Linear Speed from Angular Velocity Calculator

Definition of linear speed in circular motion

The linear speed v of a point moving on a circle is the speed tangent to the path. Its magnitude is tied to angular velocity by v = ω·r, where r is the distance of the point from the axis of rotation. The SI unit is the meter per second. Angular velocity ω is in radians per second; because the radian is dimensionless, the product ω·r has the dimension of length over time.

Every point of a rigid body shares the same angular velocity. The points differ in radius, so they differ in linear speed. A point on the axis has r = 0 and v = 0: it turns with the body but travels no arc. The velocity vector is tangent to the circle and perpendicular to the radius. The calculator reports the magnitude of that speed. The sign of ω describes only the sense of rotation.

History: the arc, Huygens, and Euler

Circular measure, the ratio of arc length to radius, gives s = r·θ with no conversion from degrees. Roger Cotes introduced it in 1714; it appeared posthumously in Harmonia mensurarum (1722). James Thomson proposed the name radian in 1873. From s = r·θ, at constant r, differentiation with respect to time yields v = r·dθ/dt = r·ω.

Christiaan Huygens studied circular motion and centrifugal force in Horologium oscillatorium (1673). Leonhard Euler, in Theoria motus corporum solidorum seu rigidorum (1765), described the instantaneous motion of a rigid body as rotation about an axis with an angular velocity. In vector notation the velocity of a point of the body is v = ω × r. When the radius is perpendicular to the axis, the magnitude of that product is ω·r. This calculator computes that magnitude.

Derivation of v = ω·r

Step 1. The position of a point on a circle of radius r is described by an angle θ measured in radians. The arc length from a chosen origin is s = r·θ. Step 2. Linear speed is the time derivative of distance, v = ds/dt. At constant radius, r comes out of the derivative, so v = r·dθ/dt. Step 3. By definition ω = dθ/dt, therefore v = r·ω. Step 4. For uniform rotation ω = 2π/T, so v = 2πr/T: the point travels the circumference 2πr in one period.

Step 5. For ω = 10 rad/s and r = 0.35 m the product is 3.5 m/s, or 12.6 km/h. The period T = 2π/10 ≈ 0.628 s, and the centripetal acceleration a = ω²·r = 35 m/s². Step 6. At twice the radius and the same ω the speed is twice as large, because v is directly proportional to r. At ω = 0 the speed is 0 for any radius.

Assumptions: a rigid body and a fixed radius

The formula v = ω·r is exact when the distance of the point from the axis is constant and ω is the angular velocity about that axis. A point of a rigid body satisfies it, and so does motion on a circle. When the radius changes, for example, when a point slides along an arm, a radial component of velocity appears. The calculator does not compute that component.

For a rotation that speeds up, the formula still gives the instantaneous tangential speed if you enter the instantaneous ω. It does not by itself tell how ω changes with time. It does not compute angular velocity from a period, the force F = m·v²/r, or rotational kinetic energy. A negative radius is not a distance from the axis and is rejected. The value of ω may be negative: the sense changes, and the speed does not.

Linear speed, revolutions, and centripetal acceleration

Revolutions per minute n and angular velocity are related by ω = 2π·n/60 ≈ 0.10472·n. Hence, v = 2π·n·r/60. A disc at 300 rad/s turns at about 2,865 rpm; at a radius of 0.1 m the tangential speed is 30 m/s. Centripetal acceleration has magnitude a = v²/r = ω²·r and points toward the axis. For the reference example it is 35 m/s². A separate centripetal-acceleration calculator takes v and r. This panel shows a beside the result, so the scale is visible, but the primary result remains v.

Frequency f = ω / (2π) counts turns per second. The panel shows the period T = 2π/|ω| and the revolutions per minute. Linear speed is not angular velocity: 10 rad/s at a radius of 0.35 m is 3.5 m/s, not 10 m/s. At a radius of 1 m the numerical values of v in m/s and ω in rad/s agree, because v = ω·1.

Orders of magnitude: from a record groove to orbit

Tangential speeds span a wide range because r and ω enter as a product. A bicycle rim at 10 rad/s and r = 0.35 m moves at 3.5 m/s. A carousel seat at a radius of 6 m and 0.8 rad/s moves at 4.8 m/s. A washing-machine drum at about 1,200 rpm and r = 0.2 m moves at 25.2 m/s. A tire of radius 0.3 m at 80 rad/s matches a car speed of 24 m/s, or 86.4 km/h (about 53.7 mph). The tip of a ceiling-fan blade of radius 0.6 m at 200 rpm moves at about 12.56 m/s.

At the other end of the scale, a point on the equator, with ω = 7.292·10⁻⁵ rad/s and a radius of 6,378 km, moves at 465.0838 m/s relative to the Earth’s axis. An approximate circular orbit of radius 6,771 km at ω = 0.00115 rad/s gives 7,786.65 m/s. The same formula describes a vinyl groove, where the speed is a fraction of a meter per second. The calculator does not replace a tachometer or an ephemeris: it multiplies the numbers you enter.

Sources

The definition, the derivation, and the historical note come from mechanics textbooks and metrology sources: D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, the chapter on circular motion; H. D. Young and R. A. Freedman, University Physics, the chapters on circular motion and rigid-body rotation; R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Volume I, Chapter 18; L. Euler, Theoria motus corporum solidorum seu rigidorum, 1765; C. Huygens, Horologium oscillatorium, 1673; R. Cotes, Harmonia mensurarum, 1722; BIPM, The International System of Units (SI), 9th edition (radian, meter, second); the NASA Earth Fact Sheet (equatorial radius and sidereal day). This page is for educational purposes.