ap physics c rotation cheat sheet

Olvin

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12-01-2025

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This cheat sheet provides a concise overview of key concepts and equations for the AP Physics C: Mechanics rotation unit. Remember, understanding the underlying physics is crucial; this sheet is a tool for quick reference, not a replacement for thorough study.

I. Kinematics of Rotation

• Angular Displacement (θ): The angle through which an object rotates (radians).
• Angular Velocity (ω): The rate of change of angular displacement (rad/s). ω = Δθ/Δt
• Angular Acceleration (α): The rate of change of angular velocity (rad/s²). α = Δω/Δt
• Relationship between Linear and Angular Quantities:
  • Arc Length (s): s = rθ (r = radius)
  • Linear Velocity (v): v = rω
  • Linear Acceleration (a_t): a_t = rα (tangential acceleration)
  • Centripetal Acceleration (a_c): a_c = v²/r = ω²r (always directed towards the center of rotation)
  • Total Linear Acceleration (a): a = √(a_t² + a_c²)

Key Equations:

• Uniform Angular Acceleration:
  • ω_f = ω_i + αt
  • θ = ω_it + (1/2)αt²
  • ω_f² = ω_i² + 2αθ

II. Dynamics of Rotation

• Torque (τ): The rotational analog of force. τ = rFsinθ (θ is the angle between the force vector and the lever arm).
• Moment of Inertia (I): A measure of an object's resistance to changes in its rotational motion. It depends on the object's mass distribution and its shape. (Units: kg·m²)
• Rotational Kinetic Energy (K_rot): K_rot = (1/2)Iω²
• Newton's Second Law for Rotation: Στ = Iα (Net torque equals moment of inertia times angular acceleration.)
• Angular Momentum (L): L = Iω (Units: kg·m²/s)
• Conservation of Angular Momentum: If the net external torque on a system is zero, the total angular momentum of the system remains constant. I_iω_i = I_fω_f

Common Moments of Inertia:

• Solid Cylinder/Disk: I = (1/2)MR²
• Hollow Cylinder/Ring: I = MR²
• Solid Sphere: I = (2/5)MR²
• Hollow Sphere: I = (2/3)MR²
• Rod (about center): I = (1/12)ML²
• Rod (about end): I = (1/3)ML²

III. Work and Energy in Rotation

• Work Done by a Torque: W = τθ
• Power in Rotation: P = τω

IV. Important Considerations

• Vector Nature: Angular velocity, angular acceleration, and torque are vector quantities. Use the right-hand rule to determine their direction.
• Units: Maintain consistent units throughout your calculations (radians, kg, m, s).
• Parallel Axis Theorem: This theorem allows you to calculate the moment of inertia about an axis parallel to an axis through the center of mass. I = I_cm + Md² (d is the distance between the axes).

This cheat sheet is a starting point. Make sure you thoroughly understand the derivations and applications of these concepts through practice problems and example scenarios. Remember to consult your textbook and class notes for a comprehensive understanding of rotational motion. Good luck with your AP Physics C exam!