Rotation and Linear Speed
Turn a shaft speed into the speed of what it drives, and back again, using the radius - belts, rollers, wheels, and grinding wheels.
The radius is what links turning to travelling; Watch out
The radius is what links turning to travelling
A roller turns; the belt on it travels. A wheel turns; the vehicle moves. The two speeds are not the same number, and the thing that converts one into the other is the radius.
v = ωr · a = αr
Here a is the tangential acceleration, not the rim point’s total
acceleration: it also has an inward component ω2r. Belt and
vehicle speeds match rim speed only when there is no slipping.
- v
- The straight-line speed of a point on the rim — the speed of the belt, the sheet, the vehicle.
- ω
- Angular velocity, in radians per second. See Angular Motion for the conversion from rpm.
- r
- The radius, in the same length unit you want the answer in. Rollers and wheels are almost always specified by diameter, so halve it first.
Both of these formulas are only true when the angle is in radians. Feed them rpm and the answer is wrong by a factor of about 9.55; feed them degrees per second and it is wrong by a factor of about 57.3. Convert to rad/s first, every time.
Two mistakes account for nearly every wrong answer here: using the diameter where the radius belongs, and leaving the speed in rpm. Both are worth a deliberate check before the arithmetic starts.
Why the radian is the only angle unit that works
A radian is defined as arc length divided by radius. Multiply an angle in radians by a radius and you get the arc length back — which is exactly the distance the rim travelled. Do the same thing per second and you have the rim speed. Degrees and revolutions carry an extra arbitrary count inside them, so they need a conversion factor first. The radian does not, because it was built out of the same geometry the formula describes.
Worked example: a conveyor drive roller
Worked example: a conveyor drive roller
A conveyor belt runs over a drive roller 200 mm in diameter, turning at 45 rpm. What is the belt speed, and how long does a package take to travel 15 m?
- Halve the diameter:
r = 100 mm = 0.100 m. - Convert the speed:
45 r/min × 0.1047198 rad/s per r/min = 4.712389 rad/s. - Multiply:
v = 4.712389 rad/s × 0.100 m = 0.4712 m/s. - Divide the distance by the speed:
15 m ÷ 0.4712389 m/s = 31.83 s.
0.47 m/s, and about 32 s for 15 m. That assumes the belt does not slip on the roller, which is the assumption a real drive is designed to make true.
Worked example: a roll-forming feed rate
Worked example: a roll-forming feed rate
Forming rolls 150 mm in diameter rotate at 25 rpm. What is the feed rate of the sheet in m/min, and how long does 20 m of sheet take?
r = 75 mm = 0.075 m.25 r/min × 0.1047198 rad/s per r/min = 2.617994 rad/s.v = 2.617994 rad/s × 0.075 m = 0.19635 m/s.- The question asks for metres per minute, so convert the time unit:
0.19635 m/s × 60 s/min = 11.78 m/min. 20 m ÷ 11.78 m/min = 1.70 min, which is about 1 min 42 s.
Note what step 4 did and did not do. Changing seconds to minutes in a speed takes one factor of 60. If the quantity had been an acceleration, the same change would take 602 — see Acceleration and Squared Time.
Worked example: running the conversion backwards
Worked example: running the conversion backwards
Sometimes the linear quantity is what you know. The formulas simply rearrange:
ω = v/r and α = a/r.
A wheel, from angular to linear
A tyre 28 in in diameter turns at 3.5 rad/s. How fast is the vehicle moving, in miles per hour?
r = 14 in × 0.0254 m/in = 0.3556 m.v = 3.5 rad/s × 0.3556 m = 1.2446 m/s.1.2446 m/s ÷ 0.44704 m/s per mph = 2.78 mph.
A wheel, from linear to angular
A vehicle accelerates at 5 m/s2 on tyres 78 cm in diameter. What is the angular acceleration of a wheel?
r = 39 cm = 0.39 m.α = 5 m/s2 ÷ 0.39 m = 12.8 rad/s2.
Check the unit on the answer, not just the number. Dividing
m/s2 by m leaves 1/s2, and because
the radian is a ratio of two lengths, rad/s2 is the same thing named for
what it describes.
Rim speed is why wheel size changes the safe rpm; Stop
Rim speed is why wheel size changes the safe rpm
A grinding wheel is limited by how fast its rim moves, not by how fast its centre turns. Put a bigger wheel on the same spindle at the same rpm and the rim goes faster, because every point on it travels a longer circle in the same time.
Take a wheel whose rim speed must stay at or below 35 m/s:
| Wheel diameter | Radius | ω = v/r | Same speed in rpm |
|---|---|---|---|
| 200 mm | 0.100 m | 350 rad/s | 3,342 rpm |
| 250 mm | 0.125 m | 280 rad/s | 2,674 rpm |
The larger wheel has to run slower, by about 670 rpm, to keep the same rim speed. Nothing about the machine changed; only the radius did.
Do not calculate your way to a grinding-wheel speed. The number that governs
is the maximum operating speed marked on the wheel, and OSHA requires that the
spindle speed of the machine be checked before a wheel is mounted, to be certain it
does not exceed that marking (29 CFR 1910.215(d)(1)). The arithmetic
above explains why a bigger wheel is marked slower. It does not authorise a wheel, and
it is not a substitute for reading the wheel. See Machine Hazards and Housekeeping.
Sources
Sources
Where these values were checked
Conversion factors from
NIST
SP 811, Appendix B: revolution per minute to radian per second 1.047 198
E−01, inch to meter 2.54 E−02 (exact), and mile per hour to
meter per second 4.4704 E−01 (exact).
The grinding-wheel requirement is quoted from 29 CFR 1910.215(d)(1): “The spindle speed of the machine shall be checked before mounting of the wheel to be certain that it does not exceed the maximum operating speed marked on the wheel.”