RPM to Radians Per SecondRPM to rad/s

13 RPM to Radians Per Second Converter

Instant conversion for 13 RPM to rad/s (1.361357 rad/s). Analytical π/30 formula, rotational frequency in Hz, and full kinematics.

m
Result1.361357 rad/s
Radians per Second (rad/s)
1.3613568
Degrees per Second (°/s)
78 °/s
Hertz (Hz)
0.216667 Hz
Revolutions per Second (RPS)
0.216667 rev/s
Revolutions per Hour (RPH)
780 rev/hr
Period (seconds per rev)
4.615385 s/rev
Exact Formula
13 × π/30 = 1.3613568 rad/s
Nearest Real-World Example
Ceiling fan — high (200 RPM)
Precision Drift Meter (Exact π/30 vs. 0.10472 Rounded)Zero Drift Here

Generic web calculators use the rounded 0.10472 multiplier. At this rotational speed, that approximation introduces numerical drift:

Exact Value (π/30):1.361357 rad/s
Rounded (0.10472):1.361360 rad/s
Angular Error / Hour:0.66° / hr

Worked Conversion: 13 RPM to Radians Per Second

To convert revolutions per minute (RPM) to SI angular velocity (rad/s), multiply by 2π radians per revolution and divide by 60 seconds per minute. The analytical factor simplifies to π/30.

rad/s = 13 × (π / 30) = 1.361357 rad/s

Exact fraction: 13π/30 rad/s · Reciprocal: 1.361 rad/s × (30/π) = 13 RPM

Angular Velocity
1.361357 rad/s
Frequency
0.2167 Hz
Degrees / Second
78.00 °/s
Period per Rev
4.6154 s
Engineering Context: At 13 RPM (a rotisserie spit or drum mixer), Speeds in the low tens of RPM belong to rotisseries, concrete mixers and slow display turntables — high-torque, low-speed drives.

Reverse Conversion: 1.361 rad/s to RPM

Need to convert from radians per second back to RPM? Use our reverse programmatic converter with exact 30/π precision.

Browse Reverse Conversions →

Neighbouring & Harmonic Speed Conversions

In-Depth Engineering Guides & References

5.36 Rad/s to RPM: College Physics Homework Benchmark Solution— Serway & Halliday kinematics proof and analytical derivation.
RPM to Radians Per Second Formula & Calculus Proof— Dimensional cancellation, π/30 derivation, and IEEE 754 precision.
Angular Velocity Guide & Rotational Kinematics— SI standards, tangential velocity (v = ω·r), and centripetal acceleration.
Complete Conversion Index— Browse pre-calculated conversion speeds from 1 to 20,000 RPM.

How to Convert RPM to Radians Per Second

Complete dimensional derivation and multi-step conversion method

Step 1 · Unit Setup
ω = RPM × (π / 30) rad/s

Choose the rotational speed unit you want to convert from:

Visual Rotation Simulation (Proportional Speed)
3,000RPM
Angular:314.16 rad/s
Rotational:50.0 Hz
  1. 1
    Unit Setup

    Select Your Input Unit (RPM, rad/s, Hz, or °/s)

    Choose your starting unit: Revolutions Per Minute (RPM), Radians Per Second (rad/s), Frequency in Hertz (Hz, rev/s), or Degrees Per Second (°/s). The converter dynamically adapts in real time.

    Currently simulating:
  2. 2
    Parameters

    Input Rotational Velocity & Optional Radius

    Enter your numeric speed value or pick a machinery benchmark. If you are analyzing a rotating rotor, shaft, pulley, wheel, or centrifuge, specify the radius in meters to instantly calculate linear tangential surface speed (m/s, km/h, mph).

    Click to simulate
  3. 3
    Analysis

    Inspect Complete Analytical Kinematics Breakdown

    Review instant results across all rotational metrics: exact rad/s, RPM, cycle period per revolution, rotational frequency, and peripheral linear speed.

    Click to simulate

How to Convert RPM to Rad/s & Rad/s to RPM (Formula & Derivation)

Quick Answer: To convert RPM (revolutions per minute) to rad/s (radians per second), multiply by π and divide by 30 (or multiply by approximately 0.1047197551): ω = RPM × (π / 30) rad/s To convert radians per second (rad/s) back to RPM, multiply by 30 and divide by π (or multiply by approximately 9.5492965855): RPM = ω × (30 / π) Mathematical Derivation From First Principles: 1. One complete revolution equals 360 degrees, which is exactly 2π radians (~6.2831853 rad). 2. One minute equals exactly 60 seconds. 3. Multiplying RPM by 2π radians per revolution and dividing by 60 seconds per minute gives: ω = RPM × (2π / 60) rad/s. 4. Simplifying 2π/60 by dividing numerator and denominator by 2 yields the exact analytical conversion factor: ω = RPM × (π / 30) rad/s. Notations, Common Synonyms & Unit Variations: Rotational velocity is expressed across literature in multiple equivalent notations: • Revolutions per minute: RPM, rev/min, rev per min, rev min, or r/min. • Radians per second: rad/s, rad/sec, rad sec, rads, or rad per second. All of these notations describe the same quantities, so the same π/30 factor applies to each. Practical Engineering Guidance: Avoid using rounded decimals like 0.10472 in multi-stage engineering code. In gear trains, electric motor torque estimations (P = τ·ω), and torsional vibration models, using truncated constants causes cumulative floating-point error. Retaining the symbolic fraction π/30 guarantees exact scientific precision.

RPM to radians per second rotational velocity formula and geometry diagram
RPM to rad/s:
  ω (rad/s) = RPM × 2π ÷ 60
            = RPM × π/30 ≈ RPM × 0.1047197551

rad/s to RPM:
  RPM       = ω × 60 ÷ 2π
            = ω × 30/π ≈ ω × 9.5492965855

Related Rotational Relationships:
  Frequency f (Hz)            = RPM ÷ 60 = ω ÷ 2π
  Degrees per Second (°/s)    = rad/s × 180/π = RPM × 6
  Period T (s/rev)            = 60 ÷ RPM = 2π ÷ ω
  Tangential Velocity v (m/s) = ω × r = (RPM × π ÷ 30) × r
Example

Worked Step-by-Step Examples: Example 1: Convert 1,800 RPM (standard 4-pole synchronous motor) to rad/s: ω = 1,800 × π / 30 = 60π ≈ 188.49556 rad/s. Frequency: f = 1,800 / 60 = 30 Hz. Period: T = 60 / 1,800 ≈ 0.0333 seconds per rev. Example 2: Convert 100 rad/s to RPM: RPM = 100 × 30 / π = 3,000 / π ≈ 954.93 RPM. Example 3: Tangential linear speed of a 0.25 m radius flywheel at 3,000 RPM: ω = 3,000 × π / 30 = 100π ≈ 314.159 rad/s. v = ω × r = 314.159 rad/s × 0.25 m = 78.54 m/s (282.74 km/h or 175.69 mph).

Common RPM to rad/s Conversion Examples

Exact analytical results across mechanical engineering, electric motors, and physics benchmarks.
Engineering ScenarioCalculationKinematic Result
Automotive Engine — 3,000 RPM → rad/s3,000 × π/30314.159265 rad/s · 50 Hz · 18,000 °/s · 0.02 s/rev
AC Induction Motor — 1,800 RPM → rad/s1,800 × π/30 = 60π188.495559 rad/s · 30 Hz · 10,800 °/s · 0.0333 s/rev
Industrial Motor — 1,500 RPM → rad/s1,500 × π/30 = 50π157.079633 rad/s · 25 Hz · 9,000 °/s · 0.04 s/rev
Enterprise Hard Drive — 7,200 RPM → rad/s + Hz7,200 × π/30 = 240π753.982237 rad/s · 120 Hz · 43,200 °/s · 0.00833 s/rev
Angle Grinder — 11,000 RPM → rad/s11,000 × π/301,151.917306 rad/s · 183.33 Hz · 66,000 °/s
Reverse Conversion — 100 rad/s → RPM100 × 30/π954.929659 RPM · 15.915 Hz · 5,729.578 °/s

RPM to Radian/Second Conversion Reference Table

Calculated with exact analytical rad/s = RPM × π/30. Radians per second rounded to five decimal places; frequency in Hertz (Hz); angular displacement in degrees per second (°/s); period in seconds per revolution (s/rev). Includes 1–100 integer values plus standard industrial motor speeds.
RPMrad/sHz°/sPeriod (s)
10.104720.0167660
20.209440.03331230
30.314160.051820
40.418880.06672415
50.523600.08333012
60.628320.13610
70.733040.1167428.5714
80.837760.1333487.5
90.942480.15546.6667
101.047200.1667606
111.151920.1833665.4545
121.256640.2725
131.361360.2167784.6154
141.466080.2333844.2857
151.570800.25904
161.675520.2667963.75
171.780240.28331023.5294
181.884960.31083.3333
191.989680.31671143.1579
202.094400.33331203
212.199110.351262.8571
222.303830.36671322.7273
232.408550.38331382.6087
242.513270.41442.5
252.617990.41671502.4
262.722710.43331562.3077
272.827430.451622.2222
282.932150.46671682.1429
293.036870.48331742.069
303.141590.51802
313.246310.51671861.9355
323.351030.53331921.875
333.455750.551981.8182
343.560470.56672041.7647
353.665190.58332101.7143
363.769910.62161.6667
373.874630.61672221.6216
383.979350.63332281.5789
394.084070.652341.5385
404.188790.66672401.5
414.293510.68332461.4634
424.398230.72521.4286
434.502950.71672581.3953
444.607670.73332641.3636
454.712390.752701.3333
464.817110.76672761.3043
474.921830.78332821.2766
485.026550.82881.25
495.131270.81672941.2245
505.235990.83333001.2
515.340710.853061.1765
525.445430.86673121.1538
535.550150.88333181.1321
545.654870.93241.1111
555.759590.91673301.0909
565.864310.93333361.0714
575.969030.953421.0526
586.073750.96673481.0345
596.178470.98333541.0169
606.2831913601
616.387911.01673660.9836
626.492621.03333720.9677
636.597341.053780.9524
646.702061.06673840.9375
656.806781.08333900.9231
666.911501.13960.9091
677.016221.11674020.8955
687.120941.13334080.8824
697.225661.154140.8696
707.330381.16674200.8571
717.435101.18334260.8451
727.539821.24320.8333
737.644541.21674380.8219
747.749261.23334440.8108
757.853981.254500.8
767.958701.26674560.7895
778.063421.28334620.7792
788.168141.34680.7692
798.272861.31674740.7595
808.377581.33334800.75
818.482301.354860.7407
828.587021.36674920.7317
838.691741.38334980.7229
848.796461.45040.7143
858.901181.41675100.7059
869.005901.43335160.6977
879.110621.455220.6897
889.215341.46675280.6818
899.320061.48335340.6742
909.424781.55400.6667
919.529501.51675460.6593
929.634221.53335520.6522
939.738941.555580.6452
949.843661.56675640.6383
959.948381.58335700.6316
9610.053101.65760.625
9710.157821.61675820.6186
9810.262541.63335880.6122
9910.367261.655940.6061
10010.471981.66676000.6
12012.5663727200.5
18018.8495631,0800.3333
20020.943953.33331,2000.3
30031.4159351,8000.2
50052.359888.33333,0000.12
60062.83185103,6000.1
75078.5398212.54,5000.08
90094.24778155,4000.0667
1,000104.7197616.66676,0000.06
1,200125.66371207,2000.05
1,450151.8436424.16678,7000.0414
1,500157.07963259,0000.04
1,750183.2595729.166710,5000.0343
1,800188.495563010,8000.0333
2,400251.327414014,4000.025
2,850298.4513047.517,1000.0211
3,000314.159275018,0000.02
3,450361.2831657.520,7000.0174
3,600376.991126021,6000.0167
5,000523.5987883.333330,0000.012
5,400565.486689032,4000.0111
7,200753.9822412043,2000.0083
10,0001047.19755166.666760,0000.006
15,0001570.7963325090,0000.004
20,0002094.39510333.3333120,0000.003

Key Rotational Kinematics Facts & Constants

  • The exact analytical conversion factor is π/30 rad/s per RPM (approx. 0.1047197551). Retaining π/30 avoids rounding error accumulation in multi-step physics modeling.
  • The exact reciprocal multiplier is 30/π RPM per rad/s (approx. 9.5492965855). Multiply any rad/s speed by 30/π to obtain exact RPM.
  • 1 Hertz (Hz) is 1 complete cycle per second, which equals 60 RPM and exactly 2π rad/s (~6.283185 rad/s).
  • A standard 4-pole AC induction motor on a 60 Hz grid has a synchronous speed of 1,800 RPM (60π rad/s ≈ 188.496 rad/s), slipping to ~1,750 RPM under mechanical load.
  • A 2-pole AC induction motor on a 60 Hz grid runs at 3,600 RPM synchronous speed (120π rad/s ≈ 376.991 rad/s).
  • An engine redlining at 8,000 RPM spins at 837.76 rad/s, sweeping through 48,000 degrees per second.
  • High-performance enterprise hard drive platters at 7,200 RPM rotate at 120 Hz and 753.98 rad/s.
  • Dental air-turbine drills rotate between 350,000 and 450,000 RPM, reaching angular velocities over 41,888 rad/s.
  • Centrifuges spinning at 100,000 RPM reach 10,472 rad/s and generate accelerations exceeding 800,000 × g.
  • Linear tangential velocity at a radius r is v = ω·r. Doubling shaft radius or rotational speed doubles peripheral velocity.

Editorial & Engineering Team

Engineering & Applied Kinematics Specialists

  • Mechanical Systems Engineering
  • Applied Rotational Kinematics

Our engineering and editorial specialists build, review, and mathematically verify rotational kinematics tools and technical datasets against international SI metrology standards.

Sources & References

  1. SI Brochure: The International System of Units (9th Edition)Bureau International des Poids et Mesures (BIPM)
  2. NIST Guide to the SI, Chapter 4: Two Classes of SI Units and the RadianNational Institute of Standards and Technology (NIST)
  3. IEEE Standard 754-2019 for Floating-Point ArithmeticIEEE Standards Association
  4. ISO 80000-3: Quantities and Units — Space and TimeInternational Organization for Standardization (ISO)

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