1.4.1 Lagrangian and Energy
1.4.2 Uniform Precession
1.4.3 Nutation
1.5 Motion of the Spinning Top
1.6 Stability of the Spinning Top
1.7 Conclusion
1.8 Bibliography
1.9 Exercises
2. Tippe Top
2.1 Motion of a Tippe Top
2.2 Role of Friction
2.3 A Simple Model for Turn Over of a Tippe Top
2.3.1 Forces and the Coordinate Axes
2.3.2 Frictional Torque
2.3.3 Significance of Frictional Torque
2.4 Conclusion
2.5 Bibliography
2.6 Exercises
3. PhiTOP
3.1 Geometry and Notation of the Spinning Object
3.2 Dynamical Equations
3.3 Equation of Motion for θ
3.4 Steady Precession
3.5 Rising of a PhiTOP
3.6 Rising of a Spinning Egg
3.7 Conclusion
3.8 Bibliography
3.9 Exercises
4. Euler’s Disk
4.1 Euler’s Disk in Rolling Motion
4.1.1 Kinematics
4.1.2 Expressions for Spin and Precession Rates
4.2 Finite-Time Singularity
4.3 Friction at Small Angle and Finite-Time Singularity
4.4 Conclusion
4.5 Bibliography
4.6 Exercises
5. Fidget Spinner
5.1 Physics of the Fidget Spinner
5.2 Spin Time of a Fidget Spinner
5.3 Calculation of the Moment of Inertia
5.4 Conclusion
5.5 Bibliography
5.6 Exercises
6. Rattleback
6.1 Spin Reversal Requirements
6.2 Equation of Motion
6.3 Rattleback with a Symmetric Distribution of Mass
6.4 Rattleback with an Asymmetric Distribution of Mass
6.5 Inertial Asymmetry and Rotation of the Principal Moment of Inertia Axes
6.6 Conclusion
6.7 Bibliography
6.8 Exercises
7. Hurricane Balls
7.1 Qualitative Analysis
7.2 Quantitative Analysis
7.2.1 Lagrangian of the System
7.2.2 Condition for Lift-Off
7.2.3 Steady State Solution
7.3 Role of Ω on the Steady State Solution
7.4 Two Balls Separated by a Distance
7.5 Conclusion
7.6 Bibliography
7.7 Exercises
8. Gee-Haw Whammy-Diddle (The Notched Stick)
8.1 Physical Explanation
8.2 A Theoretical Model
8.2.1 Tight Contact
8.2.2 Loose Contact
8.3 Gravity-Spinning Magnet
8.4 Conclusion
8.5 Bibliography
8.6 Exercises
9. Gyroscope
9.1 Tricks with the Chandler Gyroscope
9.2 A Four-Mass Gyroscope Model
9.2.1 Forces on Rotating Masses Due to Precession
9.2.2 Force on Precessing Masses Due to Rotation
9.3 Dipping of Gyroscope and its Precession Motion
9.4 Stability Analysis of the Gyroscopic Motion
9.5 Conclusion
9.6 Bibliography
9.7 Exercises
10. The Buzzer
10.1 Theoretical Model
10.1.1 Energy Sources and Equation of Motion of ϕ
10.1.2 Input Torque
10.1.3 Drag Torque
10.2 Twisting Torque
10.3 Conclusion
10.4 Bibliography
10.5 Exercises
11. Yo-Yo
11.1 Linear and Rotational Motions
11.2 Physical Mechanism of Working of Yo-Yo
11.3 Motion of the Unwinding Yo-Yo
11.3.1 Forces and an Expression for Acceleration
11.3.2 Expression for I
11.3.3 Expression for T
11.3.4 Energy of Yo-Yo
11.3.5 Expression for h and Time Required for Complete Unwinding
11.3.6 Equation for the Vertical Displacement
11.4 Conclusion
11.5 Bibliography
11.6 Exercises
12. Astrojax
12.1 Orbital Motions of a Ball of an Astrojax
12.2 Some Other Tricks
12.3 Conclusion
12.4 Bibliography
12.5 Exercises
13. Hula Hoop
13.1 Hula Hoop is a Sport/Game
13.2 Physical Basis of the Hula Hooping Skill
13.3 Theoretical Study
13.3.1 Model Equations
13.4 Stability Analysis
13.5 Conclusion
13.6 Bibliography
13.7 Exercises
Part IV: Flying Toys
1. Balsa Gliders
1.1 Aerodynamic Forces
1.2 How Does a Glider Fly?
1.3 Significant Factors of Bulsa Gliders
1.4 Determination of Glide Angle and Lift and Drag Forces
1.5 Maximum Time Aloft
1.6 Conclusion
1.7 Bibliography
1.8 Exercises
2. Magnus Glider
2.1 Magnus Force
2.2 Vector Potential for Incompressible Irrotational Flow
2.3 Theoretical Model
2.3.1 Forces Acting on a Magnus Glider
2.3.2 Velocity Potential
2.3.3 Lift (Magnus) Force
2.3.4 Equation of Motion
2.4 Conclusion
2.5 Bibliography
2.6 Exercises
3. Kite
3.1 Forces Acting on a Kite
3.2 The Four Phases of Flight
3.3 A Physical Model
3.3.1 Lagrangian
3.3.2 Generalized Forces
3.3.3 Lagrange’s Equations
3.4 Physics of Kite Strings
3.5 Conclusion
3.6 Bibliography
3.7 Exercises
Part V: Throwable Toys
1. Frisbee
1.1 How Does a Frisbee Fly?
1.2 Gyroscopic Stability
1.3 Theoretical Treatment
1.4 Numerical Simulation
1.5 Conclusion
1.6 Bibliography
1.7 Exercises
1.8 Program
2. Boomerang
2.1 Throwing a Boomerang
2.2 Forces Acting on a Boomerang
2.3 Why Does a Boomerang Fly?
2.4 Crescent Shape of the Boomerang and its Stability
2.5 How Does a Boomerang Come Back?
2.6 Conclusion
2.7 Bibliography
2.8 Exercises
3. Skipping Stones
3.1 Why and How Do Stones Skip Across the Water?
3.2 Mathematical Theory
3.2.1 Basic Assumptions
3.2.2 Equations of Motion
3.2.3 Square Stone
3.3 Maximum Number of Skips and the Bounce-Off Condition
3.4 Trajectory of the Skipping Stone
3.5 The Need to Spin the Stone and the Magic Value of α
3.6 Conclusion
3.7 Bibliography
3.8 Exercises
4. Bouncing Popper
4.1 Explanation of High Bouncing of a Popper
4.2 Calculation of Initial Speed and the Time to Rise to a Maximum Height
4.3 Conclusion
4.4 Bibliography
4.5 Exercises
Part VI: Heat and Thermodynamic Toys
1. Drinking Bird
1.1 Description of the Drinking Bird Setup
1.2 Working and Performance of the Toy
1.3 Period of Oscillation
1.4 Sunbird
1.5 Drinking Bird of Second Kind
1.6 Conclusion
1.7 Bibliography
1.8 Exercises
2. Putt-Putt Boat
2.1 Principle of Operation of the Putt-Putt Boat – Cyclic Ejection and Suction of Water
2.2 Origin of the Putt-Putt Sound and the Need of Two Pipes
2.3 Mechanism of Propulsion
2.4 Conclusion
2.5 Bibliography
2.6 Exercises
3. Crookes Radiometer
3.1 Working Principle of the Radiometer
3.2 Radiometric Forces
3.3 Thermodynamic Explanation
3.4 Backward Motion of Vanes
3.5 Conclusion
3.6 Bibliography
3.7 Exercises
Part VII: Electric and Magnetic Toys
1. The Gauss Rifle
1.1 Working of a Gauss Rifle
1.1.1 Number of Balls to be Used
1.1.2 Field, Force and Magnetic Moment
1.2 Optimization of a Single Stage Gauss Rifle
1.3 Maximum Kinetic Energy for N Successive Rifles
1.4 Conclusion
1.5 Bibliography
1.6 Exercises
2. The Levitron
2.1 Qualitative Analysis of Levitation
2.2 Dynamical Origin of Stable Float
2.2.1 Total Force and Total Potential
2.2.2 Equilibrium Point
2.2.3 Stability Analysis
2.2.4 Stability region for a Ring Dipole
2.3 Conclusion
2.4 Bibliography
2.5 Exercises
3. Electric Train
3.1 Mechanical Description
3.2 Equations of Motion
3.3 Determination of the Parameters
3.4 Conclusion
3.5 Bibliography
3.6 Exercises
4. A Simple Electromagnetic Train
4.1 Principle of Working
4.2 Forces Opposing the Motion of the Train
4.3 Forces on the Train
4.3.1 Force Due to the Current in the Solenoid
4.3.2 Force due to Eddy Currents
4.4 Conclusion
4.5 Bibliography
4.6 Exercises
5. Fun-Fly-Stick
5.1 Triboelectricity, Electrostatic Induction, Corona Discharge
5.2 Operation Inside the Stick
5.3 Floating of a Fun-Flyer
5.4 Some Other Fun Activities with the Fun-Fly-Stick
5.5 Conclusion
5.6 Bibliography
5.7 Exercises
6. Plasma Ball
6.1 Description
6.2 Some Fun Tricks With a Plasma Globe
6.3 Conclusion
6.4 Bibliography
6.5 Exercises
Part VIII: Miscellaneous Toys
1. Some Other Toys
1.1 Balancing Toys
1.2 Gyro-Ring
1.3 Toy Guns
1.4 Floating a Ping-Pong Ball
1.4.1 Mechanism of Floating of the Ball
1.4.2 Minimum Volumetric Flow Rate for Stable Floating
1.5 Simple Helicopter Toys
1.6 Pinwheel
1.7 Tornado Bottle
1.8 Magnetic Levitation of Iron Balls
1.9 Magnetic Seal and Ball
1.10 Simple DC Motors
1.11 Kaleidoscope
1.12 Conclusion
1.13 Bibliography
1.14 Exercises
Index