Physics: Rotational Kinematics & Dynamics

Rotational Kinematics

• Angular displacement (θ): The angle through which a body rotates. Measured in radians (rad).
• Angular velocity (ω): The rate of change of angular displacement. Measured in radians per second (rad/s).
• Angular acceleration (α): The rate of change of angular velocity. Measured in radians per second squared (rad/s²).
• Equations for constant angular acceleration:
  • ωf = ωi + αt
  • θ = ωit + ½ αt²
  • ωf² = ωi² + 2αθ

Rotational Dynamics

• Torque (τ): The rotational equivalent of force. Calculated as τ = rFsinθ, where r is the lever arm, F is the force, and θ is the angle between the force and lever arm. Measured in Newton-meters (Nm).
• Moment of Inertia (I): A measure of a body's resistance to rotational acceleration. Depends on mass distribution relative to the axis of rotation. Measured in kg⋅m².
• Newton's Second Law for Rotation: τ = Iα. This relates torque, moment of inertia, and angular acceleration.

Rotational Energy

• Rotational kinetic energy (KE_rot): The energy associated with rotation. Calculated as KE_rot = ½ Iω². Analogous to linear kinetic energy.

Relationships Between Linear and Rotational Quantities

• Connecting linear and rotational quantities for a point on a rotating object:
  • v = rω
  • a_t = rα (tangential acceleration)
  • a_c = v²/r = ω²r (centripetal acceleration)
• These equations apply to points on a rotating body; linear motion (velocity, tangential acceleration, centripetal acceleration) depends on distance from the axis and angular motion.

Different Moment of Inertia Calculations

• Different shapes have different formulas for calculating moment of inertia:
  • Point mass: I = mr²
  • Solid cylinder (central axis): I = ½mr²
  • Solid sphere (central axis): I = (2/5)mr²
  • Hollow cylinder (central axis): I = mr²
  • Hoop (central axis): I = mr²

Angular Momentum

• Angular momentum (L): A measure of rotational inertia. Calculated as L = Iω. Often associated with conservation of angular momentum in isolated systems. Measured in kg⋅m²/s.
• Conservation of Angular Momentum: In the absence of external torques, the angular momentum of a system remains constant.

Applications of Rotational Motion Principles

• Engineering and physics applications:
  • Designing machinery (gears, pulleys).
  • Understanding planetary/satellite motion.
  • Analyzing everyday rotating objects (amusement park rides, sports).