How do I convert RPM to rad/s?

ω (rad/s) = N (RPM) × 2π / 60. So 1800 RPM = 1800 × 2π / 60 = 188.5 rad/s. Going the other way: N (RPM) = ω × 60 / (2π). 1 Hz = 60 RPM = 2π rad/s ≈ 6.283 rad/s. The calculator does all four units in one pass.

What is the formula for pulley RPM?

Pulleys conserve linear belt speed at the rim, so D₁·N₁ = D₂·N₂. A 100 mm driver at 1800 RPM driving a 200 mm driven pulley gives N₂ = 1800 × (100 / 200) = 900 RPM. The smaller pulley always spins faster for the same belt. Subtract 1–2 % for V-belt slip on real installations.

How does a gear train change RPM?

Each gear pair has a ratio T_driver / T_driven. Multi-stage compound trains multiply the ratios. Example: 20T → 60T (ratio 1/3) followed by 15T → 45T (ratio 1/3) gives total ratio 1/9 — input 1750 RPM becomes output 194 RPM, a 9:1 reduction. Tooth counts must be integers; gear stages use real tooth numbers, not simulated.

How do I calculate spindle RPM for a target cutting speed?

N (RPM) = (v × 1000) / (π × D) with v in m/min and D in mm. Or in SFM: N = (v_sfm × 12) / (π × D_in). A 10 mm carbide endmill at 200 m/min cutting speed needs N = (200 × 1000) / (π × 10) ≈ 6 366 RPM. Always confirm v against the tool manufacturer — values vary 5× between materials.

What does ω·r tell me?

It's the linear velocity of any point on the rotating body — tyre tread on a wheel, a lathe workpiece at the cutter, the rim of a pulley. v = ω·r. A 30 cm-diameter tyre at 1000 RPM has v = (1000·2π/60) × 0.15 = 15.7 m/s = 56 km/h. Doubling RPM doubles linear speed; doubling radius doubles linear speed.

What is centripetal acceleration?

It's the inward radial acceleration of a rotating point: a = ω²·r = v²/r. A 1 m-radius rotor at 3000 RPM has a = (3000·2π/60)² × 1 ≈ 98 700 m/s² ≈ 10 060 g — enough to fail a poorly-balanced rotor. Centrifugal force on a mass m is F = m·ω²·r. Doubling RPM quadruples the stress.

How is RPM related to motor frequency?

Synchronous speed of an AC motor is N_s (RPM) = 120 · f / P, where f is line frequency (60 Hz in North America, 50 Hz in Europe / Australia / Asia) and P is the number of poles. A 4-pole motor on 60 Hz runs at 1800 RPM synchronous (1750 RPM under load due to slip). 2-pole motors hit 3600 / 3000 RPM; 6-pole motors 1200 / 1000 RPM.

Calculator · Mechanical · Machinery's Handbook · Shigley

RPM calculator

Five solve modes for rotational speed: unit conversion (RPM ↔ rad/s ↔ Hz ↔ deg/s), pulley / belt ratio with optional slip, multi-stage gear train, cutting / surface speed (SFM and m/min), and linear velocity of a rotating point with centripetal acceleration. PDF report attached to each calculation. Reviewed by a licensed PE.

Use the calculator

Pick a mode at the top — Convert, Pulley, Gear, Cutting, or Linear. Each mode has its own input set and emits a result with the calculation breakdown plus the equivalent values in adjacent units (km/h, ft/min, sfm, m/s) for easy cross-checking.

CALC.011 RPM · 5 modes · Conversion · Pulley · Gear · Cutting · Linear

Convert between rotational units

Value[ω]

Unit[U]

Returns equivalents in all four units. 1 RPM = 2π/60 ≈ 0.1047 rad/s = 1/60 Hz = 6 deg/s.

Result

— RPM

Select a mode above and enter values.

FORMULA · ω = 2π · N / 60 SOURCE · MACHINERY'S HANDBOOK · SHIGLEY

RPM — at a glance

RPM to rad/s: ω = 2π·N / 60. So 1800 RPM = 188.5 rad/s; 1 RPM ≈ 0.1047 rad/s.
RPM to Hz: Hz = RPM / 60. 3600 RPM = 60 Hz; 1800 RPM = 30 Hz.
AC motor synchronous RPM: N_s = 120·f / P (f = line frequency, P = poles). A 4-pole motor on 60 Hz runs at 1800 RPM synchronous (~1750 RPM under load).
Pulley / belt RPM: D₁·N₁ = D₂·N₂. A 100 mm driver at 1800 RPM with a 200 mm driven pulley gives 900 RPM.
Gear ratio formula: Output RPM = input RPM × (T_driver / T_driven). Multi-stage trains multiply ratios — two 3:1 stages give a 9:1 reduction.
Torque from RPM and HP: HP = T·N / 5252 (T in lb-ft, N in RPM). At 1750 RPM, 1 HP equals 3.0 lb-ft of torque.
MPH from RPM: v_mph = π·D·N·60 / 63 360 (D in inches, N in RPM). A 26-inch tyre at 1000 RPM ≈ 77 mph.
Cutting speed → spindle RPM: N = (v × 1000) / (π × D_mm). A 10 mm carbide endmill at 200 m/min needs ~6 366 RPM.

The five RPM formulas

Eq. 01 — Rotational unit conversion SI · IEEE Std 945 · BIPM SI 2019

\omega = \frac{2\pi \cdot N}{60} \quad [\text{rad/s}], \qquad N = \frac{60 \cdot \omega}{2\pi} \quad [\text{RPM}]

ω: angular velocity, rad/s
N: revolutions per min, RPM

The four common rotational units are linked by simple constants. RPM and rad/s differ by 2π/60 ≈ 0.1047. Hz is just rev/s — divide RPM by 60. Deg/s is rad/s × 180/π. The calculator returns all four in one pass so you never have to look up a constant.

Eq. 02 — Pulley / belt drive SI / Imperial · Shigley, Mechanical Engineering Design

D_{1} \cdot N_{1} = D_{2} \cdot N_{2}, \qquad N_{2} = N_{1} \cdot \frac{D_{1}}{D_{2}} \cdot (1 - s)

D₁, D₂: driver and driven pitch diameter, any
N₁, N₂: driver and driven rotational speed, RPM
s: belt slip factor (0 = none, 0.02 = typical V-belt), —

The smaller pulley always spins faster, by exactly the inverse diameter ratio. V-belts and flat belts slip 1–2 % under full load; synchronous belts and roller chains slip zero. Use pitch diameter (centre of belt groove) for V-belts, not the outer-most diameter.

Eq. 03 — Gear train (multi-stage) SI · Shigley · Machinery's Handbook

\omega_{out} = \omega_{in} \cdot \prod_{i=1}^{n} \frac{T_{driver,i}}{T_{driven,i}}

ω_in, ω_out: input and output angular speed, rad/s
T_driver: tooth count of driving gear in each stage, —
T_driven: tooth count of driven gear in each stage, —
n: number of stages, —

A "stage" is one mating gear pair. Compound trains stack stages on a common shaft and multiply ratios — a 1/3 reduction followed by another 1/3 reduction gives 1/9 total. Epicyclic / planetary trains use a different sun-planet-ring formula and can give very high ratios (50:1+) in compact form.

Eq. 04 — Cutting / surface speed Imperial / SI · Machinery's Handbook · ASME B5.50

v = \pi \cdot D \cdot N \quad \implies \quad N \, [\text{RPM}] = \frac{v_{sfm} \cdot 12}{\pi \cdot D_{in}} = \frac{v_{m/min} \cdot 1000}{\pi \cdot D_{mm}}

v: surface speed at the cutting edge, sfm or m/min
D: tool diameter (mill / drill) or workpiece diameter (lathe), in or mm
N: spindle speed, RPM

The "12" in the imperial form converts feet to inches; the "1000" in the metric form converts metres to millimetres. Surface speed is the linear speed at which the cutting edge meets the workpiece — what drives heat, tool wear, and surface finish. Manufacturer datasheets list recommended v for each tool/material combination; you compute N from that.

Eq. 05 — Linear and centripetal motion SI · Halliday, Resnick & Walker — Physics

v = \omega \cdot r, \qquad a_{centripetal} = \omega^{2} \cdot r = \frac{v^{2}}{r}

v: tangential linear velocity, m/s
a: centripetal (inward) acceleration, m/s²
ω: angular velocity, rad/s
r: radius from rotation axis, m

Tangential velocity scales linearly with both ω and r. Centripetal acceleration scales as ω² — doubling RPM quadruples the radial stress on a rotor. This is why turbine and motor design are dominated by tip-speed limits (typically 200–500 m/s) and balance-grade requirements.

How to compute rotational speed, step by step

Decide what you are solving for. Five common problems: (1) convert one rotational unit to another (RPM ↔ rad/s ↔ Hz ↔ deg/s); (2) find the driven RPM of a belt or pulley; (3) find the output RPM of a multi-stage gear train; (4) set spindle RPM for a target cutting speed (or vice versa); (5) find the linear speed of a point at radius r on a rotating body.
Pick the right rotational unit. RPM for mechanical work (motors, spindles, drive shafts). rad/s for physics and dynamics (torque · ω = power). Hz for synchronous machines tied to the AC line (60 Hz × 2 / poles = synchronous RPM). deg/s for control systems and gimbals.
Enter geometry in any consistent unit. For pulleys: pitch diameter (not outer diameter) of each sheave, in mm or inches. For gears: tooth count of each gear in each stage. For cutting speed: tool diameter (turning) or workpiece diameter (lathe), in mm or inches. The calculator converts internally.
Apply the correct formula. Pulley: D₁·N₁ = D₂·N₂ — driven speed scales as the inverse diameter ratio. Gear: ω_in × T_driver = ω_out × T_driven, multi-stage compounds multiply. Cutting: v = π·D·N. Linear: v = ω·r tangential, ω²·r centripetal. The calculator picks and applies the formula based on your selected mode.
Check belt slip and gear backlash if accuracy matters. V-belts slip 1–2 % at full load — multiply driven speed by (1 − s) for the working figure. Synchronous belts and chains have zero slip but cannot start under load without a clutch. Gear trains have backlash (~0.05–0.2 mm at the pitch line) which limits servo positioning accuracy, not steady-state speed.
For machining, cross-check with manufacturer data. The cutting-speed formula v = π·D·N is exact, but the target v depends on tool, material, coolant, depth of cut, and feed. Use the calculator to set RPM, then verify against the manufacturer's recommended range (HSS: 30–60 m/min for steel; carbide: 150–250 m/min for steel; aluminium: 200–600 m/min). Surface finish and tool life are far better at the low end of the range, surface finish drops at the high end.

Worked example: belt-driven lathe spindle

A 1.5 kW induction motor at 1750 RPM drives a lathe headstock through a 75 mm pulley on the motor and a 200 mm pulley on the spindle. What is the spindle RPM, and what cutting speed does it give on a 40 mm-diameter steel workpiece?

Step	Calculation	Result
Pulley ratio	75 / 200	0.375
Spindle RPM (no slip)	1750 × 0.375	656 RPM
Spindle RPM (2 % V-belt slip)	656 × 0.98	643 RPM
Cutting speed v on 40 mm Ø	π × 40 mm × 643 RPM / 1000	80.8 m/min (265 sfm)
Recommended v for steel HSS	30–60 m/min	Too fast! ~1.5×
Adjustment: change pulleys to 50 mm / 200 mm	1750 × 50/200 × 0.98	429 RPM → v = 53.9 m/min

The original ratio gives roughly the right speed for carbide tooling on steel (150–250 m/min), but is well above the recommended range for HSS. Reducing the motor pulley to 50 mm — or adding a back-gear stage — drops the spindle into the HSS-friendly range. The same motor mechanically supports both ranges via a step pulley.

Reference values

Synchronous AC motor RPM (N = 120 · f / P)

The exact synchronous speed of a three-phase induction motor depends on the line frequency and the number of stator poles. Under load the motor slips 1–4 % below synchronous; the values below are no-load.

Poles (P)	60 Hz (USA / NA / Japan)	50 Hz (Europe / AU / NZ / Asia)	Typical use
2	3 600 RPM	3 000 RPM	High-speed motors, pumps, blowers, small compressors
4	1 800 RPM	1 500 RPM	General-purpose industrial motors (most common)
6	1 200 RPM	1 000 RPM	Conveyors, mills, mixers
8	900 RPM	750 RPM	Crushers, low-speed elevators, large pumps
12	600 RPM	500 RPM	Marine propulsion, large compressors
16	450 RPM	375 RPM	Hydroelectric generators, slow industrial drives

Cutting speed by tool / material

Recommended surface speed v at the cutting edge for general machining. Values are mid-range; always confirm with the tool manufacturer for the specific operation, depth of cut, and feed.

Workpiece material	HSS (m/min)	HSS (sfm)	Carbide (m/min)	Carbide (sfm)
Aluminium 6061	120–250	400–800	300–600	1 000–2 000
Brass / bronze	60–90	200–300	150–250	500–800
Mild steel (1018, A36)	25–35	80–120	120–180	400–600
Medium-carbon steel (1045)	20–30	65–100	90–150	300–500
Stainless 304 / 316	10–18	35–60	60–120	200–400
Tool steel (hardened)	5–10	15–30	30–80	100–250
Cast iron (grey)	25–40	80–130	90–200	300–650
Titanium (Ti-6Al-4V)	8–12	25–40	30–60	100–200
Plastics (acrylic, nylon)	100–200	300–650	200–500	650–1 600

Pick the spindle RPM with N = (v × 1000) / (π × D_mm) once v is chosen. The calculator above does this conversion automatically when you set the cutting-speed mode to "Spindle RPM".

Pulley vs gear vs chain — drive comparison

Three common ways to transmit rotation between shafts. Pick by the load, ratio range, accuracy, and noise tolerance of the application.

Aspect	V-belt / flat belt	Synchronous belt (HTD / GT)	Roller chain	Gear pair
Slip / accuracy	1–2 % slip under load	Zero slip	Zero slip	Zero slip + backlash 5–30 arc-min
Practical ratio per stage	1:1 – 8:1	1:1 – 8:1	1:1 – 7:1	1:1 – 10:1 (spur), 60:1 (worm)
Power capacity	≤ 100 kW typical	≤ 200 kW	≤ 300 kW	Unlimited (industrial)
Linear speed limit	~30 m/s rim	~50 m/s rim	~12 m/s pin	Pitch-line ≤ 80 m/s
Efficiency	92–96 %	96–99 %	96–98 %	97–99 % (spur), 50–90 % (worm)
Noise	Low	Medium	Medium-high	Low to high (depends on quality)

Frequently asked questions

How do I convert RPM to rad/s?: ω (rad/s) = N (RPM) × 2π / 60. So 1800 RPM = 1800 × 2π / 60 = 188.5 rad/s. Going the other way: N (RPM) = ω × 60 / (2π). 1 Hz = 60 RPM = 2π rad/s ≈ 6.283 rad/s. The calculator does all four units in one pass.
What is the formula for pulley RPM?: Pulleys conserve linear belt speed at the rim, so D₁·N₁ = D₂·N₂. A 100 mm driver at 1800 RPM driving a 200 mm driven pulley gives N₂ = 1800 × (100 / 200) = 900 RPM. The smaller pulley always spins faster for the same belt. Subtract 1–2 % for V-belt slip on real installations.
How does a gear train change RPM?: Each gear pair has a ratio T_driver / T_driven. Multi-stage compound trains multiply the ratios. Example: 20T → 60T (ratio 1/3) followed by 15T → 45T (ratio 1/3) gives total ratio 1/9 — input 1750 RPM becomes output 194 RPM, a 9:1 reduction. Tooth counts must be integers; gear stages use real tooth numbers, not simulated.
How do I calculate spindle RPM for a target cutting speed?: N (RPM) = (v × 1000) / (π × D) with v in m/min and D in mm. Or in SFM: N = (v_sfm × 12) / (π × D_in). A 10 mm carbide endmill at 200 m/min cutting speed needs N = (200 × 1000) / (π × 10) ≈ 6 366 RPM. Always confirm v against the tool manufacturer — values vary 5× between materials.
What does ω·r tell me?: It's the linear velocity of any point on the rotating body — tyre tread on a wheel, a lathe workpiece at the cutter, the rim of a pulley. v = ω·r. A 30 cm-diameter tyre at 1000 RPM has v = (1000·2π/60) × 0.15 = 15.7 m/s = 56 km/h. Doubling RPM doubles linear speed; doubling radius doubles linear speed.
What is centripetal acceleration?: It's the inward radial acceleration of a rotating point: a = ω²·r = v²/r. A 1 m-radius rotor at 3000 RPM has a = (3000·2π/60)² × 1 ≈ 98 700 m/s² ≈ 10 060 g — enough to fail a poorly-balanced rotor. Centrifugal force on a mass m is F = m·ω²·r. Doubling RPM quadruples the stress.
How is RPM related to motor frequency?: Synchronous speed of an AC motor is N_s (RPM) = 120 · f / P, where f is line frequency (60 Hz in North America, 50 Hz in Europe / Australia / Asia) and P is the number of poles. A 4-pole motor on 60 Hz runs at 1800 RPM synchronous (1750 RPM under load due to slip). 2-pole motors hit 3600 / 3000 RPM; 6-pole motors 1200 / 1000 RPM.

Sources and methodology

Oberg, E. et al. Machinery's Handbook, 31st Edition. Industrial Press, 2020.
Shigley, J.E., Budynas, R.G., Nisbett, J.K. Mechanical Engineering Design, 10th Edition. McGraw-Hill, 2015.
ASME B5.50 / ISO 25-2 — Tool nomenclature and cutting speeds.
BIPM. The International System of Units (SI), 9th Edition, 2019.