Calculate the angular frequency (ω) of rotating or oscillating objects. Enter either angle change (Δθ) and time for rotational motion, or use frequency (f) / period (T) for oscillating systems. Get back angular frequency in rad/s, along with equivalent frequency in Hz and period in seconds.
Angular Frequency (ω)
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Frequency (f)
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Period (T)
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Rotational Speed
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Angular frequency (ω) is a scalar quantity that describes how fast an object rotates or oscillates with respect to time. It is measured in radians per second (rad/s) and is commonly denoted by the Greek letter ω (omega). It differs from ordinary frequency by a factor of 2π.
There are several angular frequency formulas depending on the context. For rotating objects: ω = Δθ / Δt, where Δθ is the change in angle (in radians) and Δt is the elapsed time. For oscillating systems: ω = 2πf, where f is the frequency in Hz, or equivalently ω = 2π / T, where T is the period.
Regular frequency (f) measures how many complete cycles occur per second (in Hz). Angular frequency (ω) measures the rate of change of angle and equals 2π times the regular frequency: ω = 2πf. While frequency counts cycles, angular frequency counts radians traversed per second.
The SI unit of angular frequency is radians per second (rad/s). It can also be expressed in degrees per second or converted to revolutions per minute (RPM) for practical engineering applications.
To convert RPM to angular frequency, use the formula: ω = (RPM × 2π) / 60. For example, a motor spinning at 120 RPM has an angular frequency of (120 × 2π) / 60 = 4π ≈ 12.566 rad/s.
For a simple pendulum, the angular frequency is ω = √(g / L), where g is the acceleration due to gravity (9.81 m/s²) and L is the length of the pendulum in meters. This formula assumes small-angle oscillations.
The period T and angular frequency ω are inversely related through the equation ω = 2π / T. A longer period means a lower angular frequency, and vice versa. For example, a system with a period of 0.5 s has an angular frequency of 2π / 0.5 = 4π ≈ 12.57 rad/s.
In physics, angular frequency itself is always positive since it represents a magnitude of rotation rate. However, angular velocity (a vector quantity) can be negative, indicating rotation in the opposite direction. When using ω in wave equations, a negative sign may indicate a wave traveling in the opposite direction.