Acceleration - Angular Converter

Radian/Square Second (rad/s²)

The SI unit of angular acceleration. It represents the rate of change of angular velocity. α = dω/dt.

Common uses: International standards, physics, engineering, rotational dynamics calculations.

Revolution/Square Second (rev/s²)

Angular acceleration in revolutions per second squared. 1 rev/s² ≈ 6.283 rad/s². Used for high-speed rotating equipment.

Common uses: Turbine acceleration, high-speed machinery, rotating system analysis.

Revolution/Minute/Second (rev/min/s)

Change in RPM per second. 1 rev/min/s ≈ 0.1047 rad/s². Commonly used in motor and engine specifications.

Common uses: Motor acceleration specs, engine acceleration, industrial equipment startup time.

Radian/Square Minute (rad/min²)

Angular acceleration using minutes. 1 rad/min² = 0.000278 rad/s². Used for very slow rotations.

Common uses: Astronomical calculations, slow mechanical systems, long-duration rotations.

Revolution/Square Minute (rev/min²)

Change in revolutions per square minute. 1 rev/min² ≈ 0.01745 rad/s². Used for very slow rotating systems.

Application: Long-term acceleration analysis, planetary motion, slow machinery.

Angular Acceleration Fundamentals

Key equations for angular motion with acceleration:

• Definition: α = dω/dt (change in angular velocity / time)
• Rotational motion analogy: τ = I × α (torque = moment of inertia × acceleration)
• Linear relationship: a = α × r (linear acceleration = angular × radius)
• Constant acceleration: ω = ω₀ + α×t

Typical Angular Acceleration Values

• Typical electric motor startup: 5-50 rad/s² (0.5-5 m/s² at 1m radius)
• Rapid motor startup: 50-500 rad/s² (high-torque motors)
• Vehicle acceleration (wheel angular): 5-20 rad/s² at wheel
• Centrifuge acceleration: 100-1000 rad/s² (very fast acceleration)
• Gentle machinery: 0.1-1 rad/s² (gradual acceleration)
• Turbine ramp-up: 1-10 rad/s² (controlled acceleration)
• Flywheel spin-up: 10-100 rad/s² (fast energy storage)

Angular Kinematics Equations

For constant angular acceleration:

• Angular velocity: ω = ω₀ + α×t
• Angular displacement: θ = ω₀×t + ½×α×t²
• Final angular velocity squared: ω² = ω₀² + 2×α×θ
• Average angular velocity: ω_avg = (ω₀ + ω) / 2

Torque and Angular Acceleration (Newton's Second Law for Rotation)

The fundamental relationship in rotational dynamics:

• Linear motion: F = m × a
• Rotational motion: τ = I × α
• Where: τ = torque (N·m), I = moment of inertia (kg·m²), α = angular acceleration (rad/s²)
• Relationship: Greater torque or smaller moment of inertia produces greater angular acceleration

Power in Rotational Systems

Power relates to torque and angular velocity:

• Power: P = τ × ω
• During angular acceleration: P = τ × ω = I × α × ω
• Rotational kinetic energy: KE = ½ × I × ω²
• Work-energy relationship: W = ΔKE = ½ × I × (ω² - ω₀²)

Common Applications

Angular acceleration calculations are essential in:

• Motor Design: Starting torque, acceleration time, startup profiles
• Vehicle Performance: Acceleration from standstill, gear change dynamics
• Rotating Equipment: Spindles, turbines, fans, compressors startup analysis
• Flywheel Systems: Energy storage spin-up time calculations
• Robotics: Joint accelerations, motion planning, torque requirements
• Mechanical Design: Shaft loading, bearing design for accelerated rotation
• Control Systems: Acceleration limiting, soft starting, speed ramps

Relationship between Angular and Linear Acceleration

For motion on a circular path:

• Tangential acceleration: a_t = α × r (along the path)
• Centripetal acceleration: a_c = ω² × r (toward center)
• Total acceleration: a_total = √(a_t² + a_c²)
• Example: A wheel accelerating at 10 rad/s² with radius 0.3 m has tangential acceleration of 3 m/s²