Angular Velocity Converter

Quick Reference Table

From	To	Multiply By	Example
RPM	Rad/s	0.10472	60 RPM = 6.28 rad/s
RPM	Deg/s	6	100 RPM = 600 deg/s
RPM	Hz	0.01667	60 RPM = 1 Hz
Rad/s	RPM	9.5493	1 rad/s = 9.55 RPM
Rad/s	Deg/s	57.2958	1 rad/s = 57.3 deg/s
Deg/s	RPM	0.16667	360 deg/s = 60 RPM
Hz	RPM	60	1 Hz = 60 RPM
Hz	Rad/s	6.28319	1 Hz = 6.28 rad/s

Common Angular Velocities

Object/Device	RPM	Rad/s	Hz
Earth's rotation	0.000694	0.0000727	0.0000116
Clock second hand	1	0.105	0.0167
Clock minute hand	0.0167	0.00175	0.000278
Electric power (60 Hz)	3,600	377	60
Car engine (idle)	600-1,000	63-105	10-17
Car engine (highway)	2,000-3,000	209-314	33-50
Car wheel (60 mph)	~800	~84	~13
Hard drive (7200 RPM)	7,200	754	120
Washing machine spin	800-1,600	84-168	13-27
Turbine engine	10,000-30,000	1,047-3,142	167-500
Dentist drill	~400,000	~41,888	~6,667
Centrifuge (lab)	3,000-20,000	314-2,094	50-333

Understanding Angular Velocity Units

Revolutions Per Minute (RPM)

Definition: The number of complete rotations (360-degree turns) completed in one minute. One RPM equals one full revolution every 60 seconds, or 0.10472 radians per second.

History: Emerged with the development of rotating machinery during the Industrial Revolution. James Watt used RPM measurements for steam engines. Became standardized with tachometers in the late 19th century.

Current Use: Universal standard for rotating machinery: car engines (idle: 600-1,000 RPM, redline: 5,000-8,000 RPM), motors, turbines, hard drives (5,400-15,000 RPM), washing machines, fans, and pumps. Tachometers in vehicles display engine speed in RPM. CD-ROM drives originally rated by maximum RPM (up to 52x = ~10,400 RPM).

Radians Per Second (rad/s)

Definition: The SI unit of angular velocity. Measures the angle swept per second in radians (one complete rotation = 2π radians ≈ 6.28 radians). One radian is approximately 57.3 degrees.

History: Based on the radian, a natural mathematical unit defined by the radius of a circle. Became standard in physics and engineering because it simplifies rotational motion equations and calculus.

Current Use: Primary unit in physics, engineering, robotics, and scientific calculations. Essential for rotational dynamics, angular momentum, and torque calculations. Used in control systems, gyroscopes, aerospace engineering, and anywhere precise mathematical analysis of rotation is needed. Preferred in academic and research contexts.

Degrees Per Second (deg/s or °/s)

Definition: The angular velocity measured in degrees swept per second. One complete rotation = 360 degrees per second. One degree per second equals 0.01745 radians per second.

History: Based on the Babylonian division of circles into 360 degrees. More intuitive than radians for non-technical applications because people commonly understand degrees.

Current Use: Used in navigation, astronomy, aviation (aircraft turn rates), cameras (panning speed), and consumer applications where degrees are more familiar than radians. Gimbal systems and camera stabilizers specify rotation rates in degrees per second. Video game rotation speeds often expressed in deg/s.

Hertz (Hz)

Definition: Frequency unit measuring complete cycles per second. For rotation, 1 Hz = 1 revolution per second = 60 RPM = 2π radians per second. Named after Heinrich Hertz.

History: Originally used for electromagnetic waves (radio frequencies). Extended to any periodic phenomenon including rotation. Officially adopted as SI unit of frequency in 1960.

Current Use: Electrical systems (AC power: 50 Hz Europe, 60 Hz Americas), processor speeds (formerly megahertz/gigahertz), refresh rates (monitors: 60-240 Hz), sound (440 Hz = A note), and vibration analysis. Sometimes used for very high-speed rotation (turbines, centrifuges). Power grid frequency directly related to generator rotation speed.

Revolutions Per Second (rev/s or RPS)

Definition: Complete rotations per second. Identical to Hertz when applied to rotation. 1 rev/s = 1 Hz = 60 RPM.

History: Natural extension of RPM for faster rotations where per-minute is less convenient.

Current Use: High-speed machinery, turbines, centrifuges, and physics problems. Less common than RPM or Hz but occasionally used in technical specifications. Useful for very fast rotations where RPM numbers become unwieldy (e.g., 120,000 RPM = 2,000 rev/s).

Degrees Per Minute (deg/min)

Definition: Angular displacement in degrees per minute. 360 deg/min = 1 RPM.

History: Used in contexts where slow, precise rotations are specified.

Current Use: Astronomy (telescope tracking of stars due to Earth's rotation: ~15 deg/hour = 0.25 deg/min), slow-turning equipment, and scientific instruments requiring precise angular positioning.

Revolutions Per Hour (RPH)

Definition: Complete rotations per hour. 1 RPH = 1/60 RPM = 0.01667 RPM.

History: Used for extremely slow rotations.

Current Use: Rare; used for very slow processes like rotisserie cooking, display turntables, astronomical observations (Earth's rotation is ~1 RPH), and long-period mechanical timers.

Frequently Asked Questions

How do I convert RPM to rad/s?

Multiply RPM by 0.10472 (or π/30). Formula: rad/s = RPM × (2π/60) = RPM × 0.10472. Example: 100 RPM × 0.10472 = 10.47 rad/s. This is because one revolution = 2π radians, and one minute = 60 seconds. To convert rad/s to RPM, multiply by 9.5493 (or 30/π).

What is a good RPM for a car engine?

Depends on context: Idle: 600-1,000 RPM. City driving: 1,500-2,500 RPM. Highway cruising: 2,000-3,000 RPM. Peak power: 5,000-7,000 RPM (varies by engine). Redline (maximum safe): 5,500-8,500 RPM (gasoline), 4,000-5,500 RPM (diesel). Modern engines with efficient gearing can cruise at lower RPM for fuel economy.

Why are radians used instead of degrees?

Radians are mathematically natural and simplify calculus equations. In radians, the derivative of sin(x) is simply cos(x). With degrees, extra conversion factors appear. Radians make angular velocity, acceleration, and rotational dynamics equations cleaner. Arc length = radius × angle (in radians), which is simple and direct. Physics and engineering prefer radians for mathematical elegance.

How many RPM is 60 Hz?

60 Hz = 3,600 RPM. Calculation: 60 cycles per second × 60 seconds per minute = 3,600 revolutions per minute. This is why AC synchronous motors on 60 Hz power naturally run at multiples of 3,600 RPM (or divisors: 1,800 RPM for 2-pole, 1,200 RPM for 4-pole, etc.). European 50 Hz power yields 3,000 RPM base speed.

What RPM should my hard drive be?

Desktop HDDs: 7,200 RPM standard for performance, 5,400 RPM for quiet/cool operation. Laptop HDDs: 5,400 RPM typical (balance of speed, heat, power). High-performance/servers: 10,000-15,000 RPM. SSDs have no moving parts (0 RPM). Higher RPM = faster data access but more heat/noise. Modern SSDs make HDD RPM less relevant.

How do you measure RPM?

Methods: Tachometer (contact or non-contact laser/optical). Strobe light (flash synchronized with rotation). Hall effect sensor (magnetic pickup). Optical sensor (reflective tape). Smartphone apps (using camera to detect rotation). Acoustic analysis (sound frequency). Engine vehicles have built-in tachometers displaying RPM on dashboard.

What's centrifugal force at different RPMs?

Centrifugal acceleration = ω²r = (2πRPM/60)² × radius. Example at 1 meter radius: 60 RPM = 4 g's, 100 RPM = 11 g's, 1,000 RPM = 1,118 g's, 10,000 RPM = 111,847 g's. This is why high-speed centrifuges can separate materials and why engine components at high RPM experience enormous stress.

Can angular velocity be negative?

Yes. Positive indicates counterclockwise rotation (by convention in physics); negative indicates clockwise rotation. Direction matters for physics calculations involving vectors. In practical applications like engines, RPM is usually given as positive absolute value. Angular acceleration can also be negative (deceleration).

Common Uses for Angular Velocity Conversion

• Automotive: Understanding engine RPM and optimal shift points
• Mechanical Engineering: Motor and machinery specifications
• Physics Education: Rotational dynamics problems and calculations
• Robotics: Servo motor and wheel speed control
• Manufacturing: CNC machine spindle speeds and tool selection
• Aerospace: Turbine and propeller rotation rates
• Electronics: Fan speeds and cooling system specifications
• Science Labs: Centrifuge and mixer settings

Angular Velocity Conversion Tips

1. Remember key ratios: 60 RPM = 1 Hz = 6.28 rad/s = 360 deg/s
2. Quick RPM to rad/s: Divide RPM by 9.55 (or multiply by 0.105)
3. Quick RPM to deg/s: Multiply RPM by 6
4. Hz to RPM: Simply multiply by 60
5. Use radians for math: Physics equations require rad/s, not RPM
6. Watch units in formulas: Mixing RPM and rad/s causes errors
7. Consider direction: Clockwise vs counterclockwise matters in physics

Interesting Angular Velocity Facts

• Earth's rotation: 0.000694 RPM (one rotation every 24 hours)
• Saturn's rings: Inner rings ~1,800 RPM equivalent orbital velocity
• Pulsar (neutron star): Up to 43,000 RPM (716 Hz)
• Formula 1 engines: Peak ~18,000 RPM (300 Hz)
• Top fuel dragster: 8,000-9,000 RPM peak
• Jet engine turbine: 10,000-30,000 RPM
• Dentist drill: ~400,000 RPM (6,667 Hz)
• Gas turbine: Up to 100,000 RPM in some designs
• Ultracentrifuge (lab): Up to 150,000 RPM (2,500 Hz)
• Flywheel energy storage: 20,000-50,000 RPM
• Helicopter main rotor: 200-500 RPM typically
• Wind turbine: 10-20 RPM (blade tip speed is linear velocity)

Angular Velocity in Different Applications

Automotive Engines

Gasoline engines: Idle 600-1,000 RPM, peak power 5,000-7,500 RPM, redline 6,000-9,000 RPM. Diesel engines: Lower RPM range, peak power 3,000-4,500 RPM, redline 4,000-5,500 RPM. Motorcycle engines: Can exceed 15,000 RPM. Electric motors: Can achieve 10,000-20,000 RPM, instant torque at low RPM.

Computer Hardware

Hard drives: 5,400, 7,200, 10,000, 15,000 RPM standard speeds. Cooling fans: 800-3,000 RPM typical. Optical drives: CD 200-500 RPM (52x), DVD up to 10,000 RPM, Blu-ray lower for stability. Floppy drives: 300-360 RPM.

Home Appliances

Washing machine wash: 50-100 RPM. Spin cycle: 800-1,600 RPM. Blender: 10,000-30,000 RPM. Food processor: 1,500-3,000 RPM. Ceiling fan: 50-300 RPM. Microwave turntable: 5-6 RPM.

Industrial Equipment

CNC spindle: 5,000-40,000 RPM. Lathe: 50-4,000 RPM. Drill press: 250-3,000 RPM. Grinder: 3,000-10,000 RPM. Centrifugal pump: 1,500-3,600 RPM. Conveyor rollers: 20-100 RPM.

Power Generation

Steam turbine: 3,000-3,600 RPM (50/60 Hz). Gas turbine: 3,000-10,000 RPM. Hydro turbine: 80-600 RPM (depends on head/flow). Wind turbine: 10-20 RPM blades, gearbox increases to 1,500-1,800 RPM generator. Nuclear reactor coolant pumps: 1,000-1,800 RPM.

Relationship Between Angular and Linear Velocity

Linear velocity (v) at a radius (r) from rotation axis relates to angular velocity (ω) by: v = ω × r (ω must be in rad/s).

Examples:

• Car wheel: 26-inch diameter (~0.33 m radius) at 800 RPM = 800 × 0.105 × 0.33 = 27.7 m/s ≈ 62 mph
• Hard drive platter: 3.5-inch radius (~0.044 m) at 7,200 RPM = 7,200 × 0.105 × 0.044 = 33.3 m/s
• Centrifuge tube: 0.15 m radius at 10,000 RPM = 10,000 × 0.105 × 0.15 = 157.5 m/s (352 mph!)
• Earth's equator: 6,378 km radius at 0.000694 RPM = 465 m/s (1,040 mph)

This explains why objects farther from rotation center move faster despite same RPM, and why high-RPM rotors experience extreme forces.

Safety Considerations at High RPM

• Over-revving engines: Can cause valve float, connecting rod failure, catastrophic damage
• Imbalanced rotation: Even small imbalance creates huge forces at high RPM
• Centrifugal forces: Increase with square of RPM - doubling RPM quadruples force
• Gyroscopic effects: High-RPM rotors resist orientation changes
• Vibration: Resonant frequencies can cause failure
• Material stress: Components must withstand enormous forces
• Grinding wheels: Must not exceed rated RPM - can explode
• Centrifuges: Proper balancing critical to prevent destruction