MotionLab

Angular Velocity Calculator

How fast something spins — radians turned per second.

Angle turned (rad)

Time (s)

Angular velocity (rad/s)

3.14

ω = θ / t

What you entered

Angle turned (rad): 6.28Time (s): 2

1
ω = θ ÷ t
6.28 ÷ 2= 3.14 rad/s

Result

Angular velocity (rad/s): 3.14

Rotating 6.28 rad in 2s gives an angular velocity of 3.14 rad/s.

Angular velocity from angle swept over time

Angular velocity is the angle in radians divided by the time taken, in radians per second. One full turn is 2 pi, about 6.2832 radians.

Angle (rad)	Time (s)	Angular velocity (rad/s)	Equivalent
1	1	1.0000	1 radian per second
3.1416	1	3.1416	Half turn per second
6.2832	2	3.1416	One turn in 2 seconds
6.2832	1	6.2832	One turn per second, 60 rpm
12.5664	1	12.5664	Two turns per second, 120 rpm

Rows two and three give the same angular velocity from different angles and times, since only the rate matters. To convert to rpm, multiply by 60 and divide by 2 pi - so 6.2832 rad/s is exactly 60 rpm. Angular velocity is the same everywhere on a rotating rigid body, which is what distinguishes it from linear velocity: a point on the rim of a wheel moves much faster than one near the hub, but both sweep the same angle per second. Multiply angular velocity by the radius to get the linear speed at that radius.

Radians, not degrees

This formula expects the angle in radians (2π radians = 360°), the standard unit for angular motion formulas — convert from degrees first if that's what you're starting with (degrees × π/180).

Connecting to linear speed

A point on a spinning object has a linear speed equal to angular velocity times its distance from the center (v = ωr) — that's why the outer edge of a spinning disk moves faster than a point near its center, despite both having the same angular velocity.

Frequently asked questions

A bicycle wheel makes 3 full revolutions per second. What's its angular velocity?

ω = 2π × 3 = 18.85 rad/s. The outer edge of a 0.35 m radius wheel has linear speed v = ωr = 18.85 × 0.35 = 6.6 m/s (about 24 km/h).

A hard drive spins at 7,200 RPM. What's that in rad/s?

ω = 7200 × 2π/60 = 753.98 rad/s. The edge of a 3.5-inch platter (r ≈ 0.044 m) moves at v = ωr ≈ 33.2 m/s — about 120 km/h. This is why head crashes are so destructive.

How do I convert between RPM, degrees/s, and rad/s?

1 revolution = 360° = 2π rad. So: RPM × 2π/60 = rad/s. Degrees/s × π/180 = rad/s. This calculator uses rad/s (the SI standard), which keeps all the rotational physics formulas clean.

What's the connection between angular velocity and centripetal force?

Centripetal force F = mω²r. Higher angular velocity means much more centripetal force (it scales with ω²). A washing machine's spin cycle at 1,200 RPM needs enormous structural strength. See the centripetal force calculator.

How is angular velocity different from angular acceleration?

Angular velocity (ω) is how fast something is spinning right now (rad/s). Angular acceleration (α) is how fast the spin rate is changing (rad/s²). Constant ω means zero α — the object spins steadily. See the angular acceleration calculator.