Angular Momentum (L

Classical Mechanics: Rotational Kinematics & Angular Momentum

Rotational motion is the direct angular analogue to linear Newtonian translation:

• Linear vs. Rotational Analogues:

• Mass m Mass Moment of Inertia \

• Velocity v Angular Velocity \

• Linear Momentum \ Angular Momentum \

• Force \ Torque \

• Kinetic Energy (1) / (2)mv^2 Rotational Energy (1) / (2)Iω^2

• Conservation of Angular Momentum: In the absence of external net torque), total angular momentum is strictly conserved). When a spinning ice skater pulls their arms inward, their moment of inertia I decreases, causing rotational speed ω to dramatically increase.

• Flywheel Energy Storage Systems (FESS): High-speed flywheels utilize high angular momentum and energy density (E ω^2) to provide instantaneous grid frequency regulation and regenerative braking on hybrid powertrains.

How to Use This Calculator

Enter your Rotating Body Geometry & Moment of Inertia (I) — choose from Solid Cylinder / Flywheel Disc (I = ½ M R²), Thin Cylindrical Shell / Rim (I = M R²), Solid Sphere (I = ⅖ M R²), Rotating Body Mass (M in kg) (e.g. 15 (15 kg flywheel)), Radius / Half-Length (R or L in meters) (e.g. 0.25 (250 mm radius)), Rotational Velocity (RPM - Revolutions per Minute) (e.g. 1,800 RPM (Standard motor speed)), and Angular Acceleration (α in rad/s²) (e.g. 12.0). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Rotational Dynamics \& Angular Momentum equations:

Angular Velocity: ω = (2 π × RPM) / (60) (rad/s)

Angular Momentum: L = I · ω (kg·m^2/s = J·s)

Newton's 2nd Law for Rotation (Torque): τ = I · α = (dL) / (dt) (N·m)

Rotational Kinetic Energy: K_rot = (1) / (2) I ω^2 (Joules)

Moments of Inertia: cases Solid Disc: I = (1) / (2) M R^2 \\ Hollow Ring: I = M R^2 \\ Solid Sphere: I = (2) / (5) M R^2 \\ Rod Center: I = (1) / (12) M L^2 cases

Worked Example

For example, evaluating a solid steel flywheel disc of mass \ and radius \ spinning at 1,800 RPM with an angular acceleration Mass moment of inertia is 0.25^2) = 0.46875 kg·m^

• Angular velocity is . Angular momentum is . Net required torque is . Stored kinetic energy is 188.50^2) = 8,327.6 Joules (8.33 kJ) with outer rim speed .

Understanding Your Result

Automotive benchmarks for reference: engine displacement affects power density (target 100+ hp/liter for performance), power-to-weight ratio above 0.1 hp/lb is considered sporty. Tire sizing: the middle number (e.g., 60 in 225/60R17) is the aspect ratio — lower numbers indicate sportier, wider tires with less cushioning. Braking distance from 60 mph: average car 120–140 ft on dry pavement; wet roads add 40–60%. Fuel efficiency varies significantly by driving style — aggressive acceleration and braking can reduce MPG by 15–30%.