Chapter 10 — Rotation of a Rigid Body

📚 Reference: Halliday, Resnick, Krane — Physics (4th ed.), Vol 1, Chapter 10. Independent treatment; no text or figures reproduced. All scenarios are original.

What this chapter is about

Chapters 2 and 4 treated translation — an object moving from place to place. Now we turn to rotation — an object spinning about an axis. A spinning wheel, a planet on its axis, a rotating gyroscope, a tumbling asteroid — all are described by the same laws we are about to build.

The core insight: rotation obeys the same principles as translation, but with different variables. Where translation has velocity v, rotation has angular velocity ω. Where translation has force F, rotation has torque τ. The mathematics is parallel — sometimes called the rotational analogue to linear mechanics.

Chapter 10 is the first of two chapters on rotation:

• Ch 10 (this chapter) — rigid bodies, rotational kinematics, moment of inertia, torque
• Ch 11 — rotational dynamics (rotational Newton's second law), angular momentum, conservation

Sections

1. 10.1 — Rotational variables: angle, angular velocity, angular acceleration — the radian as a dimensionless measure, θ(t), ω = dθ/dt, α = dω/dt, vector nature of \vec ω and \vec α (right-hand rule).

2. 10.2 — Relating angular and linear quantities — the arc-length relation s = rθ, tangential velocity v = rω, tangential and centripetal acceleration a_t = rα, a_c = rω², velocity and acceleration vectors in circular motion.

3. 10.3 — Rotational kinematics with constant angular acceleration — the four angular kinematic equations (analogues of §2.4), angle swept, time to reach a target rotation, angular velocity from angular displacement.

4. 10.4 — Moment of inertia — defining I = ∑ m_i r_i², rotational inertia as the resistance to angular acceleration, computing I for simple shapes (disk, rod, ring, sphere), parallel-axis theorem, why I depends on axis choice.

5. 10.5 — Rotational kinetic energy — kinetic energy of rotation KE_rot = (1/2)Iω², rolling without slipping (combination of translation and rotation), total kinetic energy KE_total = (1/2)Mv² + (1/2)Iω².

6. 10.6 — Torque and the cross product — definition τ = r × F, magnitude τ = rF sin(φ), moment arm, the cross product in detail, finding the net torque on an object, torque sign convention (right-hand rule).

7. 10.7 — Newton's second law for rotation — Στ = Iα, deriving from \vec F = m\vec a, applying to spinning disks and pulleys, static equilibrium (both ΣF = 0 and Στ = 0).

8. 10.8 — Worked problems — eight complete step-by-step problems: rotating pulley-mass system, torque on a disk, moment of inertia by integration, rolling down an incline, gyroscope precession setup, each linked to the §10.x tool being exercised.

9. 10.9 — Practice problems — 25 problems across three levels (⭐ easy to ⭐⭐⭐ hard), covering kinematics, moment of inertia, energy, torque, without step-by-step solutions, with final answers for self-check.

10. 10.10 — Q&A — 20 frequently asked questions: "Why does angular velocity use radians?", "Does torque always cause angular acceleration?", "Why does a spinning figure skater spin faster when arms are pulled in?", each linked to relevant sections.

Prerequisites

From earlier chapters, you need:

• Vectors and cross product (§3.5–3.6) — essential for torque as \vec τ = \vec r × \vec F
• Circular motion (§4.5) — centripetal acceleration a_c = v²/r, foundation for relating linear and angular
• Newton's laws (§5) — \vec F = m\vec a generalizes to \vec τ = I\vec α
• Work and energy (§7) — rotational kinetic energy parallels translational

If shaky on cross products or circular motion, review those sections briefly before starting.

Reading suggestion

§10.1-10.4 are the foundation — read these in order. The angular kinematic equations (§10.3) and moment of inertia (§10.4) are the hardest parts; spend time there.

If pressed for time:

• Master §10.1, §10.2, §10.4 (the core concepts)
• Skip the parallel-axis theorem (§10.4) on first pass; come back when needed
• Solve at least one worked problem from each section

Key sections that unlock later chapters:

• §10.2 (relating angular/linear) — used in every rolling problem
• §10.4 (moment of inertia) — non-negotiable; even advanced physics relies on these I values
• §10.6 (torque as cross product) — linchpin for dynamics and electromagnetism

Connection to later chapters

• Chapter 11: angular momentum, rotational dynamics, conservation laws, gyroscopes, precession
• Chapter 12: static equilibrium (both linear and rotational)
• Chapter 15-16: simple harmonic motion, angular formulation
• Chapter 27-32: electromagnetism — magnetic torque on a current loop (\vec τ = \vec μ × \vec B), same cross product structure

Key concepts by section

Section	Concept	Formula	~Estimate
10.1	Angular position, velocity, acceleration	θ, ω = dθ/dt, α = dω/dt	3–4 pages
10.2	Arc length, tangential velocity, centripetal acceleration	s = rθ, v_t = rω, a_c = rω²	4–5 pages
10.3	Angular kinematic equations	ω = ω₀ + αt, θ = ω₀t + (1/2)αt², etc.	3–4 pages
10.4	Moment of inertia (various shapes, parallel-axis theorem)	I = ∑ m_i r_i², table of I for 10+ shapes	5–6 pages
10.5	Rotational kinetic energy, rolling without slipping	KE_rot = (1/2)Iω²	4–5 pages
10.6	Torque, cross product, moment arm	τ = r × F, τ = rF sin(φ)	4–5 pages
10.7	Rotational Newton's 2nd law, static equilibrium	Στ = Iα, ΣF = 0 ∧ Στ = 0	4–5 pages
10.8	Worked problems (pulley, disk, rolling, gyroscope)	8 complete step-by-step	8–10 pages
10.9	Practice problems (graded difficulty)	25 problems, 3 levels	6–8 pages
10.10	Q&A (conceptual + common mistakes)	20 questions	5–6 pages

Interactive elements planned

• Rotational kinematics simulation — adjust initial angle, angular velocity, angular acceleration; see θ(t), ω(t), α(t) curves and a spinning disk visual
• Moment of inertia calculator — enter mass and radius of disk/rod/sphere/ring, compute I, then compare two objects rotating with same torque
• Torque puzzle — given a rod with two forces applied at different points, find the net torque; visual feedback on the cross product
• Rolling without slipping interactive — a disk rolls down an incline; slider shows split between translational and rotational kinetic energy
• Flashcards — 40–50 cards covering all definitions, formulas, and conceptual checkpoints

Worked problem scenarios (original)

1. Accelerating pulley-mass system — a mass hangs from a string wrapped around a spinning disk; find the angular acceleration and compare to linear analysis
2. Disk with two torques — a disk receives torque from an applied force and friction; net angular acceleration?
3. Moment of inertia by integration — build I for a disk from scratch using ∫ r² dm
4. Racing down the ramp — disk vs hoop vs sphere rolling down the same incline; which reaches the bottom first and why?
5. Spinning platform collision — person stands on a rotating platform, throws an object outward; effect on angular velocity (preview to angular momentum)
6. Door torque and acceleration — push a door at its edge vs at the hinge; relate the force, distance, and angular acceleration
7. Torsional pendulum (preview) — a disk hangs from a twisted wire; oscillates angularly; relate the angular displacement to the restoring torque (leads to §15)
8. Gyroscope precession setup — a spinning wheel mounted on a pivot; apply a gravitational torque; describe what happens (detailed treatment in Ch 11)

Why rotation matters — a simple example

Two cyclists:

• Cyclist A: light wheel of radius 0.3 m, mass 2 kg
• Cyclist B: heavy wheel of radius 0.3 m, mass 4 kg

Both wheels spin at ω = 20 rad/s. Same rotational speed, same radius. But:

• Cyclist A's kinetic energy: KE = (1/2)I_A ω²
• Cyclist B's kinetic energy: KE = (1/2)I_B ω²

Since I_B = 2 I_A, Cyclist B's wheel stores twice the rotational kinetic energy. To spin it up, you push twice as hard at the pedals. To stop it, you brake twice as hard. Moment of inertia quantifies this physical resistance to spinning — it's why a lightweight wheel feels nimble and a heavyweight wheel feels sluggish, even at the same rotation rate.

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📖 Open reference: OpenStax University Physics Vol 1 — Chapter 10: Rotation of Rigid Bodies. 📖 Feynman Lectures Vol I — Ch 18–20: Rotational motion. 🧪 Related experiment: Simple pendulum experiment — connects angular kinematics to the pendulum period (§15).