Preface

Physics tells stories about change. Newtonian physics, in particular, gives a precise language to describe how things move and why. This book aims to be rigorous where it matters and friendly everywhere: figures first, formulas second, with short proofs or precise definitions when they sharpen understanding.

Guide for Readers

Pick a path that matches your goal and time.

Conceptual Tour (intuition first).

Follow Parts I–VI for the story of motion, relying on figures, analogies, and summaries. Skim the heavier ODE and numerical derivations (e.g., Chapter 2 Sections 2.4–2.5; Chapter 7 Section 7.5). Skip Part VII and treat the appendices as optional reference, glancing at the glossary when a symbol or term is new.

Mathematical Deep Dive (rigor and computation).

Read everything in Parts I–VII, including the ODE structure and numerical preview in Chapter 2, the force‑to‑motion component work in Chapter 7, and the modeling and simulation focus in Chapters 15 and 16. Pair chapters with the math appendices: start with Appendices A–B (calculus, vectors), use Appendix C for gradients/curl and the conservative field test \(\nabla \times \mathbf {F} = \mathbf {0}\) and \(\mathbf {F} = -\nabla U\), and lean on Appendices D–E for ODE stability, error scaling, and why symplectic Euler behaves better on energy.

Style and Approach

We keep a light‑hearted tone while treating the math with respect. Figures and short callouts lead the way; precise equations and compact proofs follow where they add clarity. We use the International System of Units (SI), check dimensions as a habit, and favor analogies and worked examples that tie the symbols back to everyday experience. Exercises mix quick conceptual checks with approachable real‑world tasks.

What This Book Covers

The book is self‑contained and spans the essentials of Newtonian mechanics—concepts, methods, and a compact math appendix. Here is a preview so you can choose your path:

• • Part I builds intuition and language: Chapter 1 frames the scope and scales; Chapter 2 develops graphs, vectors, calculus ideas, ODEs, and a numerical preview.

• • Part II moves along a line: Chapter 3 ties slopes/areas to \(x(t),v(t),a(t)\); Chapter 4 links forces to motion and introduces simple numerical updates.

• • Part III adds dimensions: Chapter 5 develops vector kinematics; Chapter 6 applies the toolkit to projectile and circular motion.

• • Part IV unifies forces and energy: Chapter 7 (FBDs, components, friction/drag/tension) and Chapter 8 (work, kinetic energy, power) plus Chapter 9 (potential energy and conservation).

• • Part V treats many‑particle systems and rotation: Chapter 10 (COM, momentum, collisions) and Chapter 11 (torque, rotational dynamics, energy, angular momentum).

• • Part VI samples gravity, oscillations, and a taste of continua: Chapters 12 to 14.

• • Part VII adds cross‑cutting methods: Chapter 15 (dimensions, scaling) and Chapter 16 (numerical time‑stepping and stability/accuracy intuition).

• • Appendices collect just the math we actually use: calculus (definitions + visuals), vectors/linear algebra, multivariable essentials (gradient/divergence/curl; line integrals), ODEs (separable/linear; oscillations; slope fields), and numerics (Euler flavors; error/stability/energy checks).

Who This Book Is For

Learners with high‑school algebra/trigonometry who want a clear, visual path into mechanics; engineers and technically minded professionals (including quantitative analysts, financial modellers, algorithmic traders, and researchers) who want a compact, simulation‑ready refresher with reproducible figures; and autodidacts who prefer a friendly tone without sacrificing correctness.

Objectives

By the end, you should be able to model motion with clean diagrams and equations; choose between force‑ and energy‑based methods; reason about orders of magnitude; and make pragmatic numerical predictions while checking accuracy and stability.

Notation and Conventions

We use the International System of Units (SI) throughout. Scalars appear in italic (e.g., \(m, t\)); vectors in bold lower‑case (e.g., \(\bm v, \bm a\)); matrices and tensors in bold upper‑case (e.g., \(\bm R\)). Unit vectors carry hats (e.g., \(\hat {\bm e}_x\)). Time derivatives use an overdot (e.g., \(\dot x\)), and generic derivatives use prime when context is clear (\(f'(x)\)).

Acknowledgments

With gratitude, I acknowledge the artificial intelligence research community and OpenAI for advancing tools such as Codex and GPT‑5. Their work has made projects like this book both feasible and joyful. It is a privilege to witness—and to contribute during—this period of rapid progress.

I am not a physicist by formal training, but I have been fascinated by physics since school. Leveraging modern AI systems lets me write—and learn from—books I wish already existed. The style and presentation reflect my preferences and experience: figures first, clear prose, and just enough formalism to sharpen intuition.

I would be grateful for feedback on any of my AI‑powered projects. You can find ways to get in touch at https://linktr.ee/dyjh.

Disclaimer. This book is intended solely for educational and illustrative purposes. It has not been formally peer-reviewed or vetted as a textbook, and it may contain inaccuracies or omissions. It is shared in the hope that readers may benefit from a different way of presenting these topics to a broader audience, but it should not be regarded as an authoritative reference. Readers are encouraged to cross-check important results and interpretations with standard physics texts or qualified instructors.

Contents

List of Figures

Part I Conceptual & Mathematical Foundations

Chapter 1 What Is Newtonian Physics?

Welcome to a rigorous yet light-hearted tour of classical mechanics. We will keep the math honest, the plots neat, and the jokes strictly inertial.

1.1 The Scope of Newtonian Mechanics

Classical mechanics applies when speeds are small compared to the speed of light and when quantum effects can be neglected. It is the theory of everyday motion: bicycles and buses, tennis balls and elevators. Prototypical models include point particles (e.g., a marble), rigid bodies (e.g., a book or a door), and idealized continua (e.g., a uniform rod or a beam).

In practice, we choose a model that captures the essence of a question and ignores details that do not matter at the scale of interest. A thrown ball, for example, can be treated as a point mass for its flight, but as a rotating rigid body when spin matters.

A Friendly Analogy

Think of mechanics like a recipe: the ingredients are states (positions and velocities), the instructions are the laws (forces), and the cake is the motion you observe after time evolves.

1.2 Basic Physical Quantities

We use position and time to describe motion, mass to measure inertia, and derived quantities like velocity, acceleration, force, energy, momentum, and angular momentum to analyze behavior. These quantities form a small vocabulary that we will reuse in many contexts.

A First Visualization: Constant Acceleration

As previewed in Figures 1.1 and 1.2, we sketch position and velocity under constant acceleration (think of a car smoothly pressing the accelerator).

1.3 Units, Dimensions, and Orders of Magnitude

We adopt the International System of Units (SI) throughout and routinely check dimensional consistency. Typical scales for everyday objects are shown in Figure 1.4. Developing a sense for scale helps you sanity-check results: if a car ”accelerates” at \(50\,\mathrm {m/s^2}\) for ten seconds, your estimate should raise an eyebrow.

1.4 The Structure of Physical Theories

Mechanics organizes knowledge as: states, laws, and solutions. A state summarizes what matters now (positions and velocities). Laws tell how the state changes (forces produce acceleration). With initial conditions, we compute a solution—a trajectory through time.

In practice, most Newtonian models are initial‑value problems (IVPs): given \((x(0),v(0))\) and a force law \(\bm F(x,v,t)\) that is smooth (or at least Lipschitz) in its arguments, the motion is locally unique. When forces change discontinuously (e.g., impacts or switching friction regimes), we treat those moments as events and restart the IVP with updated conditions.

1.5 Math vs. Physics in This Book

The main text favors intuition and application; short formal notes supply precise definitions and statements when helpful. Use both: intuition to guide, and math to verify. When you feel lost, scan the callouts and figures first, then read the surrounding text for the logic that binds them.

1.6 Exercises

Try a few light, insight‑building tasks.

• 1. Newtonian or not? For each scenario decide if Newtonian mechanics is appropriate and justify in one sentence: a passenger jet at \(900\,\mathrm {km/h}\); an electron in a chip; a satellite in low Earth orbit.

• 2. Back‑of‑the‑envelope. A commuter train at \(50\,\mathrm {m/s}\): estimate \(v/c\) and comment on whether relativistic effects matter. Repeat for a racing car at \(100\,\mathrm {m/s}\).

• 3. Dimensional check. Which is dimensionally consistent for force: (a) \(F=m v\), (b) \(F= m a\), (c) \(F = m a^2\)? Explain using unit symbols.

• 4. Practical: Elevator ride. Time a start–stop elevator segment with your phone. Sketch a rough \(v(t)\) based on sound/feel; mark where acceleration is positive/negative.

• 5. Practical: Everyday scales. List three objects (fruit, backpack, bicycle) and guess their masses and typical timescales for an action (falling, lifting, rolling). Place your guesses on a hand‑drawn log–log mass–time plot similar to Figure 1.4.

1.7 Summary and Review

Quick checklist of what you should now recognize:

• • The Newtonian regime and when it breaks (relativity/quantum).

• • Core quantities: position, velocity, acceleration, force, energy, momentum.

• • Dimensional analysis as a guardrail for sanity checks.

• • The structure state + laws + initial data \(\Rightarrow \) solution.

• • Figures as arguments: slope \(\leftrightarrow \) velocity; area \(\leftrightarrow \) accumulation (preview).

1.8 Where We’re Heading Next

In Chapter 2 we develop the mathematical language behind the pictures: functions and their graphs, vectors, derivatives and integrals, and ordinary differential equations. This vocabulary lets us turn words like “speeding up” into equations we can analyze and, later, solve.

Chapter 2 Mathematical Language of Newtonian Physics

This chapter builds the math we use throughout the book—functions, vectors, derivatives, integrals, and ordinary differential equations (ODEs). We keep it intuitive and visual, with analogies and small figures to anchor each idea. The goal is not symbol-pushing for its own sake, but a compact language that makes motion problems easy to set up and reason about. If you are new to some concepts, do not worry: each one will be motivated by a physical picture first and formalized second.

2.1 Functions and Graphs

Functions are rules that take an input and produce an output. In mechanics, the most common function is a trajectory \(x(t)\): position as a function of time. When the function is smooth, the graph carries two built-in “tools”: the slope of the curve (velocity) and how fast that slope changes (acceleration). Reading a graph is thus a mechanical skill: eyes for slope, eyes for area, eyes for curvature.

As shown in Figure 2.1, a smooth \(x(t)\) lets us talk about slopes (instantaneous velocity) and curvature (how acceleration feels). The corresponding velocity function \(v(t)\) appears in Figure 2.2.

2.2 Vectors in 2D and 3D

Vectors capture magnitude and direction. In two dimensions we write \(\bm r = (x,y)\); in three, \(\bm r=(x,y,z)\). Unit vectors \(\hat {\bm i},\hat {\bm j},\hat {\bm k}\) point along the coordinate axes and let us decompose motion into simple, independent pieces. A good habit: sketch arrows, label their components, and check that magnitudes and directions make sense before computing. The dot product encodes alignment (\(\bm a\cdot \bm b=\|\bm a\|\,\|\bm b\|\cos \theta \)); in 3D the cross product captures oriented area (\(\|\bm a\times \bm b\|=\|\bm a\|\,\|\bm b\|\sin \theta \), right‑hand rule for direction). Checking units after vector operations is as important as in scalar algebra.

Figure 2.4 shows vector addition via the parallelogram rule and highlights the dot product angle.

2.3 Differentiation and Integration of Motion

Differentiation measures instantaneous change; integration measures accumulated effect. If velocity is the rate at which position changes, then position is the accumulated effect of velocity. The two are inverse operations when everything is smooth. When motion is piecewise smooth (e.g., sudden braking), the graph still tells the story: slopes change abruptly, and areas still add up.

Suppose acceleration is a square pulse in time. As previewed in Figure 2.5, the area under \(a(t)\) adds to velocity \(v(t)\). This is the graphical way to remember the fundamental theorem of calculus in a mechanics setting.

2.4 Ordinary Differential Equations in Mechanics

An ordinary differential equation (ODE) relates a function to its derivatives. In mechanics, Newton’s second law \(m\,\ddot x = F(x,\dot x,t)\) is an ODE for motion in one dimension. An initial value problem (IVP) specifies \(x(0)\) and \(\dot x(0)\) and asks for the future.

For constant acceleration \(a\), the IVP solution is \(x(t)=x_0+v_0 t+\tfrac 12 a t^2\) and \(v(t)=v_0+a t\)—consistent with Figure 2.2. When forces depend on position or velocity, the ODE encodes feedback (e.g., drag slows you more when you go faster).

2.5 Numerical Approximation of Motion (Preview)

When we cannot solve an ODE exactly, we approximate it over small time steps. The Euler method updates

\[ \begin {aligned} v_{n+1} &= v_n + a(t_n, x_n, v_n)\,\Delta t,\\ x_{n+1} &= x_n + v_n\,\Delta t. \end {aligned} \]

As illustrated schematically in Figure 2.7, the numerical path follows the exact curve in small forward steps. Smaller steps improve accuracy but cost time; too-large steps can wander off the true solution. Later we will compare simple schemes to better ones and discuss stability (numerical solutions that behave physically even for larger steps).

2.6 Snell’s Law from Least Time (Lifeguard Analogy)

A lifeguard runs faster on sand than in water and wants to reach a swimmer as fast as possible. The fastest path is not a straight line—it bends at the shoreline so that

\[ \frac {\sin \theta _1}{v_1} = \frac {\sin \theta _2}{v_2}, \]

where \(v_1\) and \(v_2\) are speeds on sand and in water, and angles are measured from the normal. Defining a refractive index proportional to \(1/v\) recovers Snell’s law \(n_1\sin \theta _1=n_2\sin \theta _2\).

Before Figure 2.8, picture two straight segments meeting the shoreline at one point; we optimize that meeting point to minimize total time. The derivative step mirrors those in Chapter A.

2.7 Exercises

Short, friendly tasks to practice the math language.

• 1. Read a graph. For \(x(t)=1+0.5 t + 0.1 t^2\), compute \(v(t)\) and \(a(t)\). At \(t=2\), estimate the slope from a sketch and compare with the formula.

• 2. Vector playground. Take \(\bm a=(2,1)\) and \(\bm b=(1,2)\). Compute \(\bm a+\bm b\), \(\bm a\cdot \bm b\), and the angle \(\theta \) between them. Sketch the parallelogram as in Figure 2.4.

• 3. Practical: Two‑axis walk. Walk 20 steps east, then 15 steps north. Mark your start and end on graph paper. Measure the displacement vector and its magnitude.

• 4. Area and change. Suppose \(a(t)=0\) except \(a=2\,\mathrm {m/s^2}\) from \(t=1\) to \(t=3\). If \(v(0)=0\), what is \(v(4)\)? Use area under \(a(t)\) as in Figure 2.5.

• 5. One Euler step. For free fall without air \(\ddot x= -g\), take \(g=9.8\), \(\Delta t=0.2\), \(x_0=0\), \(v_0=0\). Do two Euler updates by hand and compare \(x(0.4)\) with the exact \(\tfrac 12 g t^2\).

2.8 Summary and Review

Checklist of ideas and tools:

• • Functions and their graphs encode motion; slopes and areas have direct physical meaning.

• • Vectors represent magnitude and direction; addition and dot product capture geometry and work.

• • Differentiation measures change; integration accumulates effects.

• • Newtonian motion fits naturally as ODEs with initial data.

• • Simple numerical methods (Euler) approximate motion step by step.

2.9 Where We’re Heading Next

In Chapter 3 we specialize to one‑dimensional kinematics. We will relate \(x(t)\), \(v(t)\), and \(a(t)\) in detail, master constant‑acceleration motion, and interpret slopes/areas on data‑like plots—building on the language you developed here.

Part II One-Dimensional Motion

Chapter 3 Kinematics in One Dimension

Motion along a straight line is the laboratory where we learn to read graphs and translate between position \(x(t)\), velocity \(v(t)\), and acceleration \(a(t)\). We emphasize pictures, consistent sign conventions, and short formulas that say exactly what the pictures say. Our aim is fluency: seeing a graph and immediately knowing what its slope means physically; seeing a formula and picturing its curve.

3.1 Position, Displacement, Velocity, Acceleration

Choose a coordinate axis, a positive direction, and a reference point \(x=0\). Displacement is change in position \(\Delta x = x_2 - x_1\) (a signed quantity), whereas distance traveled is always non‑negative. Average velocity is \(\Delta x/\Delta t\); instantaneous velocity is the slope \(v(t) = \dot x(t)\). Acceleration is the slope of velocity, \(a(t) = \dot v(t) = \ddot x(t)\). A handy mental model is the car dashboard: the trip counter estimates average velocity over an interval; the speedometer reads the instantaneous velocity; how ”hard” you feel pressed into the seat hints at acceleration.

In Figures 3.1 to 3.3 we show a consistent triplet: a simple motion with steadily increasing velocity (constant acceleration).

3.2 Graphical Kinematics

When you face a graph, ask two questions: “What is the slope here?” and “What is the area between these times?” Slope translates to an instantaneous rate (e.g., velocity from \(x(t)\)), and area translates to an accumulated effect (e.g., displacement from \(v(t)\)). Remember that areas under \(v(t)\) are signed: if \(v\) dips below zero, the negative area reduces the net displacement. Two conversions matter most:

• • Slope of \(x(t)\) \(\Rightarrow \) \(v(t)\); slope of \(v(t)\) \(\Rightarrow \) \(a(t)\).

• • Area under \(v(t)\) between \(t_1\) and \(t_2\) \(\Rightarrow \) displacement \(\int _{t_1}^{t_2} v(t)\,\mathrm {d}t\).

As previewed in Figure 3.4, the shaded area under a linearly increasing velocity gives the change in position. Picture laying down thin rectangular tiles under the curve between \(t_1\) and \(t_2\)—their combined area equals the displacement.

3.3 Uniform and Non‑Uniform Motion

Uniform motion has constant velocity (\(a=0\)); non‑uniform motion has \(a\neq 0\). In uniform motion, \(x(t)\) is a straight line; in uniformly accelerated motion, \(v(t)\) is a straight line. For constant acceleration \(a\), the classic relations are

\[ v(t)=v_0+a t,\qquad x(t)=x_0+v_0 t+\tfrac 12 a t^2,\qquad v^2 = v_0^2 + 2 a (x-x_0). \]

Each is a compact way to say what the graphs already show in Figures 3.1 to 3.3.

3.4 Real‑World Examples

Consider a car that accelerates smoothly from rest, cruises, then brakes to a stop. The \(v(t)\) sketch looks like a hill: up, flat, down. The displacement is the total area under that hill. If your daily commute has two such hills (stop‑and‑go), the day’s distance is the sum of the two areas. A moving walkway and your walking speed add linearly in \(v(t)\)—a nice reminder that the graph is a physical story.

3.5 Exercises

Practice translating words to graphs and graphs to formulas.

• 1. Slope and area. From the line \(v(t)=3+0.5 t\), find \(x(t)\) with \(x(0)=0\). Mark the area on a sketch for \(t\in [0,4]\) and compare with your formula.

• 2. Units and signs. A jogger runs west and slows down: \(v\) is negative and \(a\) is positive/negative? Explain.

• 3. Constant acceleration. With \(a=\,\mathrm {const}\), derive \(v^2 = v_0^2 + 2 a (x-x_0)\) from \(v\,\mathrm {d}v = a\,\mathrm {d}x\).

• 4. Practical: Escalator timing. Time a ride up an escalator while walking. Sketch \(v(t)\) for “stand” versus “walk” and compare areas (displacements).

• 5. Practical: Bike start/stop. From rest, pedal to cruising speed, then brake gently. Have a friend film a wheel marker and estimate \(v(t)\) from frame spacing; identify where \(a(t)\) changes sign.

3.6 Summary and Review

A quick checklist of ideas to keep on one page:

• • \(x(t)\), \(v(t)\), \(a(t)\) form a triangle of ideas: slopes go down the chain, areas climb up.

• • Constant acceleration yields straight \(v(t)\) and parabolic \(x(t)\).

• • Displacement is area under \(v(t)\); sign conventions make predictions consistent.

3.7 Where We’re Heading Next

In Chapter 4 we introduce forces and Newton’s laws in one dimension, connect \(F=ma\) to the \(v(t)\) and \(x(t)\) stories from this chapter, and compare analytic solutions with simple numerical updates.

Chapter 4 Dynamics in One Dimension

In Chapter 3 we learned to read motion; now we learn to cause it. Dynamics links pushes and pulls (forces) to changes in motion (accelerations) via Newton’s laws. In one dimension the story is crisp: pick an axis, assign signs, add forces along that axis, then use \(F_\text {net}=m a\) to connect to \(v(t)\) and \(x(t)\).

4.1 Newton’s Laws in 1D

Read the laws with your sign convention in mind and keep everything along your chosen axis:

• • First law (inertia): if \(F_\text {net}=0\), then \(v\) is constant (including the case \(v=0\)).

• • Second law: \(F_\text {net}=m a\) with \(a=\dot v=\ddot x\); signs follow your axis choice.

• • Third law: forces between interacting bodies come in equal‑and‑opposite pairs. These act on different bodies, so they never cancel within a single free‑body diagram.

We keep track of signs with a consistent axis. If “to the right” is positive, then a leftward push is a negative force. A simple mental check: if the net force arrow points right, acceleration should be positive.

4.2 Mass and Common Forces

Mass measures inertia (resistance to changes in motion). Typical 1D force components include weight \(W=mg\) (downward), normal forces from surfaces, friction (static/kinetic), and simple driving forces. In 1D problems we often project a multi‑D situation onto an axis along the motion. A compact way to organize thinking is the free‑body diagram: isolate the object and draw all forces with signs according to your axis. A minimal example appears in Figure 4.1.

4.3 Basic 1D Force Models

Three simple models already cover many situations; each makes a different prediction for how velocity should evolve:

• • Constant force \(F=F_0\) gives \(a=F_0/m\) and reproduces the constant‑acceleration results from Chapter 3.

• • Linear drag \(F= - c v\) opposes motion; with a constant drive it produces an exponential approach to a terminal speed \(v_\mathrm {term}=F_0/c\) with time constant \(\tau =m/c\).

• • Hooke spring \(F= -k (x-x_\ast )\) pulls toward an equilibrium point \(x_\ast \) (undamped motion is simple harmonic; see Chapter 13).

4.4 From Force to Motion

Once \(F_\text {net}(x,v,t)\) is specified, Newton’s law \(m\,\ddot x = F_\text {net}\) is an ODE. Sometimes we can solve it exactly; often we can only solve or visualize it approximately. As a case study, Figure 4.2 shows a constant drive opposed by linear drag: the velocity rises quickly at first and then flattens toward a terminal value.

4.5 Numerical Integration in 1D

For \(m\,\ddot x = F(x,\dot x,t)\), one forward Euler step of size \(\Delta t\) reads

\[ \begin {aligned} v_{n+1} &= v_n + \tfrac {1}{m} F(x_n, v_n, t_n)\,\Delta t,\\ x_{n+1} &= x_n + v_n\,\Delta t. \end {aligned} \]

Smaller steps improve accuracy; dimensionless groupings (e.g., \(c\,\Delta t/m\) in linear drag) help judge stability. A simple pseudocode template is: choose \(x_0,v_0\), then loop updates for \(n=0,1,2,\dots \) with your force model.

4.6 Everyday 1D Dynamics

Many familiar motions are effectively one‑dimensional after projection onto an axis. Start with a sentence describing the situation, then sketch the forces and predict \(v(t)\) qualitatively before computing.

• • Car on a straight road in light traffic: engine push roughly constant at first, air drag increases with speed; \(v(t)\) rises toward a plateau (terminal speed).

• • Sliding box on a shallow incline: component of weight competes with kinetic friction; if the component wins, \(v(t)\) grows linearly; if they match, motion steadies.

• • Braking to a stop: roughly constant braking force gives a constant negative acceleration; \(v(t)\) is a straight line sloping down to zero and \(x(t)\) a concave‑down parabola.

4.7 Exercises

Practice going from a free‑body diagram to equations of motion, and from force models to qualitative \(v(t)\) and \(x(t)\) shapes.

• 1. Sign conventions. A cart is pushed left with \(3\,\mathrm {N}\) while friction exerts \(1\,\mathrm {N}\) to the right. If right is positive, write \(F_\text {net}\) and the acceleration for \(m=1\,\mathrm {kg}\).

• 2. Free‑body sketch. Draw a free‑body diagram for a block on a rough horizontal surface pushed to the right; label \(N\), \(mg\), \(F\), and kinetic friction \(f\).

• 3. Linear drag. For \(m\dot v = F_0 - c v\), show that \(v(t)=\tfrac {F_0}{c}(1-e^{-ct/m})\) for \(v(0)=0\).

• 4. Practical: Pull test. Gently pull a small object across a table with a rubber band. Note the start (overcoming static friction) versus the sliding (kinetic friction). Sketch the implied \(F\) versus \(t\).

• 5. One Euler step. With \(m\ddot x = F_0 - c\dot x\), take \(m=1\), \(F_0=2\), \(c=0.5\), \(\Delta t=0.2\), \(v_0=x_0=0\). Compute \(v_{1},x_{1},v_{2},x_{2}\).

4.8 Summary and Review

A quick checklist before moving on:

• • Newton’s three laws specialize cleanly in 1D with signed forces along an axis.

• • Common models: constant force, linear drag, and springs already describe many systems.

• • From \(F_\text {net}\) to motion: solve \(m\ddot x=F\) analytically when possible, numerically when needed.

4.9 Where We’re Heading Next

In Chapter 5 we lift the same ideas into the plane and space, building free‑body diagrams and trajectories that use the vector language introduced in Chapter 2.

Part III Two- and Three-Dimensional Motion

Chapter 5 Vectors and Kinematics in 2D/3D

In one dimension, arrows could hide; in two and three dimensions, arrows are the language. This chapter develops vector kinematics: we describe position with \(\bm r(t)\), velocity with \(\bm v(t)=\dot {\bm r}(t)\), and acceleration with \(\bm a(t)=\dot {\bm v}(t)\). We keep the narrative visual—sketches first, formulas second. By the end, you should be able to sketch a path, draw tangents and normals confidently, and move between components and geometry with ease.

5.1 Vector Representation of Motion

We write \(\bm r(t)=(x(t),y(t))\) in the plane or \((x(t),y(t),z(t))\) in space. Unit vectors \(\hat {\bm i},\hat {\bm j},\hat {\bm k}\) point along the coordinate axes. Differentiation and integration act componentwise, so everything you learned in Chapter 2 carries over with arrows on top.

Two immediate “vector calculus” facts are worth keeping on a sticky note:

• • Speed is the magnitude of velocity: \(v=\|\bm v\|=\|\dot {\bm r}\|\). The unit tangent is \(\hat {\bm t}=\bm v/\|\bm v\|\).

• • Arc length \(s\) satisfies \(\tfrac {\mathrm d s}{\mathrm d t}=v\). Using \(\tfrac {\mathrm d}{\mathrm d t}=\tfrac {\mathrm d s}{\mathrm d t}\tfrac {\mathrm d}{\mathrm d s}=v\tfrac {\mathrm d}{\mathrm d s}\) makes tangent/normal formulas compact.

As shown in Figure 5.1, a parametric curve with a few velocity arrows already tells a story: the red arrows hug the path and tilt as the slope changes. The length of each arrow represents speed; their directions give the tangent.

Before we state the reflection rule, it helps to see how any incoming direction at a surface splits into a component parallel to the surface and one along the normal. The geometric split in Figure 5.2 is the picture we will reuse when we derive the vector reflection law in Figure 5.5.

5.2 Components, Projection, and Geometry

Vectors project naturally onto axes. The dot product \(\bm a\cdot \bm b=\|\bm a\|\,\|\bm b\|\cos \theta \) measures alignment; in mechanics it yields work: \(W=\int \bm F\cdot \mathrm {d}\bm r\). The cross product (in 3D) encodes perpendicularity and areas; it will reappear for angular momentum and torque.

To make components concrete, Figure 5.4 shows a vector with its \(x\)- and \(y\)-projections. Reading this picture is a skill: eyes move horizontally and vertically to match lengths with components before any computation.

5.3 Reflection as a Vector Rule

Mirrors are a perfect vector application. At a flat surface with unit normal \(\hat {\bm n}\), an incoming unit direction \(\hat {\bm s}_{\!\text {in}}\) reflects to

\[ \boxed {\;\hat {\bm s}_{\!\text {out}} = \hat {\bm s}_{\!\text {in}} - 2\,(\hat {\bm s}_{\!\text {in}}\cdot \hat {\bm n})\,\hat {\bm n}\;} \]

This formula says “flip the normal component.” It encodes the law of reflection (angle in equals angle out) without trigonometry.

Before Figure 5.5, note how the equal angles are measured from the normal.

5.4 Straight-Line and Curvilinear Motion

For straight motion at constant direction, \(\bm v\) is constant in direction and \(\bm a\) is parallel (speeding up) or antiparallel (slowing down). Curvilinear motion bends the path: \(\bm v\) points along the tangent; \(\bm a\) generally has a tangent part (changing speed) and a normal part (turning). At constant speed, acceleration is purely normal and points inward.

As illustrated in Figure 5.6, the normal component indicates how sharply we turn. Quantitatively,

\[ \bm a = \underbrace {\dot v}_{a_\mathrm t}\,\hat {\bm t} \;\; + \;\; \underbrace {v^2\,\kappa }_{a_\mathrm n}\,\hat {\bm n},\qquad \kappa =\Big \|\frac {\mathrm d\hat {\bm t}}{\mathrm d s}\Big \|\,. \]

Here \(a_\mathrm t=\dot v\) changes speed and \(a_\mathrm n=v^2\kappa \) bends the path. For a circle of radius \(R\), the curvature \(\kappa =1/R\) so \(a_\mathrm n=v^2/R\)—the centripetal acceleration used in Chapter 6.

5.5 Examples

Let’s connect the pictures to daily motion. Each item hints at \(\bm r(t)\), \(\bm v(t)\), and \(\bm a(t)\) without equations first, encouraging a sketch‑then‑compute workflow:

• • A pedestrian turns a corner at nearly constant speed—\(\|\bm v\|\) is steady but \(\bm a\) points toward the center of the turn.

• • A drone flies north while rising: \(x(t)\) and \(y(t)\) evolve independently; the path is a tilted line in 3D.

• • A robot traces a parametric curve: \(\bm r(t)=(t,\sin t)\); the tangent arrows flip as the sine wave crests and troughs.

5.6 Exercises

Practice translating between pictures, components, and geometric statements.

• 1. Components. For \(\bm r(t)=(2t, t^2)\), compute \(\bm v\) and \(\bm a\). At \(t=1\), sketch the tangent.

• 2. Dot product. For \(\bm a=(2,1)\) and \(\bm b=(1,3)\), compute \(\bm a\cdot \bm b\) and the angle between them.

• 3. Practical: Corner turn. Walk at steady speed and turn a corner; note the inward “pull”. Sketch \(\bm v\) and the inward normal direction.

• 4. Parametric reading. For \(\bm r(t)=(t,\sin t)\), mark where \(\bm v\) is horizontal or vertical.

• 5. Area/work preview. A constant force \(\bm F=(1,0)\) moves a point along \(\bm r(t)=(t, t^2)\). Compute \(\int _0^1 \bm F\cdot \mathrm {d}\bm r\).

5.7 Summary and Review

A quick checklist before moving on:

• • Vector kinematics: \(\bm r\), \(\bm v=\dot {\bm r}\), \(\bm a=\dot {\bm v}\); components evolve independently.

• • Tangent/normal picture: \(\bm v\) tangent; \(\bm a\) can turn you even at constant speed.

• • Dot product measures alignment and powers work; projections simplify problems.

5.8 Where We’re Heading Next

In Chapter 6 we apply vector kinematics to parabolic trajectories without air and to uniform/non‑uniform circular motion with centripetal acceleration and geometric decompositions. The tangent/normal picture from Figure 5.6 will become your compass.

Chapter 6 Projectiles and Circular Motion

This chapter applies vector kinematics to two iconic trajectories: parabolas (projectiles without air) and circles (uniform and non‑uniform circular motion). We connect algebraic formulas to clean geometric pictures and keep the tangent/normal viewpoint from Chapter 5 front and center. Our strategy is Galileo’s: split motion into independent directions, solve the easy parts, and then recombine.

6.1 Projectile Motion Without Air

Resolve the initial velocity \(\bm v_0\) into components: \(v_{0x}=v_0\cos \alpha \), \(v_{0y}=v_0\sin \alpha \). With \(x\) horizontal and \(y\) vertical, the equations are

\[ \begin {aligned} x(t)&= x_0 + v_{0x}\, t,\\ y(t)&= y_0 + v_{0y}\, t - \tfrac 12 g t^2,\qquad g>0. \end {aligned} \]

Eliminating \(t\) gives a parabola. The time of flight (for \(y_0=0\)) is \(T=\tfrac {2 v_0\sin \alpha }{g}\); the range is \(R=\tfrac {v_0^2\sin 2\alpha }{g}\). Two independent “clocks” are running: a steady horizontal clock \(x(t)=x_0+v_{0x}t\) and a vertical clock slowed by gravity \(y(t)=y_0+v_{0y}t-\tfrac 12 g t^2\).

For level launch and landing (\(y_0=0\)), the range is maximized at \(\alpha =45^\circ \) (no air). If launch and landing heights differ, the optimal angle shifts; independence of horizontal and vertical motion holds only when air resistance is neglected and gravity is uniform.

As illustrated in Figure 6.1, different launch angles trace a family of parabolas; complementary angles share the same range.

6.2 Uniform Circular Motion

Parametrize a circle of radius \(R\) by \(\bm r(t)=\big (R\cos (\omega t),\, R\sin (\omega t)\big )\). Then

\[ \bm v(t)=\dot {\bm r}(t)=\omega R\big (-\sin \omega t,\cos \omega t\big ),\qquad \bm a(t)=\dot {\bm v}(t)=-\omega ^2 R\big (\cos \omega t,\sin \omega t\big )=-\omega ^2\bm r(t). \]

The acceleration points inward (centripetal) with magnitude \(a=\omega ^2 R = v^2/R\). Use radians for \(\omega \) and any calculus with angles. At constant speed, the compass (direction) turns even while the speedometer reading stays fixed—acceleration measures turning, not only speeding up.

Before Figure 6.3, note that velocity is tangent and acceleration is inward at every point. A tight rope swing or a car on a roundabout are everyday versions of the same geometry: something must pull inward to bend the path.

6.3 Non‑Uniform Circular Motion

When speed changes along a curve, acceleration splits into tangent and normal parts,

\[ \bm a = a_\mathrm {t}\,\hat {\bm t} + \frac {v^2}{R}\,\hat {\bm n},\qquad a_\mathrm {t}=\dot v, \]

where \(\hat {\bm t}\) is the unit tangent and \(\hat {\bm n}\) points inward. As previewed in Figure 6.5, pressing the accelerator while turning adds a forward (tangent) component to the always‑inward normal part.

6.4 Exercises

Practice reading vector motion in the plane and circle.

• 1. Parabola checks. For \(v_0=20\,\mathrm {m/s}\) and \(\alpha =30^\circ \), compute \(T\), \(R\), and the maximum height; compare with the curve in Figure 6.1.

• 2. Complementary angles. Show that \(\sin 2\alpha \) is unchanged by \(\alpha \mapsto 90^\circ -\alpha \); interpret the equal‑range property.

• 3. Uniform circle. For \(R=2\,\mathrm {m}\) and \(\omega =1.5\,\mathrm {rad/s}\), compute \(v\) and \(a\); mark the vectors on a sketch like Figure 6.3.

• 4. Practical: Hose arc. Observe a water stream from a garden hose (low speed, calm day). Sketch the curve and mark where the vertical component of velocity changes sign.

• 5. Practical: Cornering. On a bike at steady speed, ride a gentle arc; identify the inward “pull” as the normal component of \(\bm a\).

6.5 Summary and Review

A quick checklist before moving on:

• • Projectiles: split into components, solve, and recombine; parabolas and range emerge cleanly.

• • Uniform circles: \(\bm v\) is tangent; \(\bm a\) is inward with magnitude \(v^2/R\).

• • Non‑uniform turning: decompose acceleration into tangent and normal parts.

6.6 Where We’re Heading Next

In Chapter 7 we blend vector kinematics with force models to analyze motion in two and three dimensions—free‑body diagrams in 2D, constraints like tension and normal forces, and energy methods as an alternative lens.

Part IV Forces, Work, and Energy

Chapter 7 Forces in Multiple Dimensions

We now combine Newton’s laws with vectors. In two dimensions the recipe is clear: isolate the object, draw a free‑body diagram (FBD), write the vector balance \(\sum \bm F = m\,\bm a\), and then resolve into axes that make the geometry simple.

7.1 Newton’s Laws in Vector Form

We keep the vector law in sight and then read its “shadows” on the axes we choose. The second law reads

\[ \sum \bm F = m\,\bm a\quad (\text {vectors}). \]

Pick axes and project:

\[ \sum F_x = m a_x,\qquad \sum F_y = m a_y. \]

The art is choosing axes aligned with the geometry—on an incline, take one axis along the surface (Figure 7.1). With a good choice, one component often vanishes (e.g., no acceleration into a rigid surface).

7.2 Free‑Body Diagrams in 2D

An FBD strips the world down to just the object and the forces on it. Draw arrows from the object’s center pointing in the directions of the forces, label magnitudes, and note your axis choice. A neat diagram saves algebra. Figure 7.1 shows a block on an incline with the usual suspects.

7.3 Resolving Components

Choose axes along the surface \(\hat {\bm t}\) (take the downhill direction as positive) and perpendicular to it \(\hat {\bm n}\) (pointing outward from the surface). In these axes, the weight \(W\) splits cleanly: a component of magnitude \(W\sin \theta \) lies along \(+\hat {\bm t}\) (downhill) and a component of magnitude \(W\cos \theta \) points into the surface (opposite \(+\hat {\bm n}\)). The angle \(\theta \) is the incline angle. The faint gray arrows in Figure 7.2 indicate the chosen axes.

7.4 Friction and Drag

Static friction adjusts to whatever is needed to prevent relative motion up to a limit:

\[ |f_\mathrm {s}|\le \mu _\mathrm {s} N,\quad \text {with}\; f_\mathrm {s}=\mu _\mathrm {s}N\; \text {at incipient slip.} \]

Once sliding, kinetic friction has nearly fixed magnitude \(f_\mathrm {k}=\mu _\mathrm {k} N\) opposing motion. Linear drag models air/fluid resistance at low speeds as \(\bm F_\mathrm {drag}=-c\,\bm v\); direction opposes \(\bm v\) and \(c\) lumps density/shape factors (see Chapters 14 and 15). Start every problem by deciding: static (sticking), kinetic (sliding), or a low‑speed drag model.

Constraint forces round‑out the toolkit: the normal force enforces no‑penetration, and ideal tension acts along a string/rope. In the common idealization of massless strings and frictionless pulleys, the tension has the same magnitude throughout a connected segment.

7.5 From Vector Law to Equations

Goal: turn a picture into two clean equations you can solve. The approach is always the same: decide on axes, write the vector law, project onto axes, and use any constraint equations (like the no–lift‑off condition) to eliminate unknown forces.

A reliable four‑step recipe (use it every time):

• 1. Draw a neat FBD and choose axes that match the geometry (e.g., along and normal to a surface).

• 2. Write the vector law \(\sum \bm F = m\,\bm a\) with a sentence about the positive directions.

• 3. Project onto axes to get \(\sum F_t = m a_t\) and \(\sum F_n = m a_n\) (or \(x/y\) in a standard frame).

• 4. Use constraints (e.g., \(a_n=0\) with no lift‑off, \(f=\mu _k N\) when sliding) to eliminate unknowns, then solve for the desired quantity (\(a_t\), \(v(t)\), or \(x(t)\)).

For the incline with kinetic friction (\(f=\mu _k N\)) and angle \(\theta \) measured from horizontal, choosing \(\hat {\bm t}\) downhill and \(\hat {\bm n}\) outward gives

\[ \begin {aligned} \hat {\bm t}:\quad & m a_{\!t} = W\sin \theta - f,\\ \hat {\bm n}:\quad & 0 = N - W\cos \theta \quad (\text {no lift‑off}). \end {aligned} \]

Eliminate \(N\) using the normal equation; then \(a_{\!t} = g \sin \theta - \mu _k g \cos \theta \). From here, constant \(a_{\!t}\) lets you reuse the kinematics of Chapter 3 for \(v(t)\) and \(x(t)\) along the slope.

7.6 Exercises

Quick practice with FBDs and components.

• 1. Axes choice. For a block on an incline, draw \(\hat {\bm t}\), \(\hat {\bm n}\), and all forces. Label signs.

• 2. Resolve weight. Show that the downhill component is \(W\sin \theta \) and the into‑surface component is \(W\cos \theta \).

• 3. Static or kinetic? A block rests on an incline and you increase the angle slowly. Describe what friction does before and after it starts sliding.

• 4. Practical: Door push. Push a door near the hinge and then near the handle with the same effort. Describe the difference in motion; sketch the force directions (torque preview).

• 5. Drag direction. For a puck moving east over a thin oil film, draw \(\bm v\) and \(\bm F_\mathrm {drag}\).

7.7 Summary and Review

A quick checklist before moving on:

• • Vector law \(\sum \bm F = m\,\bm a\) becomes two component equations once axes are chosen.

• • FBDs isolate forces; axes aligned with geometry simplify algebra.

• • Friction and drag oppose motion; normal forces constrain; tension pulls along strings.

7.8 Where We’re Heading Next

In Chapter 8 we’ll solve problems by tracking energy transfer—often shorter and more insightful than summing forces step by step.

Chapter 8 Work, Kinetic Energy, and Power

Energy is a second viewpoint on motion that often turns multi‑step force calculations into one line. When forces are tricky but you can track start and finish, energy methods shine.

8.1 Work as Force Along a Displacement

For a displacement from \(\bm r_1\) to \(\bm r_2\) under force \(\bm F(\bm r)\), the work done by the force is

\[ W = \int _{\bm r_1}^{\bm r_2} \bm F\cdot \mathrm {d}\bm r = \int _{s_1}^{s_2} F_t(s)\,\mathrm {d}s, \]

where \(F_t\) is the tangential component along the path coordinate \(s\). Geometrically, the dot captures “how much the force points along the motion.” For a constant tangential force over a straight path of length \(\Delta s\), this reduces to \(W = F_t\,\Delta s\).

Sign and direction matter: only the component of force along the displacement contributes. Forces perpendicular to the motion (e.g., an ideal normal force that enforces contact without slip) do no work because \(\bm F\cdot \mathrm d\bm r=0\).

As illustrated in Figure 8.1, work is the area under the force–displacement curve \(F_t\) vs. \(s\) between the start and end points. When \(\bm F\) is conservative, this area depends only on the endpoints and defines a potential energy difference \(\Delta U\) (Chapter 9).

8.2 Work–Kinetic Energy Theorem

The net work done on a particle changes its kinetic energy:

\[ W_{\text {net}} = \Delta K = K_2 - K_1 = \tfrac 12 m v_2^2 - \tfrac 12 m v_1^2. \]

Sketch‑first reasoning: if the net force does positive work, the speed increases; negative work reduces speed. This turns force stories into speed stories without tracking every instant.

This statement holds regardless of whether forces are conservative or not; “net work” collects contributions from all forces. When forces are conservative, we can package their work into a potential \(U\) and trade \(W_{\text {net}}\) for changes in \(K+U\) (Chapter 9).

To visualize the nonlinearity of kinetic energy with speed, Figure 8.3 plots \(K(v)\) and marks two speeds.

8.3 Power

Power is the instantaneous rate at which work is done: \(P=\dfrac {\mathrm {d}W}{\mathrm {d}t}=\bm F\cdot \bm v\). A high power delivery changes kinetic energy quickly; a low power delivery takes time. Constant power means equal areas in \(P(t)\) give equal chunks of work. In rotational problems (Chapter 11), the analogous relation is \(P=\tau \,\omega \).

As a simple illustration, Figure 8.4 shows a short interval of nearly constant power and the work accumulated as the area under \(P(t)\).

8.4 Worked Example: Same Work Two Ways

Consider a push of constant tangential force \(F_t=100\,\mathrm {N}\) over a straight \(\Delta s=3\,\mathrm {m}\). The work from the force–distance view is

\[ W = F_t\,\Delta s = 100 \times 3 = 300\,\mathrm {J}. \]

If the motion happens at a steady speed \(v=1.5\,\mathrm {m/s}\) for \(\Delta t=2\,\mathrm {s}\), the power is constant \(P=F_t v = 150\,\mathrm {W}\) and the work from the power–time view is

\[ W = \int _{t_1}^{t_2} P\,\mathrm {d}t = P\,\Delta t = 150 \times 2 = 300\,\mathrm {J}. \]

Both paths agree, as shown in Figure 8.5.

8.5 Everyday Energy Examples

Energy reasoning excels when only the start and finish matter. Three favorites:

• • Stair climb: Work \(\approx m g h\); power is higher when you sprint.

• • Bike up a hill: Same vertical gain \(h\) gives the same \(m g h\) of work, regardless of path; timing changes power.

• • Car speeding up: Engine power raises \(K\); doubling speed takes four times the energy.

8.6 Exercises

Quick practice connecting forces, work, energy, and power.

• 1. Units. Check the SI units of work, energy, and power.

• 2. Area and work. For the piecewise force in Figure 8.1, estimate the shaded area (work) by counting grid squares.

• 3. Kinetic energy growth. Using Figure 8.3, explain why doubling speed takes four times the energy.

• 4. Practical: Stair climb. Estimate the work to climb one floor; compare power for an easy pace vs. a sprint.

• 5. Power area. On Figure 8.4, compute the shaded work between \(t_1\) and \(t_2\).

8.7 Summary and Review

A quick checklist before moving on:

• • Work is the integral of tangential force along a displacement (area under \(F_t\) vs. \(s\)).

• • Net work changes kinetic energy: \(\Delta K = W_{\text {net}}\).

• • Power is the rate of doing work: areas under \(P(t)\) accumulate work (\(P=\tau \,\omega \) in rotation; Chapter 11).

• • Perpendicular forces do no work; the sign of work follows alignment with motion.

8.8 Where We’re Heading Next

In Chapter 9 we combine work and kinetic energy with potential energy and the conservation law \(K+U=\text {const}\) for conservative forces; then we use energy diagrams to read turning points.

Chapter 9 Potential Energy and Conservation of Energy

For certain forces, we can store their “influence” in a scalar function \(U\) called potential energy. When only such conservative forces act, the total mechanical energy \(E=K+U\) stays constant and the motion traces out the landscape of \(U\).

9.1 Conservative Forces and Potentials

In one dimension, a force is conservative if it can be written as \(F_x(x) = -\,\mathrm {d}U/\mathrm {d}x\) for some \(U(x)\). In multiple dimensions, conservative means \(\bm F=-\nabla U\); equivalently, the work around any closed loop is zero. In smooth, simply connected regions this is the same as \(\nabla \times \bm F=\bm 0\) (see Chapter C). Only differences in \(U\) matter; adding a constant does not change physics.

Two workhorses:

• • Near‑Earth gravity: \(F_y=-mg\) gives \(U(y)=m g y\) up to an arbitrary constant.

• • Hooke spring: \(F_x=-k x\) gives \(U(x)=\tfrac 12 k x^2\).

As previewed in Figure 9.1, one is linear, the other parabolic.

9.2 Energy Conservation and Turning Points

For conservative forces, \(E=K+U\) stays constant. On an energy diagram \(U(x)\), draw a horizontal line at \(E\); the motion is allowed where \(E\ge U(x)\), with speed \(v(x)=\sqrt {\tfrac {2}{m}(E-U(x))}\). Points where \(E=U\) are turning points (\(v=0\)).

The slope of \(U\) controls the direction of acceleration: \(a(x)=\tfrac {F_x}{m}= -\tfrac {1}{m}\,\dfrac {\mathrm d U}{\mathrm d x}\). Where \(U\) decreases with \(x\) (downhill), \(a>0\); where it increases, \(a<0\). Near a stable equilibrium \(x_0\) (a local minimum of \(U\)), a Taylor expansion \(U(x)\approx U(x_0)+\tfrac 12 k(x-x_0)^2\) yields simple harmonic motion with \(k=U''(x_0)\) and \(\omega =\sqrt {k/m}\) (see Chapter 13).

As shown in Figure 9.2, a spring potential with total energy \(E\) yields symmetric turning points and the shaded gap between \(E\) and \(U\) visually represents \(K\). A gravitational version for a vertical toss appears in Figure 9.3.

9.3 Non‑Conservative Forces (Brief)

When friction or drag act, mechanical energy changes according to the work by non‑conservative forces: \(\Delta E = W_{\!\mathrm {nc}}\). Speeds are then found from \(K_2=K_1 + W_{\!\mathrm {nc}} - \Delta U\). In practice: start with \(E_1=K_1+U_1\), add the signed work by non‑conservative forces to get \(E_2\), and solve \(K_2=E_2-U_2\) for the speed.

9.4 Examples

Before computing, read the energy picture and predict: where can it move (allowed region), where will it turn, and where is it fastest (largest \(K\))?

• • Mass–spring: With total energy \(E\), the speed at \(x\) follows \(v(x)=\sqrt {\tfrac {2}{m}(E-\tfrac 12 k x^2)}\).

• • Ball toss (no air): \(U=m g y\), so \(v(y)=\sqrt {v_0^2 - 2 g (y-y_0)}\); the peak where \(v=0\) solves \(E=U\).

• • Slide with friction: Loss \(W_{\!\mathrm {nc}}=-f \Delta s\) reduces the total; compare speeds at equal heights with/without friction.

9.5 Exercises

• 1. Units. What are the SI units of \(U\), \(K\), and \(E\)? Verify consistency for \(U=\tfrac 12 k x^2\).

• 2. Turning points. For \(U=\tfrac 12 k x^2\) and total \(E\), find the turning points and the maximum speed.

• 3. Gravitational climb. A mass rises from \(y_0\) to \(y_1>y_0\) without friction. Find \(v_1\) from energy.

• 4. Practical: Springs. Compress a small spring by measured \(x\); estimate stored energy \(\tfrac 12 k x^2\) using a catalog value of \(k\).

• 5. Friction loss. With kinetic friction \(f=\mu _k N\), estimate the percentage of initial kinetic energy lost over \(\Delta s\).

9.6 Summary and Review

A brief checklist of the main ideas from this chapter:

• • Conservative forces admit a potential \(U\) with \(F=-\nabla U\) (or \(F_x=-\,\mathrm {d}U/\mathrm {d}x\) in 1D).

• • Energy conservation: \(E=K+U\) constant; turning points where \(E=U\).

• • Energy diagrams allow quick, qualitative predictions of speed and accessible regions.

9.7 Where We’re Heading Next

In Chapter 10, we model systems of particles, center of mass, and momentum conservation—powerful ideas for collisions and collective motion.

Part V Systems, Momentum, and Rotation

Chapter 10 Systems of Particles and Momentum

Real objects are made of many parts. Treating them as systems leads to simple, powerful summaries: center of mass (COM) tracks collective position; momentum tracks collective motion and is conserved for isolated systems.

10.1 Center of Mass

Intuition first: the COM is the balancing point. If you could support the system at a single location without it tipping, that location would be \(\bm R\). For point masses in 2D,

\[ \bm R=\frac {\sum _i m_i \bm r_i}{\sum _i m_i},\qquad M=\sum _i m_i. \]

The formula says “average the positions, but weight by masses.” For two masses on a line, the COM lies on the segment joining them and sits closer to the heavier mass in the ratio of masses.

Two key dynamical facts follow directly (in any inertial frame): the total momentum equals total mass times COM velocity, and external forces move the COM like a single particle of mass \(M\):

\[ \bm P = \sum _i m_i \dot {\bm r}_i = M\,\dot {\bm R},\qquad M\,\ddot {\bm R}=\sum \bm F_\text {ext}. \]

Internal forces cancel in pairs (Newton’s third law), so they do not affect \(\ddot {\bm R}\). This is why momentum conservation is so powerful for isolated systems: if \(\sum \bm F_\text {ext}=\bm 0\), then \(\dot {\bm R}\) is constant and \(\bm P\) is conserved.

As previewed in Figure 10.1, increasing \(m_2\) pulls the COM toward \(m_2\); equal masses place the COM at the midpoint.

10.2 Momentum and Impulse

Momentum adds across parts: \(\sum \bm p\) is the system’s “motion budget.” External forces change it; internal forces cancel in pairs. Over a short time with a large force (a shove), it is easiest to think in terms of impulse

\[ \bm J=\int _{t_1}^{t_2} \bm F\,\mathrm {d}t,\qquad \sum \bm p_{\,2}=\sum \bm p_{\,1}+\bm J. \]

As Figure 10.2 shows, the shaded area under \(F(t)\) is the impulse. A narrow, tall spike and a short, wide push can deliver the same area and therefore the same change in momentum.

Only external impulse changes total momentum: \(\Delta \bm P=\bm J_\text {ext}\). Equivalently for the COM,

\[ \dot {\bm R}_2=\dot {\bm R}_1+\frac {\bm J_\text {ext}}{M}. \]

10.3 Collisions

During a brief, isolated impact, external forces are negligible compared to the internal forces between the bodies, so total momentum is conserved:

\[ m_1 u_1+m_2 u_2=m_1 v_1+m_2 v_2\quad (\text {1D}). \]

Elastic collisions also conserve kinetic energy; inelastic ones do not (some energy becomes heat/deformation). Conservation of momentum predicts directions and relative magnitudes even when details of the contact are complicated.

In 1D, the coefficient of restitution \(e\) summarizes how bouncy the collision is via relative speeds:

\[ e=\frac {v_2-v_1}{u_1-u_2},\qquad 0\le e\le 1. \]

Here \(e=1\) is elastic (kinetic energy conserved), and \(e=0\) is perfectly inelastic (stick together). Momentum plus \(e\) determines \(v_1,v_2\); use energy conservation only for \(e=1\) (see Chapter 8).

Before Figure 10.4, make a sketch with arrows: heavier objects tend to reverse lighter ones; equal masses exchange speeds in a head‑on elastic collision.

10.4 Exercises

Practice reading COM, impulse, and collision statements.

• 1. COM of two masses. Find the COM for \(m_1\) at \(x=0\) and \(m_2\) at \(x=L\); where is it when \(m_2=2m_1\)?

• 2. Impulse area. Estimate the area in Figure 10.2 and relate it to a change in momentum.

• 3. Collision direction. Two carts collide head‑on with \(m_2>m_1\); argue which way the COM moves during impact.

10.5 Summary and Review

A quick checklist before moving on:

• • Center of mass summarizes where mass “acts”; it’s a weighted average of positions.

• • Momentum adds for subsystems and changes only by external impulse (\(\Delta \bm P=\bm J_\text {ext}\)).

• • Short impacts: use conservation of momentum for quick, robust predictions.

10.6 Where We’re Heading Next

In Chapter 11, we introduce rotation: angular kinematics, torque, and rotational energy.

Chapter 11 Rotation of Rigid Bodies

Rotation extends motion to angles and torques. We mirror linear ideas with angular ones and preview energy and momentum in rotational form.

11.1 Angular Kinematics

Angular position \(\theta \) measures “how far turned,” \(\omega =\dot \theta \) how fast it turns, and \(\alpha =\dot \omega \) how quickly the turn rate changes. These mirror linear kinematics: angle plays the role of position, angular velocity that of speed, and angular acceleration that of linear acceleration. At constant \(\alpha \),

\[ \omega (t)=\omega _0+\alpha t,\qquad \theta (t)=\theta _0+\omega _0 t + \tfrac 12 \alpha t^2. \]

Reading a plot of \(\theta (t)\) or \(\omega (t)\) is like reading \(x(t)\) and \(v(t)\) from Chapter 3: slopes give rates; areas give accumulated angle.

Use radians for calculus with angles. For motion along a circle of radius \(R\), arc length and angle relate by \(s=R\,\theta \), so \(v=\dot s=R\,\omega \) and \(a_\mathrm {t}=R\,\alpha \); the inward (normal) part remains \(a_\mathrm {n}=v^2/R\) (Chapter 6).

11.2 Torque and Lever Arm

A force \(\bm F\) at position vector \(\bm r\) produces torque \(\bm \tau =\bm r\times \bm F\) about the origin. In the plane, \(\tau =r F\sin \phi \) (scalar out of page), where \(\phi \) is the angle between \(\bm r\) and \(\bm F\). Long lever arms amplify torque; perpendicular pushes are most effective. Right‑hand rule: curl fingers from \(\bm r\) toward \(\bm F\)—the thumb points along \(\bm \tau \).

In rotational dynamics, torque is to angular acceleration what force is to linear acceleration. With moment of inertia \(I\) (a measure of rotational inertia), the parallel of Newton’s second law is

\[ \tau _\text {net}= I\,\alpha \qquad (\text {about a fixed axis; preview}). \]

The rotational kinetic energy analogue is \(K_\text {rot}=\tfrac 12 I \omega ^2\).

Work and power have clean rotational forms (compare Chapter 8): an infinitesimal angular displacement \(\mathrm d\theta \) under torque \(\tau \) does work \(\mathrm dW=\tau \,\mathrm d\theta \), and the instantaneous power is \(P=\tau \,\omega \).

Before Figure 11.1, note: longer lever arms and perpendicular pushes maximize torque.

11.3 Moment of Inertia and Angular Momentum

For rotation about a fixed axis, the moment of inertia measures how mass resists angular acceleration:

\[ I=\sum _i m_i r_{\!\perp ,i}^2\quad \text {(point masses)},\qquad I=\int r_{\!\perp }^2\,\mathrm d m\quad \text {(continuous)}. \]

About an axis parallel to the center‑of‑mass (COM) axis and offset by distance \(d\), the parallel‑axis relation gives \(I = I_\text {COM}+M d^2\). The angular momentum about the axis is \(L=I\,\omega \) (for a rigid body with fixed axis), and the torque law can be written as

\[ \tau _\text {ext}=\frac {\mathrm d L}{\mathrm d t}=I\,\alpha \quad (\text {if }I\text { is constant about the axis}). \]

When \(\tau _\text {ext}=0\), angular momentum is conserved. Rotational kinetic energy mirrors the linear form: \(K_\text {rot}=\tfrac 12 I\omega ^2\).

11.4 Torque vs. Horsepower in Cars

Torque \(\tau \) is a twisting tendency (how hard the engine turns the crank). Horsepower is power—how fast work is done. The bridge is one line:

\[ \boxed {\;P = \tau \,\omega \;} \]

where \(\omega \) is the crankshaft’s angular speed (in rad/s). At the same torque, revving faster multiplies power; at the same speed, more torque multiplies power.

Before Figure 11.4, keep an eye on shapes: typical street engines have a broad torque “mesa” in the mid‑range; power rises with RPM because \(P=\tau \,\omega \).

What You Feel vs. How Fast You Go

At low speed in a short gear, wheel torque (engine torque multiplied by gear and final drive ratios, minus losses) gives strong acceleration feel. At high speed, aerodynamic drag grows rapidly and you need power to hold speed—hence peak power roughly sets top speed (all else equal). Gearing trades RPM for wheel torque: lower gears multiply engine torque more, higher gears trade torque for speed.

For a power‑based view and efficiency notes, revisit Chapter 8. The equation \(P=\bm F\cdot \bm v\) at the wheels ties road force and speed to the engine’s \(P=\tau \,\omega \) through gearing and losses.

11.5 Exercises

Practice reading angular motion and torque pictures.

• 1. Turn rate. Convert 60 RPM to rad/s.

• 2. Best push. On a wrench of length \(r\), at what angle do you push for maximum torque?

• 3. Practical: Door test. Try the door near hinge vs. handle; describe the torque difference.

11.6 Summary and Review

A quick checklist of angular ideas:

• • Angular kinematics parallels linear: \(\theta \), \(\omega =\dot \theta \), \(\alpha =\dot \omega \).

• • Torque combines force and lever arm: \(\tau =\bm r\times \bm F\); perpendicular force maximizes the twist.

• • Rotational dynamics preview: \(\tau _\text {net}=I\,\alpha \), \(K_\text {rot}=\tfrac 12 I \omega ^2\).

11.7 Where We’re Heading Next

In Chapter 12 we turn to Newtonian gravitation—fields, potential, and simple orbits—then return to oscillations and fluids with this rotational toolkit in mind.

Part VI Gravity, Oscillations, and Continuum Basics

Chapter 12 Newtonian Gravitation

Gravity connects motion on Earth with motion of the planets. In Newton’s picture, masses attract with a force directed along the line joining them and proportional to the product of masses and inversely to the square of their separation. The same rule that pulls an apple downward bends the Moon’s path into a near circle—one idea, many scales.

12.1 Newton’s Law of Universal Gravitation

Start with the rule itself, then read it in pictures. For two point masses \(m\) and \(M\) separated by distance \(r\), the magnitude of the gravitational force is For clarity we denote this distance as \(R_{\oplus M}\) (rather than using punctuation in the subscript).

\[ F= G\,\frac {m M}{r^2},\qquad \text {directed along the line joining the masses.} \]

In vector form (on \(m\) due to \(M\)):

\[ \bm F(\bm r) = -\,G\,\frac {m M}{r^2}\,\hat {\bm r} = -\,\nabla \big (-\tfrac {G m M}{r}\big ),\qquad \hat {\bm r}=\frac {\bm r}{\|\bm r\|}. \]

Superposition holds: fields and forces from multiple sources add as vectors. Because gravity is central (along \(\hat {\bm r}\)), angular momentum about the attracting mass is conserved so motion lies in a plane.

We summarize the “effect per kilogram” with the gravitational field of \(M\):

\[ \bm g(\bm r)= -\,G M\,\frac {\hat {\bm r}}{r^2} \qquad (\text {points inward, weakens with }1/r^2). \]

Before Figure 12.1, note how field arrows get shorter (weaker) as you move away.

12.2 Potential and Energy

For conservative forces, energy helps us reason quickly. Choosing \(U(\infty )=0\), the potential energy of \(m\) in the field of \(M\) is

\[ U(r)=-\,\frac {G m M}{r},\qquad \Phi (r)=\frac {U(r)}{m}=-\,\frac {G M}{r}. \]

Two features stand out: \(U\) is negative for bound states (you must do work to escape), and \(U\to 0\) as \(r\to \infty \). Forces and potentials are linked by \(\bm F=-\nabla U\) (Appendix Chapter C). As shown in Figure 12.2, the graph dives downward and creeps toward zero far away.

It is convenient to define the gravitational parameter \(\mu =GM\). The specific mechanical energy (per unit mass) is

\[ \varepsilon = \frac {v^2}{2} + \Phi (r) = \frac {v^2}{2} - \frac {\mu }{r}. \]

Bound orbits have \(\varepsilon <0\) (ellipses), escape trajectories have \(\varepsilon \ge 0\) (parabolic at \(0\), hyperbolic if \(>0\)). For a circular orbit of radius \(R\), \(\varepsilon =-\mu /(2R)\), since \(v^2=\mu /R\).

12.3 Circular Orbit Condition

An object of mass \(m\) orbits \(M\) in a circle of radius \(R\) when gravity supplies precisely the needed centripetal acceleration. Equate inward forces:

\[ \frac {m v^2}{R}=G\,\frac {m M}{R^2}\quad \Rightarrow \quad v(R)=\sqrt {\frac {GM}{R}},\qquad T=\frac {2\pi R}{v}=2\pi \sqrt {\frac {R^3}{GM}}. \]

As shown in Figure 12.3, orbit speed decreases with altitude like \(R^{-1/2}\), while the period grows like \(R^{3/2}\) (Kepler’s third law for circular orbits). Near Earth’s surface, \(g=\mu /R_\oplus ^2\) connects \(\mu \) to the familiar \(g\).

An energetic bonus: the escape speed from radius \(R\) follows from energy conservation with \(E=0\) at infinity: \(\tfrac 12 m v_{\!\text {esc}}^2 - GMm/R = 0\), so \(v_{\!\text {esc}}=\sqrt {\dfrac {2GM}{R}}=\sqrt {2}\,v_\text {circ}\).

12.4 Everyday Gravity Examples

Gravity’s inverse‑square law explains apples and orbits with the same symbols. Here are concrete, number‑driven examples with a few friendly plots.

12.5 Exercises

Practice reading gravity graphs and quick estimates.

• 1. Field magnitude. Compute \(\|\bm g\|\) at Earth’s surface from \(M_\oplus , R_\oplus \); compare with \(9.8\,\mathrm {m/s^2}\).

• 2. Orbit speed. Find the speed for a circular orbit at altitude \(h\) above a planet of radius \(R\) (assume \(h\ll R\) for an easy estimate).

• 3. Escape vs. circular. Show that \(v_{\!\text {esc}}=\sqrt {2}\,v_\text {circ}\) at the same \(R\).

12.6 Summary and Review

• • Inverse‑square attraction: \(F=G m M/r^2\), field inward with \(1/r^2\).

• • Potential energy: \(U=-G m M/r\) with \(U(\infty )=0\); \(\bm F=-\nabla U\).

• • Circular orbit: \(v=\sqrt {GM/R}\), \(T=2\pi \sqrt {R^3/(GM)}\); specific energy \(\varepsilon =-GM/(2R)\).

12.7 Where We’re Heading Next

In Chapter 13, we study simple harmonic motion and damping.

Chapter 13 Simple Harmonic Motion and Oscillations

Oscillations are everywhere: springs, pendulums, violin strings. The simplest model is the simple harmonic oscillator (SHO), where the restoring force pulls back toward equilibrium in proportion to displacement.

13.1 From Force Law to Equation

Hooke’s law states \(F=-kx\). Newton’s second law gives

\[ m\ddot x = -k x \quad \Rightarrow \quad \ddot x + \omega _0^2 x=0,\qquad \omega _0=\sqrt {\frac {k}{m}}. \]

This constant‑coefficient ODE has sinusoidal solutions

\[ x(t)=A\cos (\omega _0 t+\phi ),\qquad v(t)=\dot x(t)=-A\,\omega _0\sin (\omega _0 t+\phi ), \]

with period \(T=2\pi /\omega _0\). As previewed in Figure 13.1, \(x(t)\) oscillates smoothly with fixed amplitude and period.

13.2 Phase Portrait and Geometry

Plotting \(v\) against \(x\) eliminates time and reveals the geometry of SHO: combining \(x=A\cos \theta \) and \(v=-A\omega _0\sin \theta \) gives

\[ \left (\frac {x}{A}\right )^2+\left (\frac {v}{A\omega _0}\right )^2=1, \]

an ellipse (a circle if axes are scaled equally). Motion runs around the ellipse at constant angular speed in phase space. Figure 13.2 shows the picture.

Energy gives the same ellipse directly. Since \(E=\tfrac 12 kA^2=\tfrac 12 kx^2+\tfrac 12 m v^2\), divide by \(\tfrac 12 kA^2\) to obtain

\[ \frac {x^2}{A^2}+\frac {v^2}{(A\,\omega _0)^2}=1,\qquad \omega _0=\sqrt {\frac {k}{m}}, \]

which is the phase‑space ellipse without invoking sines and cosines. Either route—explicit solution or energy—leads to the same geometry.

13.3 Energy Picture

The total energy is constant and splits between potential and kinetic:

\[ E=\tfrac 12 k x^2+\tfrac 12 m v^2=\tfrac 12 k A^2=\text {constant}. \]

At turning points (\(x=\pm A\)) energy is all potential; at \(x=0\) energy is all kinetic. Figure 13.3 shades \(U\) and \(K\) over time.

13.4 Small‑Angle Pendulum \(\approx \) SHO

Gravity plus geometry creates another SHO. A simple pendulum of length \(L\) and small angle \(\theta \) from the vertical obeys

\[ \text {(tangential)}\quad m L\,\ddot \theta = - m g \sin \theta \;\approx \; - m g\,\theta \quad (|\theta |\ll 1), \]

so \(\ddot \theta + (g/L)\,\theta =0\) with natural frequency \(\omega _0=\sqrt {g/L}\). Thus the small‑angle pendulum is an SHO in \(\theta \). This links back to Chapter 12: gravity supplies the restoring “spring.” The linearization \(\sin \theta \approx \theta \) is accurate for modest angles (say \(|\theta |\lesssim 10^\circ \text {–}15^\circ \)); beyond that, the period grows slightly with amplitude.

Before Figure 13.4, note that the restoring component is \(mg\sin \theta \) along the tangent and that arc length \(s\approx L\theta \) for small angles.

13.5 Light Damping (Preview)

With small resistive forces \(F_\text {damp}=-b\,v\), the equation becomes \(m\ddot x+b\dot x+kx=0\). Solutions oscillate with slowly decaying amplitude \(A\,e^{-\gamma t}\) where \(\gamma =\tfrac {b}{2m}\) and \(\omega \approx \sqrt {\omega _0^2-\gamma ^2}\). It is common to define the damping ratio \(\zeta =\dfrac {b}{2\sqrt {mk}}=\dfrac {\gamma }{\omega _0}\) and the quality factor \(Q=\dfrac {1}{2\zeta }\) (how many radians of oscillation per e‑fold decay). Figure 13.5 shows the envelope.

13.6 Driven Response (Preview)

A periodic drive \(F_0\cos (\Omega t)\) leads to steady oscillations at the drive frequency \(\Omega \) with amplitude (for light damping)

\[ A(\Omega )=\frac {F_0/m}{\sqrt {(\omega _0^2-\Omega ^2)^2+(2\gamma \,\Omega )^2}}. \]

Amplitude peaks near \(\Omega \approx \omega _0\) (resonance) and broadens with larger \(\gamma \). The steady‑state motion also lags the drive by a phase \(\phi \) with \(\tan \phi =\dfrac {2\gamma \,\Omega }{\omega _0^2-\Omega ^2}\). Full derivations live in Appendix Chapter D; Figure 13.7 plots the amplitude shape.

13.7 Exercises

Practice deriving, reading, and estimating.

• 1. Period. For \(m=1\), \(k=4\), compute \(\omega _0\) and \(T\).

• 2. Phase portrait. Show that \((x/A)^2+(v/(A\omega _0))^2=1\) for the SHO.

• 3. Light damping. If \(\gamma =0.1\,\omega _0\), estimate the amplitude after \(10\) periods.

13.8 Summary and Review

• • SHO: \(m\ddot x+kx=0\) with \(x(t)=A\cos (\omega _0 t+\phi )\), \(\omega _0=\sqrt {k/m}\).

• • Phase‑space circle (scaled): motion at constant speed around the ellipse.

• • Energy swaps between \(U\) and \(K\) while \(E\) remains constant; damping leaks energy slowly.

• • Driving near \(\omega _0\) produces large amplitudes (resonance) moderated by damping.

13.9 Where We’re Heading Next

In Chapter 14, we sketch how Newtonian ideas extend to simple fluids.

Chapter 14 Fluids and Continuum Mechanics (Introductory)

Many everyday phenomena involve continuous media rather than point masses. We sample Newtonian ideas for fluids: pressure, buoyancy, and a taste of flow—enough to reason about water, air, pipes, and simple lift effects without leaving the Newtonian toolbox.

14.1 Pressure and Buoyancy

Consider a fluid at rest under gravity. A vertical column of height \(h\) carries the weight of the fluid above: balancing forces on a thin slab gives

\[ p(h)=p_0+\rho g h\quad (h\text { measured downward from the free surface}). \]

Pressure acts equally in all directions at a point. In differential form, hydrostatic balance reads \(\nabla p=\rho \,\bm g\); in a uniform \(\bm g=-g\,\hat {\bm y}\) this integrates to \(p=p_0+\rho g h\). For a submerged body, the pressure increases with depth on its bottom face relative to its top face, creating a net upward buoyant force equal to the weight of displaced fluid (Archimedes):

\[ F_\mathrm {b}=\rho g V_\mathrm {disp}. \]

Float or sink? Compare densities: if \(\rho _\text {object}<\rho _\text {fluid}\) the displaced weight can balance the object’s weight and it floats; if \(\rho _\text {object}>\rho _\text {fluid}\) it sinks. Before Figure 14.1, note the straight‑line pressure–depth plot; Figure 14.2 shows a simple buoyancy free‑body diagram.

14.2 Flow Basics (Preview)

For steady, incompressible flow in a pipe, mass conservation says \(\rho A_1 v_1 = \rho A_2 v_2\), i.e. \(A_1 v_1=A_2 v_2\)—when the pipe narrows, the fluid speeds up. Along a streamline in slowly varying flow without pumps or strong losses, Bernoulli’s principle balances energy per unit volume:

\[ p+\tfrac 12 \rho v^2 + \rho g z \approx \text {constant}. \]

We use it qualitatively to understand tradeoffs between pressure, speed, and height. Assumptions matter: Bernoulli applies along a streamline for inviscid, steady flow (or where viscous losses are small) and incompressible speeds (low Mach). When viscosity or turbulence dominate, expect departures (see Chapter 15 for Reynolds number scaling). Figure 14.3 sketches a narrowing pipe and velocity change. Figure 14.4 visualizes a faster flow over a curved top surface (lower pressure) and a slower flow beneath (higher pressure), producing a net upward force.

14.3 Exercises

Short, concrete checks:

• 1. Hydrostatics. A diver descends \(15\,\mathrm {m}\) in freshwater (\(\rho \approx 1000\,\mathrm {kg/m^3}\)). Estimate the gauge pressure increase \(\Delta p\).

• 2. Buoyancy. A cube of side \(0.2\,\mathrm {m}\) floats in oil (\(\rho \approx 800\,\mathrm {kg/m^3}\)) with \(3\,\mathrm {cm}\) above the surface. Estimate the cube’s density.

• 3. Continuity. Water flows steadily through a pipe that narrows from \(A_1=6\,\mathrm {cm^2}\) to \(A_2=2\,\mathrm {cm^2}\). If \(v_1=0.5\,\mathrm {m/s}\), find \(v_2\).

14.4 Summary and Review

• • Hydrostatics: \(\nabla p=\rho \,\bm g\); \(p=p_0+\rho g h\) in uniform gravity.

• • Buoyancy: \(F_\mathrm b=\rho g V_\mathrm {disp}\); float/sink set by density comparison.

• • Steady incompressible flow: continuity \(A v=\text {const}\); Bernoulli along a streamline when viscous losses are small.

14.5 Where We’re Heading Next

In Chapter 15 we step back and develop general tools—dimensional analysis and scaling—that sharpen intuition and provide quick checks. These ideas pair well with fluids: scaling arguments explain, for example, why small insects can walk on water whereas larger animals cannot.

Part VII Methods, Modelling, and Numerical Simulation

Chapter 15 Dimensional Analysis and Scaling

Dimensional analysis is a compact way to check equations and predict relationships before calculating. Scaling arguments explain why the same equations look different for small and large systems—why ants “feel” strong and elephants do not.

15.1 Consistency Checks

Dimensional homogeneity is the first line of defense: every term you add or compare must represent the same kind of quantity. If one side of an equation has the units of force and the other side has the units of energy, the equation cannot be right no matter what numbers you plug in.

Everyday examples help:

• • \(F=m a\) passes the test because \([m a]=\mathrm {M}\,\mathrm {L}\,\mathrm {T}^{-2}=[F]\).

• • \(v= v_0 + a t\) is fine: each term has \(\mathrm {L}\,\mathrm {T}^{-1}\).

• • \(x= x_0 + a t^2\) is fine for constant acceleration, but \(x= x_0 + a t\) fails (wrong power of time).

• • Arguments of \(\sin (\cdot )\), \(\exp (\cdot )\), \(\log (\cdot )\) must be dimensionless. Expressions like \(\exp (t)\) are illegal unless \(t\) was first non‑dimensionalized (e.g., \(t/\tau \)).

Practical habits that pay off:

• • Carry symbolic dimensions through your algebra before substituting numbers.

• • Convert mixed measurement systems early (e.g., km/h to m/s) so dimensions remain transparent.

• • When in doubt, rewrite a formula to isolate one quantity and check that the isolated quantity has the right dimensions.

Lead‑in questions to ask when scanning a formula:

• • Do all additive terms share the same dimensions?

• • Do arguments of trig/exp/log functions come out dimensionless?

15.2 Building Dimensionless Groups (by Hand)

Dimensionless groups are combinations of variables that carry no units. They often reveal the natural control knobs of a problem and collapse data taken at different scales onto a single curve.

Worked example (linear drag). Suppose the terminal speed \(v_t\) depends on \(m\) (mass), \(g\) (acceleration), and \(c\) (drag coefficient for \(F_d=c v\)). Write the dimensions

\[ [v_t]=\tfrac {\mathrm {L}}{\mathrm {T}},\;[m]=\mathrm {M},\;[g]=\tfrac {\mathrm {L}}{\mathrm {T}^2},\;[c]=\tfrac {\mathrm {M}}{\mathrm {T}}. \]

Seek \(v_t\sim m^a g^b c^d\). Matching exponents of \(\mathrm {M},\mathrm {L},\mathrm {T}\) yields \(a=1,\;b=1,\;d=-1\), so \(v_t\sim mg/c\). We did not need the exact numerical constant to know the form.

For quadratic drag \(F_d=\tfrac 12\rho C_D A v^2\), momentum exchange with the fluid sets the scale; \(v_t\) should grow with weight and shrink with fluid density and cross‑section. Try \(v_t\sim (m g/\rho A)^{1/2}\), consistent with \([\rho A] = \mathrm {M}/\mathrm {L}\).

Reading data with \(\pi \) groups. If a measured quantity \(Y\) depends on \(X_1,\dots ,X_n\), form \(\pi \) groups \(\pi _i\) and plot one against another. A flat cloud becomes a tight curve when the right non‑dimensional variables are used.

End‑to‑End Example: Drag on a Sphere

We expect the steady drag force \(F_D\) on a sphere to depend on fluid density \(\rho \), dynamic viscosity \(\eta \), characteristic length \(L\) (diameter), and speed \(v\). List dimensions

\[ [F_D]=\mathrm {M}\,\mathrm {L}\,\mathrm {T}^{-2},\quad [\rho ]=\mathrm {M}\,\mathrm {L}^{-3},\quad [\eta ]=\mathrm {M}\,\mathrm {L}^{-1}\,\mathrm {T}^{-1},\quad [L]=\mathrm {L},\quad [v]=\mathrm {L}\,\mathrm {T}^{-1}. \]

There are \(n=5\) variables and \(k=3\) base dimensions, so expect \(n-k=2\) independent \(\pi \) groups. Choose repeating variables \((\rho , v, L)\) (they span \(\mathrm {M},\mathrm {L},\mathrm {T}\)). Build

\[ \pi _1=\frac {F_D}{\rho v^2 L^2}\quad (\text {dimensionless drag coefficient } C_D),\qquad \pi _2=\frac {\rho v L}{\eta }\quad (\text {Reynolds number } \mathrm {Re}). \]

The Buckingham \(\pi \)–theorem says \(\pi _1=f(\pi _2)\), i.e.

\[ \boxed {\;C_D = f(\mathrm {Re})\;} \]

which is the classic empirical relationship: in creeping flow (small \(\mathrm {Re}\)), \(C_D\sim 24/\mathrm {Re}\) (Stokes); at larger \(\mathrm {Re}\), \(C_D\) flattens to an \(\mathcal {O}(1)\) constant. The moral: the dimensionless plot \(C_D\) vs. \(\mathrm {Re}\) collapses experiments of different sizes and speeds.

15.3 The Pendulum Period (Dimensional Guess)

At small angles, a simple pendulum’s period \(T\) depends primarily on gravity \(g\) and length \(L\); mass and small‑amplitude do not matter. With \([T]=\mathrm {T}\), \([g]=\mathrm {L}\,\mathrm {T}^{-2}\), and \([L]=\mathrm {L}\), the only combination with time units is

\[ T\sim \sqrt {\frac {L}{g}}. \]

The exact solution supplies the constant \(2\pi \), but the shape \(T\propto L^{1/2}\) falls straight out of dimensions. The message is practical: long pendulums swing slowly; short ones swing quickly. If a prediction violates this by suggesting \(T\propto L\), dimensions flag a mistake before you compute.

As shown in Figure 15.1, a log–log plot of \(T\) versus \(L\) has slope \(1/2\).

15.4 Quick \(\pi \)–Theorem Workflow

When many variables appear, the Buckingham \(\pi \)–theorem says you can rewrite a relationship among \(n\) variables with \(k\) base dimensions as a relation among \(n-k\) independent dimensionless combinations (\(\pi \) groups). A practical 4‑step recipe:

• 1. List variables and their dimensions.

• 2. Choose \(k\) repeating variables spanning the base dimensions.

• 3. Build \(n-k\) independent \(\pi \) groups (products that come out dimensionless).

• 4. Write a relation among the \(\pi \)’s; test/fit with data or limiting cases.

Lead‑in example: drag coefficient \(C_D\) is already dimensionless; Reynolds number \(\mathrm {Re}=\dfrac {\rho v L}{\eta }\) distinguishes flow regimes.

15.5 Scaling Intuition

If length scales by \(\lambda \), areas scale by \(\lambda ^2\), volumes by \(\lambda ^3\). Weight scales like \(\lambda ^3\) but cross‑sectional strength like \(\lambda ^2\)—one reason small creatures seem “stronger” for their size. In fluids, \(\mathrm {Re}\propto L\) at fixed \(v,\rho ,\eta \)—bigger systems more easily become turbulent. On free surfaces (waves, ships), the gravity–inertia balance is captured by Froude \(\mathrm {Fr}=\dfrac {v}{\sqrt {gL}}\); model tests match \(\mathrm {Fr}\) to preserve wave patterns even if \(\mathrm {Re}\) cannot be matched simultaneously (see Chapter 14).

15.6 Exercises

Short problems to practice and check:

• 1. Check it. Verify that kinetic energy \(K=\tfrac 12 m v^2\) has units of work.

• 2. Pendulum. Use dimensions to guess how the period depends on \(L\) and \(g\).

• 3. Drag guess. For quadratic drag, argue that \(v_t\sim \sqrt {mg/(\rho A)}\).

15.7 Summary and Review

A quick checklist before moving on:

• • Dimensional consistency is a must‑have filter for equations.

• • \(\pi \) groups reduce variables to dimensionless combinations.

• • Scaling clarifies how size changes behavior; power laws show straight lines on log–log plots.

15.8 Where We’re Heading Next

In Chapter 16 we add simple numerical methods and modelling workflows to turn equations into runnable predictions, with practical checks for accuracy, stability, and energy behavior.

Chapter 16 Numerical Methods for Newtonian Systems

Analytic solutions are wonderful—but many problems need approximate time‑stepping. We discretize time, update positions and velocities, and watch accuracy/stability as we go.

16.1 Discretizing Time

For \(\dot x=f(x,t)\), explicit Euler is \(x_{n+1}=x_n + f(x_n,t_n)\,\Delta t\). For mechanics, the first‑order system

\[ \dot x = v,\qquad \dot v = a(x,v,t) \]

updates as

\[ v_{n+1}=v_n + a(x_n,v_n,t_n)\,\Delta t,\qquad x_{n+1}=x_n + v_n\,\Delta t. \]

The semi‑implicit/symplectic variant uses the new velocity in the position update: \(x_{n+1}=x_n+v_{n+1}\,\Delta t\)—often more stable for oscillations.

Terminology: the local truncation error (error made in one step assuming perfect input) is \(\mathcal O(\Delta t^2)\) for Euler; accumulated global error over an interval is \(\mathcal O(\Delta t)\). Halving \(\Delta t\) should roughly halve the global error for first‑order methods (see Appendix Chapter E).

Before Figure 16.1, recall the schematic from Chapter 2: staircase vs. smooth.

16.2 Accuracy and Stability

Smaller \(\Delta t\) typically improves accuracy (local error \(\sim \Delta t^2\) for Euler; global \(\sim \Delta t\)). For oscillatory or stiff problems, naive Euler can blow up; symplectic Euler often behaves qualitatively better for conservative systems.

Linear stability lens. For the test equation \(\dot y=\lambda y\) with \(\Re (\lambda )<0\), explicit Euler gives \(y_{n+1}=(1+\lambda \Delta t) y_n\). This decays only if \(|1+\lambda \Delta t|<1\), which bounds \(\Delta t\). For oscillations (purely imaginary \(\lambda \)), \(|1+i\omega \Delta t|>1\) so Euler grows—explaining energy drift. Structure‑preserving (symplectic) schemes keep the phase‑space area and tend to bound energy oscillations in Hamiltonian systems.

A Concrete Example: Simple Harmonic Oscillator

Consider \(\ddot x + \omega ^2 x=0\) with \(\omega =1\), \(x(0)=1\), \(v(0)=0\) so the exact solution is \(x(t)=\cos t\). The total energy \(E=\tfrac 12(v^2+x^2)\) should stay constant.

Before Figure 16.2, keep in mind what “good” looks like: a nearly horizontal energy trace. A steadily rising or falling trace means the method is injecting or removing energy.

16.3 Practical Workflow and Checks

Simple habits raise confidence:

• • Start coarse, then halve \(\Delta t\) and compare: stable results should converge.

• • Track conserved or monotone quantities (e.g., energy in conservative systems) as qualitative checks.

• • Beware floating‑point roundoff with very small \(\Delta t\) or huge step counts; compare against an analytic or high‑accuracy reference when possible.

Error shrinks as \(\Delta t\) shrinks. On log–log axes, first‑order global error curves are straight lines with slope one, as shown in Figure 16.4.

16.4 Worked Python Examples

Short, self‑contained scripts illustrate how the updates look in code. You can copy–paste them to a file and run with any Python 3.x. We annotate each step so the mapping to the math is clear.

Simple Harmonic Oscillator: Euler vs. Symplectic Euler

We integrate \(\ddot x + x = 0\) for \(t\in [0,20]\) with \(\Delta t=0.05\) and compare energies.

# Simple harmonic oscillator: x'' + x = 0
# Compare explicit Euler and symplectic Euler on energy behavior


import math


def euler_sho(x0=1.0, v0=0.0, dt=0.05, t_end=20.0):
     x, v = x0, v0
     xs, vs, Es = [], [], []
     t = 0.0
     while t <= t_end + 1e-12:
         # Energy E = 0.5*(v^2 + x^2) should be constant for exact solution
         Es.append(0.5*(v*v + x*x))
         xs.append(x); vs.append(v)
         # Explicit Euler updates use acceleration a = -x evaluated at (x_n, v_n)
         a = -x
         v = v + a*dt         # v_{n+1} = v_n + a_n dt
         x = x + v*0.0 + (v - a*dt)*dt      # equivalent to x_{n+1} = x_n + v_n dt
         # (written this way to highlight it's the old v; compact form below)
         # x = x + vs[-1]*dt
         t += dt
     return xs, vs, Es


def symplectic_euler_sho(x0=1.0, v0=0.0, dt=0.05, t_end=20.0):
     x, v = x0, v0
     xs, vs, Es = [], [], []
     t = 0.0
     while t <= t_end + 1e-12:
         Es.append(0.5*(v*v + x*x))
         xs.append(x); vs.append(v)
         # Symplectic Euler uses new v to update x; still a = -x at the start
         a = -x
         v = v + a*dt              # v_{n+1} = v_n + a_n dt
         x = x + v*dt              # x_{n+1} = x_n + v_{n+1} dt
         t += dt
     return xs, vs, Es


if __name__ == "__main__":
     # Run both methods and print final energies for a quick comparison
     _, _, E_eu = euler_sho()
     _, _, E_sy = symplectic_euler_sho()
     print(f"Euler    final E: {E_eu[-1]:.6f}   (start {E_eu[0]:.6f})")
     print(f"Symplec final E: {E_sy[-1]:.6f}   (start {E_sy[0]:.6f})")

You should observe Euler’s energy drifting noticeably while the symplectic method oscillates around the initial value.

Projectile with Linear Drag (Component Form)

We model \(m\dot {\vec v}=\vec F=\langle 0, -mg\rangle - c\,\vec v\) with \(m=1\), \(g=9.81\), \(c=0.2\). This shows how to structure vector updates.

# 2D projectile under linear drag: v' = g - (c/m) v
# Semi-implicit (symplectic) Euler on velocity; then update position


import math


def projectile_drag(x0=(0.0, 0.0), v0=(20.0, 16.0), g=9.81, c=0.2, m=1.0,
                          dt=0.02, t_end=3.0):
     x, y = x0
     vx, vy = v0
     traj = []      # store (t, x, y, vx, vy)
     t = 0.0
     while t <= t_end + 1e-12:
         traj.append((t, x, y, vx, vy))
         # Acceleration components from gravity and linear drag
         ax = -(c/m)*vx
         ay = -g - (c/m)*vy
         # Update velocities first (symplectic-like)
         vx = vx + ax*dt
         vy = vy + ay*dt
         # Then update positions using the new velocities
         x = x + vx*dt
         y = y + vy*dt
         t += dt
         # Stop if projectile hits the ground (simple termination)
         if y < 0.0:
               break
     return traj


if __name__ == "__main__":
     data = projectile_drag()
     # Show a few samples to verify the trajectory trends
     for k in range(0, len(data), max(1, len(data)//5)):
         t, x, y, vx, vy = data[k]
         print(f"t={t:4.2f}   x={x:6.2f}   y={y:6.2f}   vx={vx:6.2f}   vy={vy:6.2f}")

Both examples follow the same state‑update pattern used throughout the book; only the acceleration function changes with the physics.

16.5 Exercises

Practice small steps and comparisons.

• 1. Euler step. For \(\ddot x=-\omega ^2 x\) with \(\omega =1\), \(x_0=1\), \(v_0=0\), \(\Delta t=0.1\), compute \((x_1,v_1)\) by explicit Euler and by symplectic Euler.

• 2. Step effect. Halve \(\Delta t\) and compare \(x(1)\) against the exact \(\cos 1\).

16.6 Summary and Review

A quick checklist before moving on:

• • Explicit Euler: easy but only first‑order accurate and can be unstable.

• • Symplectic Euler: similar cost, better qualitative behavior for Hamiltonian systems.

• • Smaller \(\Delta t\) improves accuracy but increases cost.

16.7 Where We’re Heading Next

These numerical tools support many chapters ahead and pair naturally with the math appendices. For a compact reference of methods and stability/error heuristics, see Chapter E.

Epilogue — What Comes After Newtonian Physics?

Newtonian mechanics is astonishingly successful. On everyday scales—with speeds much less than the speed of light, moderate gravitational fields, and objects large enough to ignore quantum granularity—it predicts motion with crisp accuracy and remarkable economy. Yet nature stretches beyond these conditions. Here is a compact map of where Newtonian ideas bend and how modern physics extends them.

Where It Breaks

• • Very fast (\(v\sim c\)). At speeds approaching the speed of light, time dilates and lengths contract. Newton’s velocity addition and absolute time fail.

• • Very strong gravity / very large scales. Near massive bodies or across cosmological distances, space and time curve. Newton’s instantaneous action at a distance is replaced by geometry.

• • Very small (atomic and subatomic). Energy comes in quanta, particles behave like waves, and measurement itself carries probabilistic structure.

Three Great Extensions

• • Special Relativity (Einstein, 1905). Mechanics and electromagnetism share the same speed limit \(c\). Space and time form spacetime; energy and momentum combine into four‑vectors. Newton’s second law survives in relativistic dress, and Newtonian results reappear when \(v\ll c\).

• • General Relativity (Einstein, 1915). Gravity is curved spacetime. Free fall is motion along geodesics. Newton’s law of gravitation emerges as a weak‑field, low‑speed approximation.

• • Quantum Mechanics (1920s–). States are waves in Hilbert space; observables are operators; outcomes are probabilistic with strict rules. Classical mechanics returns in the limit of large quantum numbers or small \(\hbar \) (the correspondence principle).

How the Pieces Fit

In practice, physics is a set of overlapping maps, each valid on its domain with clean bridges between them:

• • Classical (Newtonian) for everyday speeds, weak gravity, and macroscopic objects.

• • Relativistic mechanics for fast motion or precise timing (GPS satellites, particle beams).

• • Quantum mechanics for atoms, molecules, and materials; quantum field theory for particle physics.

• • Classical field theories (fluids, elasticity, electromagnetism) describe collective behavior where particle details are hidden.

Each newer theory contains Newtonian mechanics as a limiting case. Learning to recognize which map applies—and when to change maps—is a professional superpower.

What We Still Don’t Have

Despite a century of progress there is no single, complete theory of everything (TOE). The Standard Model of particle physics and general relativity coexist but resist unification. Dark matter and dark energy signal phenomena beyond what current theories explain. This is not a failure; it is an invitation.

Takeaway

Newton’s framework remains the right starting point for most engineered and natural systems you will encounter. It is also the common language you’ll use when stepping into relativity, quantum theory, or continuum descriptions. Master the Newtonian map; then explore the others with confidence, knowing where each begins and where the next takes over.

Part VIII Mathematics for Newtonian Mechanics

Chapter A Calculus Essentials for Motion

This appendix gathers exactly the one–variable calculus tools used throughout the book, with pictures first and formulas second. If you want a memory hook: slope means rate; area means accumulation. When in doubt, re‑draw the picture.

A.1 Differentiation

We start with precise definitions and then build geometric intuition.

Before Figure A.1, remember: \(f'\) is slope and \(f''\) is curvature.

A.2 Derivative Rules You Actually Use

We differentiate to turn position into velocity (Chapter 2). The rules you reach for most often are the ones that let you simplify expressions quickly without losing sight of the physical meaning (rate of change in time):

Lead‑in: here are the core rules, with examples you can apply immediately.

• • Linearity: \((af+bg)'=af'+bg'\). Example: \(\tfrac {\mathrm d}{\mathrm dt}(2t+3\sin t)=2+3\cos t\).

• • Product: \((fg)'=f'g+fg'\). Example: \(\tfrac {\mathrm d}{\mathrm dt}(t\sin t)=\sin t + t\cos t\).

• • Chain: \((f\circ g)'=(f'\circ g)\,g'\). Example: \(\tfrac {\mathrm d}{\mathrm dt}\sin (t^2)=\cos (t^2)\cdot 2t\).

Small‑angle reminders we actually use (Chapters 6 and 13):

• • For \(|\theta |\ll 1\) (radians): \(\sin \theta \approx \theta \), \(\cos \theta \approx 1-\tfrac {\theta ^2}{2}\). Always use radians for calculus with trig.

A.3 Limits and Continuity

Intuitively, \(f(t)\) approaches \(L\) as \(t\to t_0\) if we can make \(f(t)\) as close to \(L\) as we like by taking \(t\) sufficiently close to \(t_0\). Continuity at \(t_0\) means the graph has no jump or hole there: \(\lim _{t\to t_0} f(t)=f(t_0)\).

In practice, we rely on two habits:

• • Look left and right: do one‑sided limits agree? If yes and the point value matches, the function is continuous.

• • Replace scary algebra with a picture: zoom into the point on the graph. If the zoom looks like a straight line, a derivative likely exists.

Before Figure A.2, recall: the derivative is the limit of secant slopes.

A.4 Taylor Approximations (Local Models)

Near a point \(t_0\), \(f(t)\) is well‑approximated by a low‑order polynomial built from derivatives at \(t_0\). Think “best straight line,” then “best gentle bend.” For smooth \(f\),

\[ f(t) \approx f(t_0) + f'(t_0)(t-t_0) + \tfrac 12 f''(t_0)(t-t_0)^2. \]

Engineers read this as “straight line + gentle bend.” In Chapter 13, small‑angle pendulum motion is nothing but a Taylor approximation to the sine.

Before Figure A.3, remember: approximations are local. Step too far, and the picture bends away.

A.5 Integration

First a general definition, then a picture.

Before Figure A.4, keep in mind: the Riemann–Stieltjes sum \(\sum f(\xi _i)\,\Delta \alpha _i\) weights samples by increments of an increasing integrator \(\alpha \). Equal steps in \(\alpha \) generally mean uneven steps in \(t\).

A.6 Fundamental Theorems (How They Fit)

The two operations differentiation and integration are inverses under suitable assumptions. We collect the relationships used throughout the book and visualize the area–as–accumulation idea.

• • Fundamental Theorem of Calculus (part I). If \(F'(t)=f(t)\) and \(f\) is integrable,

  \[ \int _a^b f(t)\,\mathrm dt = F(b)-F(a). \]

• • Fundamental Theorem of Calculus (part II). If \(f\) is integrable and sufficiently nice,

  \[ \frac {\mathrm d}{\mathrm dt}\int _a^t f(\tau )\,\mathrm d\tau = f(t). \]

• • Riemann–Stieltjes with differentiable integrator. If \(\alpha \) is differentiable with \(\alpha '\in L^1,\) then

  \[ \int _a^b f\,\mathrm d\alpha = \int _a^b f(t)\,\alpha '(t)\,\mathrm dt, \]

  and the Leibniz rule extends to

  \[ \frac {\mathrm d}{\mathrm dt}\int _a^t f(\tau )\,\mathrm d\alpha (\tau ) = f(t)\,\alpha '(t) \]

  when the hypotheses hold (see also Chapter 8).

• • Integration by parts mirrors the product rule: for suitable \(f,g\),

  \[ \int _a^b f\,\mathrm dg = f(b)g(b)-f(a)g(a) - \int _a^b g\,\mathrm df. \]

Before Figure A.5, keep the picture in mind: the integral is the signed area. As introduced in Chapter 3, area under \(a(t)\) gives the change in \(v\), and area under \(v(t)\) gives the change in \(x\).

A.7 Integrals and the FTC (Mechanics Lens)

The Fundamental Theorem of Calculus (FTC) ties rates to totals. If \(F'(t)=f(t)\) and \(f\) is integrable on \([a,b]\), then

\[ \int _a^b f(t)\,\mathrm dt = F(b)-F(a). \]

In mechanics: \(v=\dot x\) so \(x(b)-x(a)=\int _a^b v(t)\,\mathrm dt\) (displacement is area under \(v\)); \(a=\dot v\) so \(v(b)-v(a)=\int _a^b a(t)\,\mathrm dt\). Piecewise‑smooth signals are fine: add the areas across pieces.

A.8 Worked Examples

The following presents worked examples to illustrate the relationship between differentiation and integration. First, application of integration to a velocity curve.

Second, application of differentiation to a position curve.

A.9 Techniques We Actually Use

Lead‑in: a tiny toolkit goes a long way in mechanics.

• • Substitution: straighten a composition. Example: \(\int 2t\cos (t^2)\,\mathrm dt = \int \cos u\,\mathrm du = \sin u + C\) with \(u=t^2\).

• • Integration by parts: trade derivative for antiderivative. Example: \(\int t e^t\,\mathrm dt = t e^t - \int e^t\,\mathrm dt = e^t(t-1)+C\).

Chapter B Vectors and Linear Algebra Essentials

This appendix gives only the vector tools we actually use in the main text. Pictures lead; formulas follow. Keep three verbs in mind: add (combine), project (compare), rotate (re‑express).

B.1 Vectors and Basic Operations

A vector in the plane is an ordered pair \(\bm a=(a_x,a_y)\); in space, \(\bm a=(a_x,a_y,a_z)\). Add components to add vectors; scale components to scale a vector. In 2D, \(\lVert \bm a\rVert =\sqrt {a_x^2+a_y^2}\); in 3D, \(\lVert \bm a\rVert =\sqrt {a_x^2+a_y^2+a_z^2}\). The unit vector is \(\hat {\bm a}=\bm a/\lVert \bm a\rVert \) when \(\bm a\ne \bm 0\).

Before Figure B.1, recall: tip-to-tail addition is geometry you can see.

B.2 Dot Product and Projection

The dot product measures alignment: \(\bm a\cdot \bm b=\lVert \bm a\rVert \,\lVert \bm b\rVert \cos \phi \), where \(\phi \) is the angle between them. Algebraically in 2D/3D, \(\bm a\cdot \bm b=a_x b_x+a_y b_y(+a_z b_z)\). The projection of \(\bm a\) onto \(\hat {\bm u}\) is \((\bm a\cdot \hat {\bm u})\,\hat {\bm u}\).

We use this for work in Chapter 8: \(W=\int \bm F\cdot \mathrm d\bm r\)—force along displacement.

Before Figure B.2, keep in mind: the dot is the signed length of the shadow of one vector on another.

B.3 Cross Product, Area, and Torque

In 3D, \(\bm a\times \bm b\) is perpendicular to both \(\bm a\) and \(\bm b\), with magnitude \(\lVert \bm a\times \bm b\rVert =\lVert \bm a\rVert \,\lVert \bm b\rVert \sin \phi \), equal to the area of the parallelogram spanned by \(\bm a\) and \(\bm b\). Right‑hand rule sets direction. Torque in Chapter 11 uses this idea: \(\bm \tau =\bm r\times \bm F\).

Before Figure B.3, visualize area and normal as two sides “sweeping” a sheet and a thumb pointing up.

B.4 Rotations and Change of Basis (2D)

In the plane, rotation by angle \(\theta \) is

\[ R(\theta )=\begin {bmatrix}\cos \theta & -\sin \theta \\[2pt] \sin \theta & \cos \theta \end {bmatrix},\qquad \bm a' = R(\theta )\,\bm a. \]

Geometrically, \(R(\theta )\) turns every arrow by \(\theta \) (angles in radians for calculus). These orthonormal rotations preserve lengths and dot products; \(\det R(\theta )=1\). Change of basis simply means describing the same arrow with a different set of unit vectors—numbers change, the arrow does not.

Before Figure B.4, note how both the components and the drawn arrow pivot together; keep labels away from overlap.

Chapter C Multivariable Calculus Lite

We collect just the multivariable ideas needed for fields and potentials: gradient (steepest ascent), directional derivatives (rate along a direction), and a light touch on divergence, curl, and line integrals.

C.1 Gradient and Directional Derivative

For a scalar field \(f:\mathbb R^n\to \mathbb R\), the gradient

\[ \nabla f = \begin {bmatrix} \partial f/\partial x_1 \\ \vdots \\ \partial f/\partial x_n \end {bmatrix} \]

points in the direction of steepest increase. The directional derivative along a unit vector \(\hat {\bm u}\) is

\[ D_{\hat {\bm u}} f = \nabla f \cdot \hat {\bm u}, \]

the instantaneous rate of change of \(f\) in the \(\hat {\bm u}\) direction.

Before Figure C.1, remember: longer arrows mean steeper uphill.

C.2 Conservative Fields and Potentials

A vector field \(\bm F\) is conservative if it is the gradient of a potential \(\phi \): \(\bm F=\nabla \phi \) (potential defined up to an additive constant). Then line integrals are path‑independent and

\[ \int _{\mathcal C} \bm F\cdot \mathrm d\bm r = \phi (\text {end})-\phi (\text {start}). \]

In 2D/3D on simple regions, a sufficient test is that \(\nabla \times \bm F=\bm 0\) and the region is simply connected. In Chapter 8, this is the condition for a force to admit a potential energy; in Chapter 12, \(\bm F=-\nabla U\) with \(U=-GMm/r\).

C.3 Divergence and Curl (Qualitative)

The divergence \(\nabla \cdot \bm F\) measures sources and sinks; positive divergence looks like fluid expanding from a point. The curl \(\nabla \times \bm F\) measures local rotation: a nonzero curl makes tiny paddles spin.

Before Figure C.3, keep the pictures in mind: arrows spreading vs. arrows swirling.

C.4 Line Integrals Along Paths

For a path \(\mathcal C\colon \bm r(t)\), \(a\le t\le b\), the line integral of a vector field is

\[ \int _{\mathcal C} \bm F\cdot \mathrm d\bm r = \int _a^b \bm F(\bm r(t))\cdot \bm r'(t)\,\mathrm dt. \]

If \(\bm F=\nabla \phi \), the integral depends only on endpoints. Orientation matters: reversing the path flips the sign.

Chapter D Ordinary Differential Equations Essentials

We keep ODE tools lean and visual: how to read direction, how to separate variables, how integrating factors tame linear equations, and how second‑order linear models (oscillators) behave.

D.1 Reading Solutions from Direction (Slope) Fields

For a first‑order ODE \(y'=f(t,y)\), a slope field sketches a short line segment of slope \(f(t,y)\) at each grid point. Solutions trace curves tangent to these segments.

Before Figure D.1, note: for the logistic law \(y'=y(1-y)\), slopes depend only on \(y\); below \(y=0\) arrows point downward (negative), between \(0\) and \(1\) they tilt upward, and near \(y=1\) they flatten.

D.2 Separable and Linear First‑Order ODEs

Separable: if \(y'=g(t)h(y)\), rearrange \(\dfrac {\mathrm dy}{h(y)}=g(t)\,\mathrm dt\) and integrate both sides. Linear: if \(y'+p(t) y = q(t)\), multiply by an integrating factor \(\mu (t)=e^{\int p(t)\,dt}\) to make \((\mu y)'=\mu q\).

D.3 Second‑Order Linear ODEs: Oscillations

The undamped oscillator \(\ddot x + \omega _0^2 x=0\) has sinusoidal solutions. Damping adds decay: \(\ddot x + 2\gamma \dot x + \omega _0^2 x=0\). Forced response adds a drive \(F_0\cos \omega t\) and exhibits resonance near \(\omega \approx \omega _0\) when damping is small.

Before Figure D.2, recall: phase portraits plot \((x,\dot x)\); undamped orbits are closed; damping spirals inward.

Before Figure D.3, keep in mind: damping flattens and broadens the resonance peak.

D.4 Existence and Uniqueness (Statement)

If \(f\) and \(\partial f/\partial y\) are continuous near \((t_0,y_0)\) (Lipschitz in \(y\) suffices), then the initial value problem \(y'=f(t,y)\), \(y(t_0)=y_0\) has a unique solution in some interval around \(t_0\). Practically: well‑behaved right‑hand sides give a single trajectory through each point; discontinuities or non‑Lipschitz points can lead to multiple or no solutions.

Chapter E Numerical Methods Quick Reference

This appendix distills the discrete tools used across the book into one friendly place. Our aim is intuition you can act on: what each update does, how accuracy improves with smaller steps, and how to sanity‑check stability and energy behavior.

E.1 Finite Differences and Euler Updates

For \(\dot x=f(x,t)\) at \(t_n\), the forward difference reads

\[ \frac {x_{n+1}-x_n}{\Delta t}\approx f(x_n,t_n), \]

meaning “new minus old equals slope times step.” In mechanics we evolve the pair \((x,v)\) via \(\dot x=v\) and \(\dot v=a(x,v,t)\):

\begin{align*} \text {Explicit Euler:}\quad & v_{n+1}=v_n + a(x_n,v_n,t_n)\,\Delta t,\\ & x_{n+1}=x_n + v_n\,\Delta t.\\ \text {Symplectic Euler:}\quad & v_{n+1}=v_n + a(x_n,v_n,t_n)\,\Delta t,\\ & x_{n+1}=x_n + v_{n+1}\,\Delta t. \end{align*} “Order” counts how fast error shrinks as you shrink \(\Delta t\). Explicit Euler is first‑order and simple. Symplectic Euler is also first‑order but uses the fresh velocity to update position, which makes a big qualitative difference for conservative systems: it tends to bound energy instead of drifting.

E.2 Error vs. Step Size

On log–log axes, first‑order global error falls with slope 1. A practical recipe is to run with \(\Delta t\) and with \(\Delta t/2\); the observed change estimates the error size. Before Figure E.1, keep that slope‑1 picture in mind: halving \(\Delta t\) roughly halves the error for order‑1 schemes.

E.3 Energy Behavior (Conservative Systems)

For \(\ddot x + \omega _0^2 x=0\), explicit Euler typically drifts in energy (frames “gain” or “lose” energy), while symplectic Euler tends to oscillate around the constant true energy. When in doubt for long conservative runs, pick the symplectic flavor.

E.4 Stability Sketches

For the test equation \(y'=\lambda y\) (the standard stability yardstick), explicit Euler is stable only if \(|1+\lambda \,\Delta t|<1\). For real \(\lambda <0\), this collapses to \(-2<\lambda \,\Delta t<0\)—too large a step turns decay into blow‑up.

When you lack a trusted reference answer, halve \(\Delta t\) and check that results change by the expected order and fall below a tolerance you set up front (units matter!).

E.5 Quick Checks

A few habits that pay off in practice:

• • Reduce \(\Delta t\) and see if answers change by \(\mathcal O(\Delta t)\) (order‑1) or faster.

• • In conservative tests, monitor energy; prefer symplectic Euler for long runs.

• • Keep units consistent; errors in scale can masquerade as “instability.”

Glossary

Short, alphabetized definitions for quick lookup. Cross‑links point to chapters where a concept features prominently.

Acceleration

Rate of change of velocity; in 1D \(a=\dot v\); see Chapter 3.

Angular acceleration

Rate of change of angular velocity \(\alpha =\dot \omega \); see Chapter 11.

Angular momentum

Rotational analogue of momentum; conserved in absence of external torque; see Chapter 11.

Area (under a curve)

Signed integral; area under \(v(t)\) gives displacement; see Chapter 3.

Buoyancy

Upward force on a body in a fluid equal to the weight of displaced fluid; see Chapter 14.

Center of mass (COM)

Weighted average of position; system moves as if mass were concentrated at COM; see Chapter 10.

Coefficient of restitution

Dimensionless measure of bounciness in collisions; ratio of relative speeds after/before; see Chapter 10.

Conservative force

One with path‑independent work and a potential \(U\); see Chapter 9.

Continuity equation

Statement of conservation (e.g., mass) for a flowing medium; see Chapter 14.

Damping ratio

Dimensionless measure of damping strength \(\zeta \); relates to decay rate and quality factor; see Chapter 13.

Dot product

Measure of alignment between vectors; work \(=\bm F\cdot \mathrm d\bm r\); see Chapter 8.

Divergence

Scalar measure of sources/sinks of a vector field; see Chapter C.

Energy

Capacity to do work. Kinetic \(K=\tfrac 12 mv^2\); potential \(U\) depends on configuration; see Chapters 8 and 9.

Force

Interaction that changes motion; Newton’s second law \(\sum \bm F = m\,\bm a\); see Chapter 7.

Froude number

Dimensionless ratio \(\mathrm {Fr}=v/\sqrt {gL}\) comparing inertia to gravity in free‑surface flows; see Chapter 15.

Gradient

Vector of partial derivatives pointing uphill; see Chapter C.

Impulse

Integral of force over time; changes momentum; see Chapter 10.

Inertia

Resistance to changes in motion; quantified by mass; rotational counterpart is moment of inertia.

Line integral

Integral of a vector field along a path; work; see Chapters C and 8.

Matrix

Rectangular array representing a linear map; rotations use \(R(\theta )\); see Chapter B.

Moment of inertia

Rotational inertia about an axis; appears in \(\tau =I\alpha \) and \(K_\text {rot}=\tfrac 12 I\omega ^2\); see Chapter 11.

Momentum

Product of mass and velocity \(\bm p=m\,\bm v\); conserved for isolated systems; see Chapter 10.

Normal force

Contact force perpendicular to a surface; appears in FBDs; see Chapter 7.

Numerical method

Discrete scheme to approximate ODE solutions (e.g., Euler, symplectic Euler); see Chapters E and 16.

Potential energy

Stored capability of conservative forces; changes by negative work; see Chapter 9.

Power

Rate of doing work \(P=\bm F\cdot \bm v\); see Chapter 8.

Projection

Component of one vector along another; dot product; see Chapter B.

Quality factor

Dimensionless \(Q=1/(2\zeta )\); higher \(Q\) means slower decay; see Chapter 13.

Resonance

Amplification when driving frequency matches a system’s natural frequency; see Chapter 13.

Resultant

Net vector sum, e.g., net force; see Chapter 7.

Reynolds number

Dimensionless ratio \(\mathrm {Re}=\rho v L/\eta \) comparing inertia to viscosity; see Chapter 15.

Rotational kinematics

Angular position \(\theta \), velocity \(\omega \), acceleration \(\alpha \); see Chapter 11.

Slope

Geometric meaning of derivative; tangent line slope; see Chapters A and 2.

Specific mechanical energy

Energy per unit mass \(\varepsilon =\tfrac 12 v^2+\Phi \); negative for bound orbits; see Chapter 12.

Symplectic method

Structure‑preserving time‑stepper for Hamiltonian systems; better long‑term energy behavior; see Chapter E.

Torque

Tendency to rotate: \(\bm \tau =\bm r\times \bm F\); see Chapter 11.

Unit vectors

Cartesian basis arrows \(\hat {\bm e}_x,\hat {\bm e}_y,\hat {\bm e}_z\) of length 1 pointing along the coordinate axes; see Chapter B.

Vector

Quantity with magnitude and direction; added tip‑to‑tail; see Chapters B and 5.

Velocity

Rate of change of position; see Chapter 3.

Work

Energy transfer by force along a path: \(W=\int \bm F\cdot \mathrm d\bm r\); see Chapter 8.

Index of Symbols

Alphabetical list of the most commonly used symbols. Units are indicated in brackets where fixed by context.

\(A\)

Area; cross‑sectional area (fluids, drag) [m\(^2\)].

\(C_D\)

Drag coefficient (dimensionless).

\(E\)

Total mechanical energy, \(E=K+U\).

\(e\)

Coefficient of restitution (dimensionless).

\(\varepsilon \)

Specific mechanical energy per unit mass [J/kg].

\(F\), \(\bm F\)

Force (scalar magnitude or vector) [N].

\(\mathrm {Fr}\)

Froude number \(\mathrm {Fr}=\dfrac {v}{\sqrt {gL}}\) (dimensionless).

\(g\)

Gravitational acceleration near Earth (\(\approx 9.81\,\mathrm {m/s^2}\)).

\(I\)

Moment of inertia about a specified axis [kg·m\(^2\)].

\(K\)

Kinetic energy, \(K=\tfrac 12 m v^2\) [J].

\(L\)

Angular momentum [kg·m\(^2\)/s].

\(m\)

Mass [kg].

\(\mu ,\;\mu _s,\;\mu _k\)

Coefficient(s) of friction (dimensionless; static/kinetic).

\(\mu \) (grav.)

Gravitational parameter \(\mu =GM\) [m\(^3\)/s\(^2\)].

\(p\), \(\bm p\)

Linear momentum \(\bm p=m\,\bm v\) [kg·m/s].

\(P\)

Power, \(P=\bm F\cdot \bm v\) [W].

\(Q\)

Quality factor \(Q=1/(2\zeta )\) (dimensionless).

\(R(\theta )\)

2D rotation matrix (see Chapter B).

\(\mathrm {Re}\)

Reynolds number \(\mathrm {Re}=\dfrac {\rho v L}{\eta }\) (dimensionless).

\(\bm r=(x,y,z)\)

Position vector [m].

\(t\)

Time [s]; \(t_n=n\,\Delta t\) in numerics.

\(\Delta t\)

Time step (see Chapter E).

\(U\)

Potential energy (context‑dependent) [J].

\(v\), \(\bm v\)

Speed/velocity (scalar/vector) [m/s].

\(W\)

Work; weight (\(W=mg\)) by context [J] or [N].

\(x, y, z\)

Cartesian coordinates [m].

\(\alpha \)

Angular acceleration [rad/s\(^2\)].

\(\eta \)

Dynamic viscosity [Pa·s].

\(\phi \)

Potential (scalar field) (see Chapter C).

\(\rho \)

Mass density [kg/m\(^3\)].

\(\zeta \)

Damping ratio (dimensionless).

\(\theta \)

Angle (radians unless stated).

\(\tau \), \(\bm \tau \)

Torque [N·m].

\(\omega \)

Angular velocity [rad/s].

\(\nabla \)

Gradient operator; \(\nabla f\), \(\nabla \cdot \bm F\), \(\nabla \times \bm F\) (see Chapter C).

\(\hat {\bm e}_x,\hat {\bm e}_y,\hat {\bm e}_z\)

Cartesian unit vectors.

Bibliography and Notes

This compact bibliography favors readable, widely available sources. Each item includes a short note on scope and style; historical remarks highlight how ideas entered the canon.

Introductory and Bridge Texts

• • D. Morin, Introduction to Classical Mechanics. Problem‑driven, witty, with careful solutions. Excellent for building intuition through practice.

• • A. P. French, Newtonian Mechanics. Clear prose and physical insight; a classic bridge between conceptual understanding and calculation.

• • R. Feynman, The Feynman Lectures on Physics, Vol. I. Big‑picture viewpoints and crisp derivations; especially good for energy and conservation ideas.

Standard Undergraduate Texts

• • D. Kleppner and R. Kolenkow, An Introduction to Mechanics. Thorough, with challenging problems; a common reference for momentum and rotation.

• • J. R. Taylor, Classical Mechanics. Gentle exposition with modern notation; accessible treatments of oscillations and central forces.

Historical Sources (Short Remarks)

• • Galileo Galilei, early kinematics (inclined planes) established constant‑acceleration motion and the decomposition of trajectories.

• • Isaac Newton, Philosophiæ Naturalis Principia Mathematica. Unified terrestrial and celestial motion; Newton’s laws and the inverse‑square gravitation law.

• • Leonhard Euler. Formalized dynamics and introduced variational and analytical methods that underpin modern mechanics and numerics.