Rotational Dynamics & Moment of Inertia

• Determine center of mass for continuous mass distributions using integration.
• Calculate moments of inertia using Parallel and Perpendicular Axis Theorems.
• Apply rotational analogue of Newton's second law: τ = Iα.
• Formulate equations for pure rolling motion on flat and inclined surfaces.

• Center of Mass Coordinates (R_cm = ∫ r dm / M)
• Torque Vector (τ = r x F) & Angular Momentum (L = r x p = Iω)
• Moment of Inertia Theorems (Parallel & Perpendicular)
• Conservation of Angular Momentum (τ_ext = 0 ⇒ L = constant)
• Kinetic Energy in Pure Rolling (K = 1/2 M v_cm^2 + 1/2 I_cm ω^2)

“Rotational dynamics mirrors linear dynamics: mass corresponds to moment of inertia, velocity to angular velocity, and force to torque.”

• •In pure rolling without slipping, instantaneous velocity of the contact point is zero: v_cm = Rω.
• •Perpendicular axis theorem applies strictly to planar (2D) laminar bodies.
• •Conservation of angular momentum explains planetary orbital speed variations.

Advanced problem collection on moment of inertia derivations, rolling on incline, and angular collision.