Rotatiedynamica

Rotatiedynamica, commonly translated as rotational dynamics, is the study of how objects rotate and how their rotational motion responds to torques. It complements translational dynamics and is essential for understanding gears, wheels, satellites, and many mechanical and physical systems. The main quantities are angular position θ, angular velocity ω, and angular acceleration α, defined as derivatives of θ with respect to time. Orientation is described by an angular velocity vector that can change in magnitude and direction.

Dynamics are governed by the relation τ = dL/dt, where τ is the net external torque and L is the angular momentum. For a rigid body, L = I · ω, with I the inertia tensor that encodes how mass is distributed relative to the rotation axes. In a coordinate system aligned with principal axes, I is diagonal and L = I ω; in general I is a symmetric tensor. The inertia about a given axis is I = ∑ m_i r_i^2 for point masses, or I = ∫ r^2 dm for continuous bodies, and the moment of inertia about different axes relates via the parallel axis theorem, I = I_cm + m d^2.

Rotational kinetic energy is KE_rot = 1/2 ω · L = 1/2 ω^T I ω. In the absence of external torques, angular momentum is conserved. Real systems often involve coupling between rotation and translation (rolling without slipping) or gyroscopic effects that cause precession.

Applications range from machine design and vehicle dynamics to astronomy and sports, and analysis typically uses conservation laws, tensor algebra, and sometimes numerical simulation.