Rotational Motion Calculator

Rotational motion describes the movement of objects around a fixed axis, analogous to linear motion but in circular paths.

Key Concepts:

• Angular Velocity (ω): Rate of change of angular displacement, ω = θ/t
• Angular Acceleration (α): Rate of change of angular velocity, α = Δω/Δt
• Torque (τ): Rotational force, τ = rF
• Moment of Inertia (I): Rotational mass, depends on shape and mass distribution
• Angular Momentum (L): Rotational momentum, L = Iω

Key Formulas:

• ω = θ/t - Angular velocity
• α = Δω/Δt - Angular acceleration
• τ = rF - Torque
• τ = Iα - Newton's second law for rotation
• K = ½Iω² - Rotational kinetic energy
• L = Iω - Angular momentum
• P = τω - Rotational power

Moments of Inertia:

• Point Mass: I = mr²
• Ring: I = mr²
• Solid Disk: I = ½mr²
• Solid Sphere: I = ⅖mr²
• Hollow Sphere: I = ⅔mr²
• Rod (Center): I = ¹⁄₁₂mL²
• Rod (End): I = ⅓mL²
• Solid Cylinder: I = ½mr²

Units:

• Angular velocity: rad/s
• Angular acceleration: rad/s²
• Torque: N·m
• Moment of inertia: kg·m²
• Angular momentum: kg·m²/s
• Rotational power: W

Applications:

• Wheels and gears
• Gyroscopes and spinning tops
• Planetary motion
• Centrifuges
• Flywheels and energy storage