Spring Work and the Meaning of the Sign

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1Motion as Slopes and Areas on Graphs 2Velocity as the Time-Derivative of Position in 1D 3Acceleration as the Time-Derivative of Velocity 4Displacement as the Integral of Velocity With Initial Conditions 5Velocity Change as the Integral of Acceleration 6Constant-Acceleration Motion Derived From Integrals 7When High-School Kinematics Shortcuts Break

8Displacement, Velocity, and Acceleration as Vectors 9Components, Unit Vectors, and Coordinate Choices 10Differentiating and Integrating Vector-Valued Functions 11Projectile Motion From a Constant Acceleration Vector 12Time of Flight and Range as Consequences of r(t)13When a 1D Shortcut Is Allowed and When It Fails 14Dot Products, Projections, and Directionality Tools

15Forces as Causes of Acceleration: The Modeling Question 16Newton’s Second Law in Vector and Component Form 17Constraints and Contact Forces: Gravity and the Normal Force 18Kinetic Friction as a Directional Model, Not a Minus Sign 19Initial Conditions and Integration Constants in Constant-Force Motion 20Position-Dependent Force: The Spring as a Non-Constant-Acceleration Model 21Diagnosing Non-Constant Acceleration From Force Dependence 22Units, Setup-Only Modeling, and the Impulse Preview

23Work as Force Accumulated Along a Displacement 24The Constant-Force Work Shortcut and Its Validity Conditions 25The Work–Energy Theorem From Newton’s Second Law in 1D

26Spring Work and the Meaning of the Sign

27Solving Variable-Force Motion With the Work–Energy Method 28Power as the Time Rate of Doing Work 29Conservative vs Non-Conservative Forces as Path (In)dependence 30Choosing Between Newton’s Laws and Energy Methods

31Rate Proportional to Amount as a Physical Model 32Separation of Variables as an Operator Workflow 33Time Constants and Initial Conditions in Exponential Decay 34Differential vs Integral Statements of Change 35Deriving the RC Differential Equation From Circuit Relations 36Solving RC Charging and Discharging With Separation of Variables 37Unit and Limit Checks for Exponential Models 38Setup-First ODE Modeling for RC Scenarios

39A Spring–Mass System as a Physical Model 40Deriving the SHM Differential Equation From Newton’s Second Law 41Solving m x¨ + kx = 0 and Defining ω 42Applying Initial Conditions to Fix Amplitude and Phase 43Period and Frequency From ω 44Energy Exchange as a Consistency Check for SHM 45Diagnosing SHM Mistakes and Previewing Damped Motion

46Electric Field and Electric Potential Difference as Distinct Quantities 47Potential Difference as the Accumulation of Field in One Dimension 48Field From Potential in One Dimension: The Meaning of a Spatial Derivative 49The Line-Integral Idea: Dot Product Along a Path 50Conservative Fields and the Meaning of Path Independence 51Partial Derivatives and the Gradient as Local Spatial Slopes 52Reading Field Direction From Contours of Constant Potential 53Sign Conventions, Scope Boundaries, and Gradient-Strength Synthesis

54From Weighted Sums to Center of Mass Integrals 55Building dm From Linear Density for a Non-Uniform Rod 56Symmetry and Limiting Cases as Setup Checks 57Moment of Inertia as a Rotational Weighted Average 58Moment of Inertia of a Rod About Different Axes 59Moment of Inertia of a Disk Using Area Density 60Common Setup Errors for Center of Mass and Moment of Inertia 61Capstone Synthesis and Exit Checklist

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Calculus-Based Physics: Kinematics, Fields and ODEs
Spring Work and the Meaning of the Sign