§11.6 — Flashcards

Flashcards for Chapter 11 to consolidate rolling motion, torque, and angular momentum.

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Rolling Motion

Card 1: No-Slip Condition

Question: If a disk of radius R rolls without slipping, what is the relationship between center-of-mass velocity and angular velocity?

Answer: \( v_{cm} = \omega R \)

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Card 2: Energy of Rolling

Question: What components make up the total kinetic energy of a rolling object?

Answer:

• Translational energy: \( \frac{1}{2}mv_{cm}^2 \)
• Rotational energy: \( \frac{1}{2}I\omega^2 \)
• Total: \( K = \frac{1}{2}m v_{cm}^2 \left(1 + \frac{I}{mR^2}\right) \)

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Card 3: Acceleration on Incline

Question: How does the acceleration of a rolling object differ from a sliding object on an incline?

Answer:

• Rolling: \( a = \frac{g\sin\theta}{1 + I/(mR^2)} < g\sin\theta \)
• Sliding: \( a = g\sin\theta \)
• Reason: Energy splits between translation and rotation

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Card 4: Which Reaches First?

Question: If a sphere and hoop with same radius and mass are released down a slope, which reaches bottom first?

Answer: Sphere — because \( I_{sphere}/(mR^2) = 2/5 < 1 = I_{hoop}/(mR^2) \)

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Torque

Card 5: Definition of Torque

Question: What is torque and what are its units?

Answer:

• Definition: \( \tau = r F \sin\theta \) (measure of force's tendency to rotate)
• Unit: Newton·meter (N·m)

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Card 6: Maximum Torque

Question: At what angle between force and lever arm is torque maximum?

Answer: 90° (force perpendicular to lever arm) → \( \tau_{\max} = rF \)

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Card 7: Rotational Newton's Law

Question: What is the rotational version of Newton's second law?

Answer: \( \tau_{net} = I\alpha \)

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Card 8: Torque and Power

Question: What is the relationship between torque, angular velocity, and power?

Answer: \( P = \tau \omega \)

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Angular Momentum

Card 9: Definition

Question: How is angular momentum defined?

Answer:

• For point mass: \( L = mvr \)
• For rigid body: \( L = I\omega \)

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Card 10: Torque-Momentum Relationship

Question: How does torque relate to angular momentum?

Answer: \( \tau_{net} = \frac{dL}{dt} \)

(Like \( F_{net} = \frac{dp}{dt} \) for linear momentum)

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Card 11: Conservation Law

Question: When is angular momentum conserved?

Answer: When no external torque acts: \( \tau_{net} = 0 \Rightarrow L = \text{constant} \)

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Card 12: Figure Skater

Question: Why does a figure skater spin faster when pulling arms in?

Answer:

• Angular momentum conserved: \( L = I_1\omega_1 = I_2\omega_2 \)
• Smaller \( I \) → larger \( \omega \)
• Energy source: work done pulling arms

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Card 13: Colliding Disks

Question: If two spinning disks collide and stick together, what happens?

Answer:

• Angular momentum conserved: \( L_{before} = L_{after} \)
• Kinetic energy decreases (converted to heat)

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Gyroscopic Motion

Card 14: Precession

Question: What is precession in a gyroscope?

Answer: The direction of angular momentum slowly rotates horizontally (instead of falling)

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Card 15: Precession Rate

Question: What is the formula for precession rate?

Answer: \( \Omega = \frac{mgr}{L} = \frac{mgr}{I\omega} \)

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Card 16: Speed and Precession

Question: If a gyroscope spins faster, does precession become slower or faster?

Answer: Slower — \( \Omega \propto 1/\omega \) (inverse relationship)

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Card 17: Why It Doesn't Fall

Question: Why doesn't a rapidly spinning gyroscope fall?

Answer: Angular momentum resists changing direction toward gravity; instead it rotates slowly (precession)

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Comparison and Applications

Card 18: Linear vs Angular

Question: Rotational equivalents of linear quantities:

Answer: | Linear | Angular | |---|---| | \( F \) | \( \tau \) | | \( m \) | \( I \) | | \( v \) | \( \omega \) | | \( a \) | \( \alpha \) | | \( p = mv \) | \( L = I\omega \) |

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Card 19: Conservation Laws

Question: What quantities are conserved in closed systems?

Answer:

• Energy (in elastic collisions)
• Linear momentum (if \( F_{net} = 0 \))
• Angular momentum (if \( \tau_{net} = 0 \))

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Card 20: Practical Applications

Question: Name three practical applications of angular momentum and gyroscopes:

Answer:

1. Bicycles: Stability and self-steering
2. Satellites: Orientation control in space
3. Gyro compasses: Navigation on ships

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📚 Content type: Self-testing flashcards for key concepts 🔗 Usage: Review before exams or problem-solving sessions

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