Angular Velocity Calculator

How to use this calculator

Three modes are available. Select the one that matches what you know.

From Revolutions and Time: Enter the number of complete revolutions and the time taken. The calculator computes ω = 2π × N / t.

From RPM: Enter revolutions per minute. The calculator converts: ω = RPM × 2π / 60.

From Period: Enter the time for one full revolution (in seconds). The calculator uses ω = 2π / T.

Radius (m): Enter the radius of rotation for any point you care about. This is used to compute the tangential velocity v = ωr. Leave at any value; only the radius field affects the tangential velocity output.

Click Calculate to get angular velocity in rad/s, RPM, tangential velocity, frequency in Hz, and period in seconds.

Angular velocity as a vector

Angular velocity ω is technically a vector, not just a number. The direction of the vector is along the rotation axis, determined by the right-hand rule: if you curl the fingers of your right hand in the direction of rotation, your extended thumb points along ω.

For a spinning top rotating counterclockwise when viewed from above, ω points upward. For a wheel rolling to the right (rotating clockwise when viewed from the right), ω points to the left (into the page from the driver’s view).

The scalar magnitude of this vector is angular speed, always non-negative. Angular velocity can be negative if you define a positive direction: clockwise vs counterclockwise around a chosen axis.

In 3D mechanics, angular velocity is written as a vector ω = (ωₓ, ω_y, ω_z). The velocity of any point P at position r from the rotation axis is:

For simple 2D rotation, this reduces to v = ωr with the tangential direction perpendicular to r.

The unit rad/s and its definition

The radian is the natural unit of angle in mathematics and physics. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius. Because the circumference is 2πr, one full revolution equals 2π radians ≈ 6.2832 rad.

Angular velocity in rad/s is defined as:

For constant angular velocity, this simplifies to ω = Δθ / Δt. For variable angular velocity, you need the instantaneous rate, which is the derivative.

Since radians are dimensionless (length / length), rad/s is equivalent to s⁻¹ in dimensional analysis. This is why ω appears without explicit units in formulas like v = ωr (the units work out as (s⁻¹)(m) = m/s).

Converting between all rotational units

Four related quantities describe rotation rate. This table converts between them.

Unit	Symbol	Conversion to rad/s
rad/s	ω	1
Revolutions per second (Hz)	f	multiply by 2π
Revolutions per minute (RPM)	n	multiply by 2π/60 ≈ 0.10472
Period (s/rev)	T	2π / T
Degrees per second	°/s	multiply by π/180 ≈ 0.017453

Key formulas:

Tangential velocity and the v = ωr relationship

Every point on a rigid rotating body shares the same angular velocity ω. But points at different distances from the axis move at different tangential (linear) speeds:

This is why a merry-go-round is more exciting at the outer edge: the same ω produces higher tangential speed farther from the center.

The connecting insight: angular velocity is a property of the whole rotating body; tangential velocity is a property of a specific point at a specific radius.

Applications of v = ωr:

• Belt and pulley systems: two pulleys connected by a belt have the same belt speed v. So ω_small × r_small = ω_large × r_large. A smaller pulley spins faster than a larger one connected to the same belt.
• Gear trains: meshing gears have the same tangential speed at the contact point. ω₁r₁ = ω₂r₂. Larger gears rotate more slowly.
• Wheels and rolling: a wheel rolling without slip has v_center = ω × r_wheel. The contact point has v = 0; the top has v = 2ωr.

Real-world angular velocities

Object	RPM	rad/s	Notes
Earth’s rotation	0.000167	7.27 × 10⁻⁵	1 revolution per sidereal day
Hour hand of clock	0.00278	2.91 × 10⁻⁴
Minute hand of clock	0.0167	1.75 × 10⁻³
Second hand of clock	1	0.1047
Vinyl record (33⅓ RPM)	33.3	3.49
Car engine (idle)	700-900	73-94
Car engine (cruise)	2000-3000	209-314
Car engine (redline)	6000-8000	628-838
Hard drive (standard)	7200	754
Dentist drill	200,000-400,000	20,900-41,900
Neutron star (pulsar)	43,000+	4,500+	Fastest observed: 716 Hz

Angular velocity in engineering

Electric motors

Motor performance is defined by the speed-torque curve, which plots torque against ω (or RPM). At synchronous speed (ω_s = 2πf/p, where f is mains frequency and p is pole pairs), an induction motor produces no torque. Maximum torque (pull-out torque) occurs at a specific slip. Efficiency and power factor both depend on operating ω.

Variable frequency drives (VFDs) control motor ω by changing the supply frequency f. Since ω_s = 2πf/p, doubling the supply frequency doubles the synchronous speed.

Turbines

Steam and gas turbines run at fixed multiples of grid frequency: 3000 RPM (50 Hz grids, 2-pole generators) or 3600 RPM (60 Hz grids). Wind turbines have variable ω and use power electronics to synchronize to the grid.

Gear systems

Gear ratios are defined as the ratio of output to input angular velocity. A 3:1 reduction gearbox has ω_out = ω_in / 3 but torque_out = torque_in × 3 × efficiency. The product ω × τ = power is conserved (minus losses).

Angular momentum and its conservation

Angular momentum is:

where I is the moment of inertia (kg·m²). For a point mass m at radius r: I = mr². For a uniform solid cylinder: I = ½mr².

Conservation of angular momentum: in the absence of external torques, L remains constant. If I decreases, ω increases, and vice versa.

This explains the figure skater’s spin: pulling arms in reduces I, so ω increases to keep L constant. A spinning neutron star rotating at 716 Hz formed when a stellar core with much larger I collapsed to a small radius, spinning up enormously as I decreased.

Measuring angular velocity

Tachometers measure RPM directly. Optical tachometers shine a light beam on a reflective mark on the shaft and count flashes per second. Contact tachometers use a friction tip pressed against the shaft.

Encoders (incremental): A slotted disc generates pulses. A microcontroller measures pulse frequency to compute ω. Resolution depends on slot count: a 1000-slot disc resolves 360/1000 = 0.36° per count.

Encoders (absolute): These output the current angle position, not just pulses. Differentiate the angle signal to get ω.

Hall-effect sensors: Used in brushless motors. Permanent magnets on the rotor trigger Hall sensors as they pass. The commutation controller uses this to know rotor position, and ω is computed from the pulse rate.

MEMS gyroscopes: Phones and drones use micro-fabricated vibrating structures. The Coriolis force on the vibrating mass shifts when the device rotates, and the shift is proportional to ω. Typical range: ±2000 °/s. Output is usually in °/s, so multiply by π/180 to get rad/s.

Converting between angular velocity units

Angular velocity appears in four common units depending on the context: rad/s (SI), RPM, Hz, and degrees/second. Converting between them is straightforward once you know the relationships.

Angular velocity in engineering reference values:

Application	RPM	rad/s
Earth’s rotation	0.000278	7.27 × 10⁻⁵
Clock minute hand	0.1	0.00175
Car engine idle	800	83.8
Car engine highway	2500	261.8
Hard drive (7200 RPM)	7200	754.0
Dentist drill	400,000	41,888
Gas turbine	20,000	2094

Angular velocity and gear systems

Gear systems convert angular velocity while trading off torque. When two meshing gears have different numbers of teeth, their angular velocities are inversely proportional to their tooth counts:

where N₁ and N₂ are the tooth counts (or radii) of gear 1 and gear 2. A gear with 10 teeth driving a gear with 40 teeth reduces angular velocity by a factor of 4, while multiplying torque by 4.

This is why transmissions in vehicles have multiple gear ratios: at low gears, the engine angular velocity translates to low wheel angular velocity (high torque for acceleration), while at high gears, the same engine speed produces higher wheel angular velocity (for highway speeds).

The product of angular velocity and torque equals power:

An ideal gear system (no friction losses) transmits the same power regardless of the gear ratio, just trading angular velocity for torque.