Relative centrifugal force is the rotor’s centripetal acceleration divided by gravity, RCF = ω²r ÷ g, which for r in centimeters and N in RPM becomes RCF = 1.118 × 10⁻⁵ × r × N². Because a rotor’s strength limits its rim speed, not its RPM, the maximum force a given rotor material allows is v² ÷ (g × r), so a smaller rotor can safely reach more g than a larger one even though it spins faster.
Part of Centrifuge Building
Centrifugal force calculations convert a rotor’s RPM and radius into multiples of gravity, rim speed and the force on a given mass, the three numbers every design and every safety check depends on.
One Formula, Three Faces
A point at radius r on a rotor turning at N revolutions per minute has angular speed ω = 2πN ÷ 60 radians per second. Its centripetal acceleration is ω²r, and dividing by gravity (9.81 m/s²) gives the relative centrifugal force, the RCF, in multiples of g:
RCF = ω² × r ÷ g = (2πN ÷ 60)² × r ÷ g
Put r in centimeters (so r in meters is r ÷ 100) and the constants collapse: (2π ÷ 60)² = 0.010966; ÷ 9.81 ÷ 100 = 1.118 × 10⁻⁵. That gives the formula used throughout this wiki, RCF = 1.118 × 10⁻⁵ × r × N².
The same acceleration can be written with the rim speed v = ωr as a = v² ÷ r, so RCF = v² ÷ (g × r). This is the more useful face for design, because a material limits v and not N (see Speed Limits and Material Stress).
Worked Example: The Reference Rotor And The Table
The cream separator bowl in Cream Separator is the reference: diameter 150 mm, radius 75 mm, at 8,000 RPM.
- ω = 2π × 8,000 ÷ 60 = 837.8 rad/s
- Acceleration: ω² × r = 837.8² × 0.075 = 701,838 × 0.075 = 52,640 m/s²
- RCF: 52,640 ÷ 9.81 = 5,366 g. Check with the shortcut: 1.118 × 10⁻⁵ × 7.5 × 8,000² = 1.118 × 10⁻⁵ × 7.5 × 64,000,000 = 5,366 g. The two agree.
- Rim speed: v = ωr = 837.8 × 0.075 = 62.8 m/s. Check: v² ÷ (g × r) = 3,948 ÷ (9.81 × 0.075) = 5,366 g.
The carrier article’s table is consistent with the formula: 10 cm radius at 3,000 RPM gives 1.118 × 10⁻⁵ × 10 × 9,000,000 = 1,006 g, and 15 cm at 10,000 RPM gives 1.118 × 10⁻⁵ × 15 × 10⁸ = 16,770 g.
Inverting for RPM: N = √(RCF ÷ (1.118 × 10⁻⁵ × r)). To reach the 5,000-8,000 g target for cream separation quoted in Centrifuge Building on a 75 mm radius: N = √(5,000 ÷ (1.118 × 10⁻⁵ × 7.5)) = √(5.963 × 10⁷) = 7,722 RPM, and for 8,000 g N = 9,768 RPM. On a 100 mm radius the same targets take 6,687 and 8,459 RPM. That is why the carrier’s 6,000-10,000 RPM range for the 150-200 mm bowls is the right order, and why a 150 mm bowl at 6,000 RPM gives only 1.118 × 10⁻⁵ × 7.5 × 6,000² = 3,019 g, below the target.
Force On A Mass, And Why Imbalance Is Violent
The force on a mass m at radius r is F = m × ω² × r, in newtons when m is in kilograms and r in meters. This is a force, not a mass: dividing by 9.81 gives kilogram-force, the weight of that many kilograms. Mixing the two is a 9.81 times error.
- 1 g at 75 mm on the reference bowl: F = 0.001 × 701,838 × 0.075 = 52.6 N, which is the weight of about 5.4 kg (52.6 ÷ 9.81).
- 10 g at 100 mm and 5,000 RPM (the carrier’s example): ω = 523.6 rad/s, ω² = 274,156; F = 0.010 × 274,156 × 0.100 = 274 N, about 28 kg-force, rotating with the rotor.
That force swings around once per revolution, which is what shakes the machine; see Dynamic Balancing.
The Design Consequence: Maximum g Falls As The Rotor Gets Bigger
If the rim speed is capped at some v_max by the material, the highest RCF is v_max² ÷ (g × r). With a safe rim speed of 103.4 m/s (derived in Speed Limits and Material Stress):
| Radius | Highest RPM at 103.4 m/s | RCF at that RPM |
|---|---|---|
| 50 mm | 19,740 | 21,800 g |
| 100 mm | 9,870 | 10,900 g |
| 300 mm | 3,290 | 3,630 g |
Each row is checked by v²/(g r): 103.4² ÷ (9.81 × 0.05) = 10,692 ÷ 0.4905 = 21,800. The table is why a basket centrifuge 300 mm in radius cannot give the 10,000+ g of a fine-particle machine: it would fly apart first.
Common Mistakes
- Newtons labelled kilograms. A force of 274 N is not 274 kg.
- Quoting RCF without the radius at which it applies.
- Using diameter for r. Halving it halves the force.
- Linearizing. Doubling the RPM is four times the force, not two.
Safety
The formula is also the load on the rotor
The acceleration that gives 5,366 g to milk gives the same acceleration to every part of the rotor, fastener and bearing. Compute the force on each fixing, not only the separation; keep the rim speed below the limit derived in Speed Limits and Material Stress and use Safety Factors on top.
What’s Next
- Learn next: Speed Limits and Material Stress — what the rim speed limit is
- Combine with: Separation Theory — what the g actually buys you