ap physics c rotation cheat sheet

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10-01-2025

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This comprehensive cheat sheet covers the key concepts and formulas for rotation in AP Physics C. Mastering these will significantly improve your performance on the exam. Remember, understanding the underlying principles is crucial, not just memorizing formulas.

I. Kinematics of Rotation

This section focuses on describing rotational motion without considering the causes.

A. Angular Displacement (θ)

• Definition: The angle through which an object rotates, measured in radians (rad). One complete revolution is 2π radians.
• Relationship to linear displacement: Δx = rΔθ, where 'r' is the radius of the rotation.

B. Angular Velocity (ω)

• Definition: The rate of change of angular displacement, ω = Δθ/Δt, measured in radians per second (rad/s).
• Relationship to linear velocity: v = rω
• Average vs. Instantaneous: Similar to linear motion, we have average (ω_avg) and instantaneous (ω) angular velocity.

C. Angular Acceleration (α)

• Definition: The rate of change of angular velocity, α = Δω/Δt, measured in radians per second squared (rad/s²).
• Relationship to linear acceleration: a_t = rα (tangential acceleration). Note that there's also a radial (centripetal) acceleration, a_c = v²/r = ω²r.

D. Equations of Motion (Analogous to Linear Motion)

These equations assume constant angular acceleration:

• θ = ω_it + ½αt²
• ω_f = ω_i + αt
• ω_f² = ω_i² + 2αθ

Where:

• ω_i = initial angular velocity
• ω_f = final angular velocity
• α = angular acceleration
• t = time
• θ = angular displacement

II. Dynamics of Rotation

This section explores the forces and torques that cause rotational motion.

A. Torque (τ)

• Definition: The rotational analogue of force. It's the tendency of a force to cause rotation. τ = rFsinθ, where:
  • r is the distance from the pivot point to the point where the force is applied.
  • F is the magnitude of the force.
  • θ is the angle between the force vector and the lever arm (the vector from the pivot point to the point of force application).
• Units: Newton-meters (Nm)

B. Moment of Inertia (I)

• Definition: A measure of an object's resistance to changes in its rotational motion. It depends on the object's mass distribution and the axis of rotation.
• Formulas: Different shapes have different formulas for I. You should memorize the moment of inertia for common shapes like:
  • Solid cylinder/disk: I = ½MR²
  • Hollow cylinder/hoop: I = MR²
  • Solid sphere: I = (2/5)MR²
  • Hollow sphere: I = (2/3)MR²
  • Rod (about its center): I = (1/12)ML²
  • Rod (about its end): I = (1/3)ML²
  • Point mass: I = mr²

C. Newton's Second Law for Rotation

• Equation: Στ = Iα. The net torque acting on an object is equal to its moment of inertia times its angular acceleration.

D. Rotational Kinetic Energy (K_rot)

• Equation: K_rot = ½Iω²

E. Work-Energy Theorem for Rotation

• Equation: W = ΔK_rot = ½Iω_f² - ½Iω_i²

III. Angular Momentum (L)

• Definition: The rotational analogue of linear momentum. It's a measure of how difficult it is to stop a rotating object.
• Equation: L = Iω
• Conservation of Angular Momentum: In the absence of external torques, the total angular momentum of a system remains constant. This is a crucial concept for many problems. If I decreases, ω must increase to conserve L, and vice-versa.

IV. Rolling Motion

This section covers objects that are both rotating and translating.

• Relationship between linear and angular velocity: v_cm = ωR (for rolling without slipping)
• Total Kinetic Energy: K_total = K_trans + K_rot = ½Mv_cm² + ½Iω²

This cheat sheet provides a concise summary. Make sure to consult your textbook and practice problems to fully grasp these concepts. Good luck with your AP Physics C exam!