Enter the total angular velocity and radius of rotation into the calculator to determine the linear tangent speed of an object in rotation. This calculator can also evaluate either the angular speed or radius when given the values of the other variables.
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Linear Speed Formula
The core formula relating angular and linear speed is:
v = r * w
- Where v is the linear (tangential) velocity in m/s
- r is the radius of rotation in meters
- w (omega) is the angular velocity in rad/s
When rotational speed is given in RPM instead of rad/s, convert first: w = 2 * pi * RPM / 60. For machining, surface speed is often expressed as SFM (surface feet per minute): SFM = pi * D (in) * RPM / 12, where D is the tool or workpiece diameter in inches.
Linear Speed vs. Angular Speed
Angular speed measures the rate of rotation (radians per second, degrees per second, or RPM), while linear speed measures the distance a point on the rotating object travels per unit time. Two points on the same spinning disk share the same angular speed, but the point farther from the center has a higher linear speed because it traces a larger circle each revolution. For a wheel spinning at 10 rad/s, a point at 0.3 m from the center moves at 3 m/s, while a point at 0.5 m moves at 5 m/s.
Linear Speeds of Common Rotating Systems
The table below shows typical linear speeds for familiar rotating objects, illustrating how widely this concept applies across scales.
| Rotating System | Typical Radius | Typical RPM or Angular Speed | Linear Speed |
|---|---|---|---|
| Clock second hand (wall clock) | 0.15 m | 1 RPM | 0.016 m/s |
| Bicycle wheel (road bike at 25 km/h) | 0.34 m | ~196 RPM | 6.94 m/s (25 km/h) |
| Car tire (highway speed, 100 km/h) | 0.33 m | ~808 RPM | 27.8 m/s (100 km/h) |
| Washing machine drum (spin cycle) | 0.25 m | ~1,200 RPM | 31.4 m/s |
| Industrial conveyor pulley | 0.10 m (200 mm dia.) | ~100 RPM | 1.05 m/s (63 m/min) |
| Typical Ferris wheel | ~25 m | ~0.5 RPM | 1.3 m/s (4.7 km/h) |
| Wind turbine blade tip (large onshore) | ~60 m | ~15 RPM | 94.2 m/s (339 km/h) |
| Helicopter main rotor tip | ~8 m | ~300 RPM | 251 m/s (904 km/h) |
| Hard disk drive platter (7,200 RPM) | 0.044 m (3.5″ drive edge) | 7,200 RPM | 33.2 m/s (119.5 km/h) |
| Dental drill burr tip | ~0.001 m (1 mm) | ~400,000 RPM | 41.9 m/s |
Earth’s Rotational Linear Speed by Latitude
Earth completes one rotation every 23 hours, 56 minutes, and 4 seconds (sidereal day). The linear speed at any latitude equals the equatorial speed multiplied by the cosine of that latitude. Earth’s equatorial radius is 6,378.1 km, yielding a surface speed of about 465.1 m/s (1,674.4 km/h) at the equator.
| Latitude | Location Example | Linear Speed (km/h) | Linear Speed (mph) | Linear Speed (m/s) |
|---|---|---|---|---|
| 0 (Equator) | Quito, Ecuador | 1,674.4 | 1,040.4 | 465.1 |
| 15 | Guatemala City | 1,617.0 | 1,004.7 | 449.2 |
| 30 | Cairo, Egypt | 1,450.2 | 901.1 | 402.8 |
| 40 | New York City, USA | 1,282.5 | 796.9 | 356.3 |
| 45 | Montreal, Canada | 1,183.7 | 735.5 | 328.8 |
| 51.5 | London, UK | 1,043.3 | 648.2 | 289.8 |
| 60 | Helsinki, Finland | 837.2 | 520.2 | 232.6 |
| 66.5 (Arctic Circle) | Rovaniemi, Finland | 667.0 | 414.5 | 185.3 |
| 90 (Pole) | North/South Pole | 0 | 0 | 0 |
This speed difference by latitude has practical consequences. Rocket launch sites are preferentially located near the equator (for example, the European Space Agency’s launch site in French Guiana at 5.2 N) because the higher surface speed provides a free velocity boost during eastward launches, reducing the fuel required to reach orbit.
Machining Surface Speed Reference
In CNC machining and manual lathe/mill work, linear speed appears as surface speed, measured in SFM (surface feet per minute) or m/min. Correct surface speed selection directly controls tool life, surface finish, and heat generation. The table below lists typical recommended cutting speeds for common materials with both HSS (high-speed steel) and carbide tooling.
| Workpiece Material | HSS Turning (SFM) | Carbide Turning (SFM) | HSS Drilling (SFM) | Carbide Drilling (SFM) |
|---|---|---|---|---|
| Aluminum (6061-T6) | 200-300 | 800-1,000 | 200-250 | 500-800 |
| Mild Steel (1018) | 80-120 | 400-600 | 70-100 | 250-400 |
| Stainless Steel (304) | 60-100 | 250-400 | 30-50 | 100-200 |
| Brass (C360) | 200-300 | 600-1,000 | 150-200 | 400-600 |
| Cast Iron (Class 30) | 60-80 | 250-400 | 50-70 | 150-250 |
| Titanium (Ti-6Al-4V) | 30-50 | 100-200 | 15-30 | 50-100 |
| Copper (C110) | 100-200 | 400-600 | 80-130 | 250-400 |
| Plastics (Acetal, Nylon) | 200-500 | 500-1,000 | 100-300 | 300-600 |
| Wood (Hardwood) | N/A | 600-1,200 | N/A | 300-800 |
To convert SFM to RPM for a given tool diameter: RPM = (SFM x 12) / (pi x D), where D is in inches. For metric: RPM = (Vc x 1000) / (pi x D), where Vc is in m/min and D in mm. Carbide tooling tolerates 3 to 5 times the surface speed of HSS because it retains hardness at much higher temperatures (carbide stays hard above 800 C, while HSS softens around 600 C).
Key Relationships Between Rotational and Linear Quantities
Linear speed connects to several other rotational quantities. Linear (tangential) acceleration equals radius times angular acceleration: a_t = r * alpha. Centripetal acceleration, which points inward toward the center, equals v^2 / r, or equivalently r * w^2. For an object on a circular path, the total linear acceleration is the vector sum of the tangential and centripetal components. Arc length traveled equals radius times angle in radians: s = r * theta. Period (T) and frequency (f) relate to angular velocity by w = 2 * pi * f = 2 * pi / T.
Unit Conversions for Angular and Linear Speed
| From | To | Multiply By |
|---|---|---|
| RPM | rad/s | 2 * pi / 60 = 0.10472 |
| deg/s | rad/s | pi / 180 = 0.01745 |
| m/s | km/h | 3.6 |
| m/s | mph | 2.23694 |
| m/s | ft/s | 3.28084 |
| ft/min (SFM) | m/min | 0.3048 |
| rev/s (Hz) | rad/s | 2 * pi = 6.28318 |
FAQ
Linear speed is the distance traveled per unit time by a point moving along a path. For a rotating object, it represents the tangential velocity at a given radius from the axis of rotation, calculated as v = r * w.
No. Speed is a scalar quantity (magnitude only), while velocity is a vector (magnitude plus direction). A car moving at 60 km/h has a linear speed of 60 km/h regardless of direction, but its velocity also specifies which direction it travels. For circular motion, the linear speed stays constant while the velocity continuously changes direction.
Surface speed determines the rate at which the cutting edge interacts with the workpiece material. Too low a surface speed causes rubbing instead of cutting, work-hardening the material and shortening tool life. Too high a surface speed generates excessive heat, accelerating tool wear. Each material and tool combination has an optimal SFM range that balances cut quality, tool longevity, and productivity.
