Velocity - Angular Converter

Complete Guide to Angular Velocity Conversion

Angular velocity conversion is fundamental to mechanical engineering, physics, robotics, automotive design, industrial automation, and any application involving rotational motion. Whether you're calculating motor speeds, analyzing rotating machinery, designing gears, programming robotics controllers, or solving physics problems, understanding how to convert between different angular velocity units like radian per second (rad/s), degree per second (°/s), degree per minute (°/min), degree per hour (°/h), degree per day (°/d), revolution per second (r/s), revolution per minute (rpm or r/min), revolution per hour (r/h), and revolution per day (r/d) enables accurate calculations and proper equipment specification. This comprehensive guide explains angular velocity concepts, unit relationships, and provides clear conversion methods for all angular velocity measurements.

Understanding Angular Velocity and Rotational Motion

Angular velocity (also called rotational speed or angular speed) measures how fast an object rotates or spins around an axis. Unlike linear velocity, which measures speed in a straight line, angular velocity measures rotation rate. Angular velocity is expressed in units of angle per unit time, such as radians per second (rad/s) or degrees per second (°/s). The faster something rotates, the higher its angular velocity. Angular velocity is crucial in mechanical systems like motors, turbines, gears, wheels, and rotating machinery. In physics, angular velocity (ω) relates to linear velocity (v) through the formula v = ω × r, where r is the radius. Understanding angular velocity conversions enables accurate calculations for rotational dynamics, machinery design, and motion control systems.

Radian per Second (rad/s) - The SI Unit

The radian per second (rad/s) is the SI (International System of Units) derived unit for angular velocity. One radian per second means an object rotates through an angle of 1 radian (approximately 57.3°) every second. Radians are a natural unit for angular measurement because they relate directly to arc length: 1 radian = the angle subtended by an arc length equal to the radius. Radian per second is standard in physics calculations, engineering specifications, and scientific applications because it simplifies mathematical formulas involving angular motion. For example, angular velocity ω = θ/t, where θ is angle in radians and t is time in seconds. Converting from rad/s to other units requires understanding the relationships: 1 revolution = 2π radians = 360°, and time conversions (1 minute = 60 seconds, 1 hour = 3600 seconds, 1 day = 86400 seconds).

Degree per Second, Minute, Hour, and Day

Degree per second (°/s) measures rotation in degrees rather than radians, with 1°/s meaning 1 degree of rotation per second. One radian equals approximately 57.296°, so 1 rad/s = 57.296°/s (or exactly 180/π °/s). Degrees are more intuitive for everyday use but less mathematically convenient. Degree per minute (°/min) equals 1/60 of a degree per second. Degree per hour (°/h) equals 1/3600 of a degree per second, useful for slow rotations like celestial motion or slow-motion mechanisms. Degree per day (°/d) equals 1/86400 of a degree per second, used for very slow rotations like Earth's rotation (approximately 0.00417°/s or 360°/day). Degree-based units are common in navigation, astronomy, and applications where intuitive angle measurement is preferred over mathematical convenience.

Revolution per Second, Minute, Hour, and Day

Revolution per second (r/s or rps) measures complete 360° rotations per second. One revolution equals 2π radians or 360°. So 1 r/s = 2π rad/s ≈ 6.283 rad/s. Revolution per minute (r/min or rpm) is extremely common in engineering and industry, especially for motors, engines, fans, and rotating equipment. One rpm = 1/60 r/s = 2π/60 rad/s ≈ 0.1047 rad/s. RPM is so widely used that motor specifications, gear ratios, and machinery ratings are typically given in rpm. Revolution per hour (r/h) equals 1/3600 r/s, used for very slow rotations. Revolution per day (r/d) equals 1/86400 r/s, used for extremely slow rotations. Converting between revolutions and radians or degrees requires remembering: 1 revolution = 2π radians = 360°.

Time Unit Conversions in Angular Velocity

Angular velocity units include a time component (per second, per minute, per hour, per day). To convert between time units while keeping the angular unit constant, use standard time conversions: 1 minute = 60 seconds, 1 hour = 60 minutes = 3600 seconds, 1 day = 24 hours = 1440 minutes = 86400 seconds. For example, to convert rad/s to rad/min, multiply by 60 (since there are 60 seconds in a minute). To convert rad/s to rad/h, multiply by 3600. To convert rad/s to rad/d, multiply by 86400. The same time conversions apply for degree-based and revolution-based units. Understanding these time relationships enables conversions like: 10 rad/s = 600 rad/min = 36000 rad/h = 864000 rad/d.

Angular Velocity Conversion Formulas

The key conversion relationships are: 1 rad/s = 57.296°/s = 3600°/min = 216000°/h = 5184000°/d. 1 rad/s = 0.1592 r/s = 9.549 r/min = 572.96 r/h = 13750 r/d. 1 r/s = 2π rad/s = 360°/s = 6.283 rad/s. 1 r/min (rpm) = 2π/60 rad/s = π/30 rad/s ≈ 0.1047 rad/s. 1 r/min = 360°/min = 6°/s. To convert degrees to radians: multiply by π/180 (or approximately 0.017453). To convert radians to degrees: multiply by 180/π (or approximately 57.296). To convert revolutions to radians: multiply by 2π. To convert radians to revolutions: divide by 2π. Our converter automatically applies these formulas to provide instant, accurate conversions between any angular velocity units, handling all the mathematical complexity for you.

Practical Applications of Angular Velocity

Angular velocity conversions are essential across numerous industries. In mechanical engineering, engineers convert rpm to rad/s when designing motors, calculating gear ratios, and specifying rotational speeds. In automotive engineering, angular velocity relates to wheel rotation, engine rpm, and transmission calculations. In robotics, angular velocity controls joint rotation, manipulator speed, and motion planning. In physics, angular velocity appears in rotational dynamics, centripetal force calculations, and moment of inertia problems. In astronomy, angular velocity describes planetary rotation, satellite motion, and celestial object spin rates. In industrial automation, angular velocity controls conveyor belts, rotating machinery, and automated processes. In aviation, angular velocity relates to propeller speed and turbine rotation. Understanding angular velocity conversion enables accurate calculations, proper equipment selection, and correct motion control across all these applications.

Angular Velocity in Motors and Machinery

Motor specifications typically provide angular velocity in revolutions per minute (rpm), while physics and engineering calculations often require radians per second (rad/s). Converting between these units is essential for motor selection, gear design, and system integration. For example, a motor rated at 3000 rpm rotates at 3000 × (2π/60) = 314.16 rad/s. When designing gear systems, angular velocity ratios determine gear ratios: if a motor at 3000 rpm drives a gear that must rotate at 1500 rpm, the gear ratio is 2:1. Understanding angular velocity conversions enables proper motor sizing, accurate speed calculations, and correct equipment matching in mechanical systems.

Converting Between Radian and Degree Units

Converting between radian-based and degree-based angular velocity requires the radian-to-degree conversion factor: 1 radian = 180/π degrees ≈ 57.296°. Therefore, 1 rad/s = (180/π)°/s ≈ 57.296°/s. To convert degree-based to radian-based units, divide by (180/π) or multiply by (π/180). For example, 360°/s = 360 × (π/180) = 2π rad/s = 1 r/s. The relationship is: angular velocity in rad/s = (angular velocity in °/s) × (π/180), and angular velocity in °/s = (angular velocity in rad/s) × (180/π). This conversion is crucial when working with mixed unit systems or converting between scientific (radian) and practical (degree) measurements.

RPM to Rad/s Conversion - Most Common Conversion

Converting revolutions per minute (rpm) to radians per second (rad/s) is one of the most common angular velocity conversions. The formula is: rad/s = rpm × (2π/60) = rpm × (π/30) ≈ rpm × 0.1047. For example, 1000 rpm = 1000 × (2π/60) = 1000 × 0.1047 = 104.72 rad/s. To convert in reverse: rpm = rad/s × (60/2π) = rad/s × (30/π) ≈ rad/s × 9.549. For example, 100 rad/s = 100 × 9.549 = 954.9 rpm. This conversion is essential when motor specifications are in rpm but calculations require rad/s, or when working with physics formulas that use radian-based angular velocity.

Angular Velocity vs Linear Velocity Relationship

Angular velocity relates to linear velocity through the radius of rotation: v = ω × r, where v is linear velocity (m/s), ω is angular velocity (rad/s), and r is radius (m). This relationship demonstrates why rad/s is mathematically convenient: when ω is in rad/s, the formula is simple. If ω were in rpm or degrees/s, additional conversion factors would be needed. For example, a wheel with radius 0.5 m rotating at 100 rad/s has a linear velocity at the rim of: v = 100 rad/s × 0.5 m = 50 m/s. Understanding this relationship helps engineers calculate belt speeds, gear tooth velocities, and rotating component linear speeds from angular velocity measurements.