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Angular Velocity — Definition, Formula & Examples

Angular velocity is how fast something spins or turns. It tells you how much angle an object sweeps through each second, usually measured in radians per second.

For an object rotating about a fixed axis, let θ(t)\theta(t) be its angular position in radians, measured from the initial side. The average angular velocity over an interval is ωˉ=ΔθΔt\bar{\omega} = \frac{\Delta\theta}{\Delta t}. The instantaneous angular velocity is ω=dθdt\omega = \frac{d\theta}{dt}. By convention, ω>0\omega > 0 for counterclockwise rotation and ω<0\omega < 0 for clockwise rotation. Its magnitude ∣ω∣|\omega| is called angular speed.

Key Formula

ω=ΔθΔtv=rω\omega = \frac{\Delta\theta}{\Delta t} \qquad\qquad v = r\omega
Where:
  • ω\omega = angular velocity (radians per unit time)
  • Δθ\Delta\theta = angle swept, in radians
  • Δt\Delta t = elapsed time
  • rr = distance from the axis of rotation (radius)
  • vv = linear (tangential) speed of the point

How It Works

Divide the angle turned by the time it took. If a problem gives revolutions per minute (rpm), multiply by 2π2\pi to turn revolutions into radians, then divide by 60 to turn minutes into seconds. Once you know ω\omega in radians per unit time, the linear speed of a point at distance rr from the center is v=rωv = r\omega. Every point on a rigid spinning object shares the same ω\omega, but points farther from the center move faster along their paths.

Worked Example

Problem: A bicycle wheel spins at 120 revolutions per minute. Find its angular velocity in radians per second. Then find the linear speed of a point on the rim, 0.5 m from the center.
Step 1: Convert revolutions to radians. One revolution is 2π2\pi radians.
120 rev×2π radrev=240π rad120 \text{ rev} \times 2\pi \ \tfrac{\text{rad}}{\text{rev}} = 240\pi \text{ rad}
Step 2: The wheel turns 240π240\pi radians in 1 minute, which is 60 seconds. Divide the angle by the time.
ω=240π rad60 s=4π rad/s≈12.57 rad/s\omega = \frac{240\pi \text{ rad}}{60 \text{ s}} = 4\pi \text{ rad/s} \approx 12.57 \text{ rad/s}
Step 3: Use v=rωv = r\omega with r=0.5r = 0.5 m.
v=(0.5)(4π)=2π m/s≈6.28 m/sv = (0.5)(4\pi) = 2\pi \text{ m/s} \approx 6.28 \text{ m/s}
Answer: The angular velocity is 4π≈12.574\pi \approx 12.57 rad/s. A point on the rim moves at 2π≈6.282\pi \approx 6.28 m/s.

Another Example

Problem: Earth rotates once every 24 hours. Find its angular velocity in radians per hour, and find the angle it turns through in 6 hours.
Step 1: One full rotation is 2π2\pi radians in 24 hours.
ω=2π24=π12 rad/h≈0.262 rad/h\omega = \frac{2\pi}{24} = \frac{\pi}{12} \text{ rad/h} \approx 0.262 \text{ rad/h}
Step 2: Rearrange ω=ΔθΔt\omega = \frac{\Delta\theta}{\Delta t} to get Δθ=ω Δt\Delta\theta = \omega\,\Delta t.
Δθ=π12×6=π2 rad=90∘\Delta\theta = \frac{\pi}{12} \times 6 = \frac{\pi}{2} \text{ rad} = 90^\circ
Answer: Earth's angular velocity is π12≈0.262\frac{\pi}{12} \approx 0.262 rad/h. In 6 hours it turns π2\frac{\pi}{2} radians, a quarter turn.

Why It Matters

In AP Physics 1, angular velocity is the starting point for rotational kinematics, torque, and angular momentum. In AP Precalculus, you model a point moving counterclockwise around a circle of radius rr, starting on the positive x-axis, with x=rcos⁡(ωt)x = r\cos(\omega t) and y=rsin⁡(ωt)y = r\sin(\omega t). Engineers use the same idea to match motor rpm to gear sizes and belt speeds.

Common Mistakes

Mistake: Using degrees per second in v=rωv = r\omega.
Correction: The formula v=rωv = r\omega works only when ω\omega is in radians per unit time. For 90°/s at r=2r = 2 m, first convert: 90∘/s=π290^\circ/\text{s} = \frac{\pi}{2} rad/s. Then v=2⋅π2=π≈3.14v = 2 \cdot \frac{\pi}{2} = \pi \approx 3.14 m/s, not 180 m/s.
Mistake: Treating rpm as if it were already rad/s, or forgetting the factor of 2π2\pi.
Correction: A revolution is 2π2\pi radians, not 1 radian. To convert rpm to rad/s, multiply by 2π2\pi and divide by 60.

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