Physics Chapter: Rotational Dynamics and Inertia

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Questions and Answers

What are the consistent SI units used when working with torque and angular acceleration?

The consistent SI units are: meters (m) for length, radians (rad) for angular displacement, kilograms (kg) for mass, and seconds (s) for time.

What is the formula for calculating the moment of inertia, I, in terms of angular acceleration, a, torque, t, and time t?

I = t/a

What is the moment of inertia of a thin hoop with radius R?

  • 1/12ML²
  • 1/2MR²
  • MR² (correct)
  • 2/5MR²

Match the following objects to their corresponding moment of inertia formulas:

<p>Thin hoop, radius R = MR² Thin hoop, radius R, width w = 1/2Mw² Solid cylinder, radius R, length l = 1/2MR² Long uniform rod, length l, through the center = 1/12ML² Hollow cylinder, inner radius R1, outer radius R2 = 1/2M(R1² + R2²) Uniform sphere, radius R = 2/5MR² Long uniform rod, length l, through one end = 1/3ML² Rectangular thin plate, length l, width w, through the center = 1/12M(l² + w²)</p>
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The moment of inertia always stays the same, regardless of the axis of rotation.

<p>False (B)</p>
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What are the two laws used to solve rotational motion problems?

<p>Newton's second law for rotation and Newton's second law for translation</p>
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Flashcards

Torque formula

Torque (τ) equals the moment of inertia (I) times the angular acceleration (α): τ = Iα.

Moment of inertia (I)

A measure of an object's resistance to changes in its rotation.

Angular Acceleration (α)

The rate of change of angular velocity over time.

Consistent Units (SI)

Using standard units for torque (newton-meters), moment of inertia (kilogram-meter squared), and time (seconds).

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Newton's 2nd Law (rotation)

Net torque equals the moment of inertia multiplied by the angular acceleration.

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Free-body diagram

A diagram showing all forces acting on an object.

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Axis of rotation

The point around which an object rotates (or is assumed to rotate).

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Torque from a force

The force multiplied by the lever arm (distance from the axis).

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Problem-solving steps (rotation)

Steps to solve rotational dynamics problems (diagram, system, free-body diagram, torques, Newton's 2nd Law, solve, estimate).

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Moment of inertia example (hoop)

Moment of inertia for a thin hoop rotating through its center is MR^2.

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Moment of inertia example (cylinder)

Moment of inertia for a solid cylinder rotating through its center is (1/2)MR^2.

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Moment of inertia example (rod through center)

Moment of inertia of a rod rotating through its center is (1/12)ML^2.

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Moment of inertia example (sphere)

Moment of inertia for a sphere rotating through its center is (2/5)MR^2.

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Rotational kinetic energy

Energy of rotation calculated as 0.5 x I x ω^2.

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Study Notes

Rotational Dynamics

  • Using torque and angular acceleration (Eq. 8-14) requires consistent units (SI: N⋅m; rad/s²; kg⋅m²).
  • Moment of inertia (I) is calculated as I = τ/α, where τ is torque and α is angular acceleration.

Moments of Inertia

  • Different shapes have distinct moment of inertia formulas.
  • Thin hoop: I = MR² (axis through center)
  • Thin hoop: I = MR² (axis through center)
  • Solid cylinder: I = ½MR² (axis through center)
  • Hollow cylinder: I = ½M(R₁² + R₂²) (axis through center)
  • Uniform sphere: I = ⅔MR² (axis through center)
  • Long uniform rod: I = (1/12)ML² (axis through center)
  • Long uniform rod: I = (1/3)ML² (axis through end)
  • Rectangular thin plate: I = (1/12)M(w² + l²) (axis through center)

Solving Rotational Motion Problems

  • Draw diagrams and choose the system.
  • Include all forces and their directions, noting the axis of rotation for torque calculation.
  • Determine torques (positive for counterclockwise).
  • Use Newton's second law for rotation (∑τ = Iα).
  • Apply Newton's second law for linear motion (∑F = ma).
  • Consider consistent units and reasonable answers.

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