Angular velocity measures how fast an object rotates around an axis, expressed in radians per second (rad/s) or revolutions per minute (RPM). Linear velocity measures how fast a point moves along a path, expressed in meters per second (m/s) or miles per hour (mph). For circular motion, linear velocity depends on both angular velocity and the distance from the rotation center (radius). Points farther from the center move faster even though the angular velocity is the same.

The formula v = ωr requires angular velocity in radians per unit time because radians are dimensionless (they’re a ratio of arc length to radius). This makes the mathematics work correctly: when you multiply rad/s by meters, the radian unit effectively cancels, leaving you with m/s. If you use degrees, you must convert to radians first: multiply by π/180. One complete revolution equals 2π radians or 360 degrees.

To convert revolutions per minute (RPM) to radians per second (rad/s), multiply by 2π and divide by 60. The formula is: ω (rad/s) = RPM × (2π/60) = RPM × 0.10472. For example, 300 RPM = 300 × 0.10472 = 31.416 rad/s. This conversion accounts for: one revolution = 2π radians, and one minute = 60 seconds.

Yes, linear velocity varies with distance from the rotation center. All points on a rigid rotating object have the same angular velocity, but points farther from the center have higher linear velocity. For example, on a rotating disk with ω = 10 rad/s, a point at radius 1 m has v = 10 m/s, while a point at radius 2 m has v = 20 m/s. The center point has zero linear velocity despite rotating.

Tangential velocity and linear velocity are the same thing in circular motion. “Tangential” emphasizes that the velocity direction is tangent to the circular path (perpendicular to the radius). The point moves in a direction perpendicular to the line connecting it to the rotation center. This is why objects released from rotating systems (like a slingshot) fly off tangent to the circle, not radially outward.

The formula v = ωr applies specifically to circular motion or motion that can be approximated as circular at an instant. For elliptical or irregular paths, the relationship becomes more complex. However, for instantaneous calculations, you can use the instantaneous radius of curvature. The formula works for any circular arc, whether it’s a complete circle, partial rotation, or momentary circular movement.

For connected gears or pulleys, the linear velocity at the contact point must be identical. If gear A (radius r₁, angular velocity ω₁) drives gear B (radius r₂, angular velocity ω₂), then r₁ω₁ = r₂ω₂. This means a smaller gear rotating faster can drive a larger gear rotating slower, or vice versa. This principle enables gear ratios to convert between speed and torque in mechanical systems.

Linear velocity is directly proportional to angular velocity (v = ωr). If you double the angular velocity while keeping the radius constant, you exactly double the linear velocity. Similarly, if you triple ω, you triple v. This linear relationship makes it straightforward to predict how changes in rotation speed affect the motion of points on a rotating object.