Flashcards — Chapter 10 🃏

Rotational Motion and Angular Dynamics — the parallel universe of rotation! Master angular variables, moment of inertia, torque, and rotational energy with these cards. ⚙️

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📋 Card deck: Key terms & formulas

Section §10.1 — Rotational kinematics

1. Angular position (θ) — angle swept by a radius vector; measured in radians; \( \theta = s/r \) where \( s \) is arc length
2. Radian — dimensionless unit; \( 2\pi \) radians = \( 360° \); one radian ≈ 57.3°
3. Angular velocity (ω) — rate of change of angle; \( \omega = d\theta/dt \) (rad/s)
4. Angular acceleration (α) — rate of change of angular velocity; \( \alpha = d\omega/dt \) (rad/s²)
5. Rotational kinematic equations — analogues to linear kinematics: \( \omega = \omega_0 + \alpha t \), \( \theta = \omega_0 t + \frac{1}{2}\alpha t^2 \), \( \omega^2 = \omega_0^2 + 2\alpha\theta \)

Section §10.2 — Relating linear and angular motion

6. Arc length relation — \( s = r\theta \) (length of arc = radius × angle in radians)
7. Tangential velocity — \( v = r\omega \) (linear speed at radius \( r \) from axis)
8. Tangential acceleration — \( a_t = r\alpha \) (linear acceleration in direction of motion)
9. Centripetal acceleration — \( a_c = r\omega^2 = v^2/r \) (acceleration toward axis)
10. Total acceleration in circular motion — \( a = \sqrt{a_t^2 + a_c^2} \) (vector sum of tangential and centripetal)

Section §10.3 — Moment of inertia

11. Moment of inertia (definition) — \( I = \sum m_i r_i^2 \) or \( I = \int r^2 \, dm \); measure of rotational inertia
12. Units of moment of inertia — kg·m² (different from mass!)
13. Solid disk/cylinder — \( I = \frac{1}{2}MR^2 \) (axis through center, perpendicular to face)
14. Thin rod (axis at center) — \( I = \frac{1}{12}ML^2 \) (length \( L \), axis perpendicular to rod)
15. Thin rod (axis at end) — \( I = \frac{1}{3}ML^2 \) (length \( L \), axis at one end)
16. Solid sphere — \( I = \frac{2}{5}MR^2 \) (axis through center)
17. Thin spherical shell — \( I = \frac{2}{3}MR^2 \) (axis through center)
18. Thin ring/hoop — \( I = MR^2 \) (all mass at radius \( R \))
19. Parallel axis theorem — \( I = I_{\text{CM}} + Md^2 \) where \( d \) is distance from CM axis to new axis
20. Why moment of inertia depends on axis — a rod rotating about its end has larger \( I \) than rotating about its center (mass farther from axis)

Section §10.4 — Torque

21. Torque (definition) — \( \vec{\tau} = \vec{r} \times \vec{F} \) (cross product of position and force vectors)
22. Magnitude of torque — \( \tau = rF\sin\phi \) where \( \phi \) is angle between \( \vec{r} \) and \( \vec{F} \)
23. Moment arm — perpendicular distance from axis to line of force; \( d_{\perp} = r\sin\phi \)
24. Alternative torque formula — \( \tau = F \cdot d_{\perp} \) (force times moment arm)
25. Units of torque — N·m (newton·meter), same dimensions as energy but different meaning
26. Positive vs negative torque — positive (counterclockwise) or negative (clockwise); right-hand rule
27. Net torque — \( \tau_{\text{net}} = \sum \tau_i \) (vector sum of all torques about an axis)
28. Torque from weight — for a body of mass \( M \) at distance \( d \) from axis: \( \tau = Mgd \) (if arm perpendicular to gravity)

Section §10.5 — Rotational energy and Newton's 2nd law

29. Rotational kinetic energy — \( KE_{\text{rot}} = \frac{1}{2}I\omega^2 \) (analogous to \( \frac{1}{2}mv^2 \))
30. Total kinetic energy (rolling) — \( KE_{\text{total}} = \frac{1}{2}Mv_{\text{CM}}^2 + \frac{1}{2}I\omega^2 \) (translation + rotation)
31. Work done by torque — \( W = \tau \cdot \theta \) (if torque is constant)
32. Power in rotation — \( P = \tau \cdot \omega \) (rate of energy transfer by torque)
33. Newton's second law (rotation) — \( \sum \tau = I\alpha \) (net torque = moment of inertia × angular acceleration)
34. Analogy: linear vs rotational — \( F = ma \) ↔ \( \tau = I\alpha \); \( m \) ↔ \( I \); \( a \) ↔ \( \alpha \)

Section §10.6 — Rolling without slipping

35. No-slip condition — \( v_{\text{CM}} = r\omega \) (center of mass speed = radius × angular velocity)
36. Kinetic energy of rolling object — \( KE = \frac{1}{2}Mv_{\text{CM}}^2 + \frac{1}{2}I\omega^2 \); split between translation and rotation
37. For rolling disk — \( KE_{\text{rot}}/KE_{\text{total}} = 1/3 \) (one-third rotational, two-thirds translational for disk rolling)
38. For rolling sphere — \( KE_{\text{rot}}/KE_{\text{total}} = 2/7 \) (rolling sphere has larger translational fraction)
39. Energy conservation on incline — \( Mgh = \frac{1}{2}Mv^2 + \frac{1}{2}I\omega^2 \) (no friction dissipation)

Section §10.7 — Static equilibrium and applications

40. Condition for static equilibrium (translational) — \( \sum \vec{F} = 0 \) (no net force)
41. Condition for static equilibrium (rotational) — \( \sum \vec{\tau} = 0 \) (no net torque about any axis)
42. Lever principle — for a lever with fulcrum, load and effort at different distances: \( F_1 r_1 = F_2 r_2 \) (torques balance)
43. Mechanical advantage — load force / effort force = \( r_{\text{effort}} / r_{\text{load}} \) (ideal lever has advantage > 1)

Section §10.8 — Angular momentum (introduction)

44. Angular momentum (for fixed axis) — \( L = I\omega \) (kg·m²/s)
45. Relation to torque — \( \sum \tau = dL/dt \) (torque is rate of change of angular momentum)
46. Conservation of angular momentum — if net external torque is zero, \( L_{\text{total}} \) is constant (e.g., spinning skater pulling arms in spins faster)
47. Figure skater example — pulling arms in decreases \( I \), so \( \omega \) must increase to conserve \( L = I\omega \)
48. Gyroscope effect — spinning object resists change in orientation of its angular momentum vector; precesses slowly under gravity

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📊 Topics covered

• Rotational kinematics — angle, angular velocity, angular acceleration, kinematic equations
• Linear-angular relations — arc length, tangential velocity, centripetal acceleration
• Moment of inertia — definition, common shapes (disk, rod, sphere), parallel axis theorem
• Torque — definition, cross product, magnitude, moment arm
• Rotational energy — rotational kinetic energy, rolling without slipping
• Newton's second law (rotation) — \( \tau = I\alpha \), static equilibrium
• Angular momentum — definition, conservation, applications

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What's next:

👉 §10.1–§10.3 — Worked problems (step-by-step examples)
👉 §10.4 — Practice problems (exercises)
👉 §10.5 — Q&A (FAQs on common misconceptions)

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