Cycle per second to Radian per second
cps
rad/s
Conversion History
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|---|---|---|
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Quick Reference Table (Cycle per second to Radian per second)
| Cycle per second (cps) | Radian per second (rad/s) |
|---|---|
| 20 | 125.66370614359172 |
| 50 | 314.1592653589793 |
| 60 | 376.99111843077516 |
| 440 | 2,764.60153515901784 |
| 1,000 | 6,283.185307179586 |
| 20,000 | 125,663.70614359172 |
About Cycle per second (cps)
Cycle per second (cps) is the older, pre-SI term for what is now called hertz. One cycle per second equals exactly one hertz. The term was in common use through the mid-20th century in electrical engineering and acoustics — specifications for audio equipment, radio equipment, and mains electricity were all stated in cycles per second. The SI formally replaced "cycles per second" with "hertz" in 1960, and the change was widely adopted through the 1960s–70s. Some older technical literature and vintage equipment datasheets still use cps.
A 1950s amplifier spec sheet listing "frequency response 20–20,000 cps" means the same as 20 Hz–20 kHz. The US mains supply was described as "60 cps" before 1960.
About Radian per second (rad/s)
Radian per second (rad/s) is the SI unit of angular velocity, measuring how fast an angle changes over time. One full rotation (360°) equals 2π radians, so one revolution per second equals 2π rad/s ≈ 6.283 rad/s. Radian per second is the preferred unit in physics and engineering for rotational dynamics, since it makes equations involving centripetal acceleration and torque work cleanly without conversion factors. Electric motors, gyroscopes, and spinning spacecraft components are analyzed using rad/s.
Earth rotates at about 7.27 × 10⁻⁵ rad/s. A wheel spinning at 10 rad/s makes about 1.6 revolutions per second. A gyroscope precessing at 1 rad/s completes one full precession cycle in about 6.3 seconds.
Cycle per second – Frequently Asked Questions
Why did the SI replace "cycles per second" with "hertz" in 1960?
The General Conference on Weights and Measures wanted consistent named units honoring key physicists, paralleling the watt, volt, and ampere. "Cycles per second" was descriptive but wordy, and it didn't follow the pattern of one-word unit names. Heinrich Hertz — who proved electromagnetic waves exist — was the obvious namesake. The swap was official from 1960, though many engineers kept saying "cps" well into the 1970s.
Are there any situations where "cycles per second" is still preferred over hertz?
In some vintage audio and ham radio communities, "cps" persists as nostalgic shorthand. More practically, it survives in teaching contexts where making the physical meaning explicit is helpful — telling a student that 440 cps means "440 complete vibrations each second" is more intuitive than "440 Hz" until they have internalised the unit. Officially, though, every standards body has switched to hertz.
If cycles per second and hertz are identical, why does this converter page exist?
Because people searching for "cycles per second to hertz" are usually reading an old textbook or datasheet that uses cps and want confirmation that it is a 1:1 equivalence — no multiplication needed. The conversion factor is exactly 1, but verifying that still saves someone a trip to the library or a forum post.
What did equipment spec sheets look like before hertz was adopted?
A 1950s oscilloscope might list its bandwidth as "DC to 5,000,000 cps." A radio receiver would specify "tuning range: 540 to 1,600 kc/s" (kilocycles per second). Turntable specs read "wow and flutter: 0.15% at 33⅓ cps." After 1960, "kc/s" became "kHz" and "Mc/s" became "MHz," but the underlying numbers stayed identical.
How is "cycles per second" different from "radians per second"?
One cycle is one full oscillation — from peak to peak. One radian is about 1/6.28 of a full circle. So 1 cycle per second = 2π radians per second ≈ 6.283 rad/s. Engineers use radians per second in equations where angular measure matters (torque, rotational inertia), and cycles per second (hertz) when counting whole oscillations. Forgetting the 2π factor is one of the most common mistakes in physics homework.
Radian per second – Frequently Asked Questions
Why do physics equations use radians per second instead of RPM or degrees?
Because radians make the maths clean. The formulas for centripetal acceleration (a = ω²r), angular momentum (L = Iω), and torque (τ = Iα) all assume ω is in rad/s. If you plug in RPM or degrees, you have to insert conversion factors of 2π/60 or π/180 everywhere. Radians are dimensionless ratios (arc length ÷ radius), so they vanish naturally from equations — no extra constants needed.
How fast does Earth rotate in radians per second?
Earth completes one full rotation (2π radians) in about 86,164 seconds (a sidereal day, slightly shorter than 24 hours). That gives approximately 7.292 × 10⁻⁵ rad/s. It sounds tiny, but at the equator it translates to a surface speed of about 465 m/s (1,674 km/h). You are always moving that fast — you just do not feel it because everything around you moves with you.
What is the difference between angular velocity and angular frequency?
They are the same number in rad/s but describe different things. Angular velocity refers to physical rotation — a wheel spinning. Angular frequency (often written ω = 2πf) describes oscillation — a vibrating spring or alternating current. A 60 Hz AC signal has ω ≈ 377 rad/s even though nothing is literally spinning. The distinction is conceptual, not mathematical.
How do you convert between rad/s and RPM?
Multiply rad/s by 60/(2π) ≈ 9.5493 to get RPM. Or divide RPM by the same factor to get rad/s. Quick shortcut: 1 rad/s ≈ 9.55 RPM, and 1,000 RPM ≈ 104.7 rad/s. If a motor spec says 3,600 RPM (common for a synchronous motor on 60 Hz mains), that is 3,600 ÷ 9.5493 ≈ 377 rad/s — the same ω as the mains frequency times 2π.
What angular velocity in rad/s does a figure skater reach during a spin?
An elite figure skater in a scratch spin pulls their arms in and reaches roughly 25–40 rad/s (about 4–6 revolutions per second). That is 240–360 RPM. The current record-holders approach 342 RPM (~35.8 rad/s). The speed increase when pulling arms in is a textbook demonstration of conservation of angular momentum — reducing the moment of inertia forces ω to increase.