November 3, 2026 · Indumor
In this chapter
Power is work per unit of time. Torque is force times lever arm. Pressure is force per unit of area. Those three definitions and a handful of constants settle almost everything a shop has to calculate away from the machine tool: the motor for a conveyor, the push of a cylinder, the gearbox ratio and the pull on each leg of a sling.
A kilogram hanging from a rope pulls with 9.81 N, because force is mass times gravity. That force at the end of a 0.5 m lever makes a torque of 4.9 N·m. If the lever turns, torque times angular speed is power; since shop speeds are counted in revolutions per minute, the constant 9550 appears: kW = N·m × rpm / 9550. In US units the same relation reads HP = lbf·ft × rpm / 5252.
F: force (N); m: mass (kg); g = 9.81 m/s²; M: torque (N·m); r: lever arm (m); ω: angular speed (rad/s) = 2·π·n/60; P: power (kW); n: speed (rpm); T: torque in lbf·ft. 9550 is 60000/2π rounded (9549.3).
The table shows what follows: at equal power, torque is inversely proportional to speed. One kilowatt at 3000 rpm delivers 3.2 N·m; the same kilowatt at 100 rpm, downstream of a gearbox, is 95 N·m. That is what gearboxes are for: a little less power comes out than goes in, because of efficiency, and the torque is multiplied by the ratio.
| Speed rpm | Torque per kW N·m | Torque per kW kgf·m | Torque per HP N·m | Torque per HP lbf·ft |
|---|---|---|---|---|
| 100 | 95.49 | 9.738 | 71.21 | 52.52 |
| 300 | 31.83 | 3.246 | 23.74 | 17.51 |
| 500 | 19.10 | 1.948 | 14.24 | 10.50 |
| 750 | 12.73 | 1.298 | 9.49 | 7.00 |
| 1000 | 9.55 | 0.974 | 7.12 | 5.25 |
| 1450 | 6.59 | 0.672 | 4.91 | 3.62 |
| 1500 | 6.37 | 0.649 | 4.75 | 3.50 |
| 2900 | 3.29 | 0.336 | 2.46 | 1.81 |
| 3000 | 3.18 | 0.325 | 2.37 | 1.75 |
1450 and 2900 rpm are the actual speeds of 4- and 2-pole motors on 50 Hz; 1500 and 3000 are synchronous. On 60 Hz the equivalents are 1750 and 3500 rpm.
Pick what to calculate and enter the other two values; the one being calculated is ignored. It works at the gearbox output as well as at the motor.
The ratio i of a drive is input speed over output speed, and it equals the diameters or tooth counts the other way round: i = n1/n2 = d2/d1 = z2/z1. In series, ratios multiply and so do efficiencies. A two-stage helical gearbox (0.97 × 0.97) loses 6 %; a 40:1 worm drive can turn a third of the power into heat. Gear formulas are in chapter 15 and belts and chains in chapter 16.
n: speed; d: pulley or pitch diameter; z: number of teeth; subscript 1 is the input (motor) and 2 the output. With i > 1 the speed drops and the torque rises.
| Stage | Efficiency % | Notes |
|---|---|---|
| Spur gears (one pair) | 97 a 99 | per stage, lubricated |
| Helical gears (one pair) | 96 a 98 | per stage; the usual industrial gearbox |
| Bevel gears | 95 a 98 | per stage |
| Worm and wheel | 50 a 90 | falls with the ratio: about 85 % at 10:1, 60 % at 60:1 and 50 % at 100:1 |
| V-belt | 94 a 97 | new and properly tensioned; lower if it slips |
| Timing belt (synchronous) | 97 a 98 | – |
| Roller chain | 96 a 98 | lubricated |
| Pair of rolling bearings on a shaft | 99 | per shaft |
| Whole drive | product of the stages | two helical stages and a V-belt: 0.97 × 0.97 × 0.95 = 0.89 |
Worked example · the gearbox of a belt conveyor
A conveyor has a 5.5 kW motor at 1450 rpm and a 20:1 gearbox with 0.94 efficiency (two helical stages). Motor torque: 9550 × 5.5 / 1450 = 36.2 N·m. At the output: 72.5 rpm and 36.2 × 20 × 0.94 = 681 N·m. On a 320 mm drum that torque pulls the belt with 681 / 0.16 = 4256 N (434 kgf, 957 lbf) at 1.21 m/s, and 4256 × 1.21 = 5.2 kW, which is what went in minus the losses. If the belt needs 600 kgf of pull, the motor is short even though the gearbox is right.
A three-phase induction motor runs a little below its synchronous speed, which depends on line frequency and the number of poles: 3600 rpm for 2 poles at 60 Hz, 1800 for 4, 1200 for 6 and 900 for 8 (3000, 1500, 1000 and 750 on a 50 Hz grid). Full-load slip runs from 1 % on large motors to 5 % or more on small ones, which is why the nameplate says 1750 or 1450 rather than 1800 or 1500.
| Poles | 60 Hz synchronous rpm | 60 Hz full load rpm, typical | 50 Hz synchronous rpm | 50 Hz full load rpm, typical |
|---|---|---|---|---|
| 2 | 3600 | 3400 to 3580 | 3000 | 2800 to 2980 |
| 4 | 1800 | 1680 to 1790 | 1500 | 1380 to 1490 |
| 6 | 1200 | 1100 to 1190 | 1000 | 900 to 990 |
| 8 | 900 | 830 to 890 | 750 | 680 to 740 |
IEC frames are named after the shaft height in millimeters (a 132 frame has its shaft centerline 132 mm above the feet) and fix the foot pattern, shaft diameter and key: a 7.5 kW 4-pole motor from any maker goes in a 132M frame with a 38 mm shaft. NEMA frames (143T, 182T, 256T and so on) follow a different system, with inch shafts. The current in the table comes from the formula with typical efficiencies and power factors at 380 V; at 460 V multiply by 0.83, at 400 V by 0.95. The nameplate current is the one that sizes the overload relay and the cable.
I: line current (A); P: shaft power (kW); U: line-to-line voltage (V); cos φ: power factor; η: efficiency (0.7 to 0.95 depending on size).
| IEC frame | Power kW | Power HP | Shaft dia. mm | Typical η % | Power factor typical | Current at 380 V A |
|---|---|---|---|---|---|---|
| 63 | 0.18 | 0.2 | 11 | 65.0 | 0.70 | 0.6 |
| 71 | 0.37 | 0.5 | 14 | 72.0 | 0.72 | 1.1 |
| 80 | 0.75 | 1.0 | 19 | 78.0 | 0.76 | 1.9 |
| 90S | 1.10 | 1.5 | 24 | 80.0 | 0.78 | 2.7 |
| 90L | 1.50 | 2.0 | 24 | 82.0 | 0.80 | 3.5 |
| 100L | 2.20 | 3.0 | 28 | 84.0 | 0.81 | 4.9 |
| 100L | 3.00 | 4.0 | 28 | 85.0 | 0.82 | 6.5 |
| 112M | 4.00 | 5.4 | 28 | 86.0 | 0.82 | 8.6 |
| 132S | 5.50 | 7.4 | 38 | 87.0 | 0.83 | 11.6 |
| 132M | 7.50 | 10.1 | 38 | 88.0 | 0.84 | 15.4 |
| 160M | 11.00 | 14.8 | 42 | 89.0 | 0.85 | 22.1 |
| 160L | 15.00 | 20.1 | 42 | 90.0 | 0.85 | 29.8 |
| 180M | 18.50 | 24.8 | 48 | 90.5 | 0.86 | 36.1 |
| 180L | 22.00 | 29.5 | 48 | 91.0 | 0.86 | 42.7 |
| 200L | 30.00 | 40.2 | 55 | 92.0 | 0.87 | 56.9 |
| 225S | 37.00 | 49.6 | 60 | 92.5 | 0.87 | 69.9 |
| 225M | 45.00 | 60.3 | 60 | 93.0 | 0.87 | 84.5 |
| 250M | 55.00 | 73.8 | 65 | 93.5 | 0.87 | 102.7 |
| 280S | 75.00 | 100.6 | 75 | 94.0 | 0.88 | 137.8 |
| 280M | 90.00 | 120.7 | 75 | 94.5 | 0.88 | 164.4 |
Current from I = P / (√3 · U · cos φ · η) with typical η and power factor; the nameplate value is the real one. At 400 V multiply by 0.95, at 460 V by 0.83, at 230 V three-phase by 1.65.
Power factor and efficiency are on the nameplate. Size the overload relay and the cable from the nameplate current, not from this estimate.
The force of a cylinder is pressure times piston area, nothing more. Extending, the pressure works on the full circle; retracting, on the annulus left once the rod is subtracted, so a cylinder pulls less than it pushes. With bar and cm², the force comes out in kN divided by 100 (and in kgf, the kN times 102); with psi and in² it comes out in lbf directly. Flow sets the speed, Q = A·v, and the pump power is L/min times bar divided by 600, in kW and before efficiencies. The usual oil is ISO VG 46 or 68 (ISO 3448).
p: pressure; A: piston area (push) or annular area (pull); v: rod speed; Q: flow. In SI units F (N) = p (Pa) · A (m²). 1 bar = 10 N/cm² = 14.5 psi.
| Bore | Area | Push in kN | Push in kgf | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Dia. mm | A cm² | 100 bar | 160 bar | 210 bar | 250 bar | 100 bar | 160 bar | 210 bar | 250 bar |
| 32 | 8.04 | 8.0 | 12.9 | 16.9 | 20.1 | 820 | 1312 | 1722 | 2050 |
| 40 | 12.57 | 12.6 | 20.1 | 26.4 | 31.4 | 1281 | 2050 | 2691 | 3204 |
| 50 | 19.63 | 19.6 | 31.4 | 41.2 | 49.1 | 2002 | 3204 | 4205 | 5006 |
| 63 | 31.17 | 31.2 | 49.9 | 65.5 | 77.9 | 3179 | 5086 | 6675 | 7947 |
| 80 | 50.27 | 50.3 | 80.4 | 105.6 | 125.7 | 5126 | 8201 | 10764 | 12814 |
| 100 | 78.54 | 78.5 | 125.7 | 164.9 | 196.3 | 8009 | 12814 | 16819 | 20022 |
| 125 | 122.72 | 122.7 | 196.3 | 257.7 | 306.8 | 12514 | 20022 | 26279 | 31285 |
| 160 | 201.06 | 201.1 | 321.7 | 422.2 | 502.7 | 20503 | 32804 | 43055 | 51257 |
| 200 | 314.16 | 314.2 | 502.7 | 659.7 | 785.4 | 32035 | 51257 | 67274 | 80088 |
| Bore | Rod | Annular area | Pull in kN | Pull in kgf | ||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Dia. mm | Dia. mm | A cm² | 100 bar | 160 bar | 210 bar | 250 bar | 100 bar | 160 bar | 210 bar | 250 bar |
| 32 | 14 | 6.50 | 6.5 | 10.4 | 13.7 | 16.3 | 663 | 1061 | 1393 | 1658 |
| 32 | 22 | 4.24 | 4.2 | 6.8 | 8.9 | 10.6 | 432 | 692 | 908 | 1081 |
| 40 | 18 | 10.02 | 10.0 | 16.0 | 21.0 | 25.1 | 1022 | 1635 | 2146 | 2555 |
| 40 | 28 | 6.41 | 6.4 | 10.3 | 13.5 | 16.0 | 654 | 1046 | 1372 | 1634 |
| 50 | 22 | 15.83 | 15.8 | 25.3 | 33.3 | 39.6 | 1615 | 2583 | 3391 | 4036 |
| 50 | 36 | 9.46 | 9.5 | 15.1 | 19.9 | 23.6 | 964 | 1543 | 2025 | 2411 |
| 63 | 28 | 25.01 | 25.0 | 40.0 | 52.5 | 62.5 | 2551 | 4081 | 5357 | 6377 |
| 63 | 45 | 15.27 | 15.3 | 24.4 | 32.1 | 38.2 | 1557 | 2491 | 3270 | 3892 |
| 80 | 36 | 40.09 | 40.1 | 64.1 | 84.2 | 100.2 | 4088 | 6540 | 8584 | 10219 |
| 80 | 56 | 25.64 | 25.6 | 41.0 | 53.8 | 64.1 | 2614 | 4183 | 5490 | 6535 |
| 100 | 45 | 62.64 | 62.6 | 100.2 | 131.5 | 156.6 | 6387 | 10219 | 13413 | 15968 |
| 100 | 70 | 40.06 | 40.1 | 64.1 | 84.1 | 100.1 | 4085 | 6535 | 8577 | 10211 |
| 125 | 56 | 98.09 | 98.1 | 156.9 | 206.0 | 245.2 | 10002 | 16004 | 21005 | 25006 |
| 125 | 90 | 59.10 | 59.1 | 94.6 | 124.1 | 147.8 | 6027 | 9643 | 12656 | 15067 |
| 160 | 70 | 162.58 | 162.6 | 260.1 | 341.4 | 406.4 | 16578 | 26525 | 34814 | 41446 |
| 160 | 110 | 106.03 | 106.0 | 169.6 | 222.7 | 265.1 | 10812 | 17299 | 22705 | 27030 |
| 200 | 90 | 250.54 | 250.5 | 400.9 | 526.1 | 626.4 | 25548 | 40877 | 53651 | 63870 |
| 200 | 140 | 160.22 | 160.2 | 256.4 | 336.5 | 400.6 | 16338 | 26141 | 34310 | 40845 |
100 bar = 1450 psi, 160 bar = 2320 psi, 210 bar = 3046 psi, 250 bar = 3626 psi. 1 kN = 225 lbf.
| Bore mm | Rod dia. mm | Area cm² | Push N | Push kgf | Annular area cm² | Pull N | Pull kgf |
|---|---|---|---|---|---|---|---|
| 32 | 12 | 8.04 | 483 | 49.2 | 6.91 | 415 | 42.3 |
| 40 | 16 | 12.57 | 754 | 76.9 | 10.56 | 633 | 64.6 |
| 50 | 20 | 19.63 | 1178 | 120.1 | 16.49 | 990 | 100.9 |
| 63 | 20 | 31.17 | 1870 | 190.7 | 28.03 | 1682 | 171.5 |
| 80 | 25 | 50.27 | 3016 | 307.5 | 45.36 | 2721 | 277.5 |
| 100 | 25 | 78.54 | 4712 | 480.5 | 73.63 | 4418 | 450.5 |
| 125 | 32 | 122.72 | 7363 | 750.8 | 114.68 | 6881 | 701.6 |
| 160 | 40 | 201.06 | 12064 | 1230.2 | 188.50 | 11310 | 1153.3 |
| 200 | 40 | 314.16 | 18850 | 1922.1 | 301.59 | 18096 | 1845.2 |
A 100 mm pneumatic cylinder at 6 bar (87 psi) pushes 481 kgf; a hydraulic one of the same bore at 210 bar (3000 psi), 16800. That is why air moves and oil clamps.
Bore and rod in mm (1 in = 25.4 mm). For a pneumatic cylinder enter the line pressure (6 bar, 87 psi). Forces are theoretical: seal friction takes 5 to 10 %.
The force to drag something is its weight times the coefficient of friction, which depends on the two materials and on the lubricant. The table values are indicative: steel on dry steel runs from 0.5 to 0.8 with finish and cleanliness, and an oil film brings it down to 0.1. The static coefficient, for starting, is always higher than the kinetic one, for keeping the motion going; that is why a load breaks free with a jerk.
| Materials | Static μ₀ | Kinetic μ | Condition |
|---|---|---|---|
| Steel on steel | 0.5 to 0.8 | 0.4 to 0.6 | dry |
| Steel on steel | 0.1 to 0.16 | 0.05 to 0.1 | lubricated |
| Steel on gray cast iron | 0.4 | 0.23 | dry |
| Steel on gray cast iron | 0.18 to 0.21 | 0.13 | lubricated |
| Steel on bronze | 0.35 | 0.3 | dry |
| Steel on bronze | 0.15 | 0.1 | lubricated |
| Steel on wood | 0.5 to 0.6 | 0.3 to 0.5 | dry |
| Rubber on concrete | 0.6 to 1.0 | 0.5 to 0.8 | dry; wet drops to 0.45 |
| Steel on concrete | 0.45 to 0.6 | 0.4 to 0.5 | dry (base plates, supports) |
Simple machines trade force for travel: the ideal mechanical advantage says how many times the force is multiplied, and the real one is lower because of friction, which on a screw eats most of the effort. The stress a part carries is force over section, σ = F/A, in N/mm² (MPa) or psi, and the safety factor is what the material withstands over what it works at. Beams, columns and buildings are not worked out here: they live in the Engineering Suite. Bolt tightening torque is in chapter 18.
σ: stress (N/mm² = MPa); F: force (N); A: section (mm²); SF: safety factor (2 to 4 on machine parts, more in lifting gear); μ: coefficient of friction; N: normal force (the weight, on a horizontal support).
| Machine | Mechanical advantage | Effort needed | Notes |
|---|---|---|---|
| First-class lever (fulcrum in the middle) | VM = a / b | F = Q · b / a | a: effort arm; b: load arm. Crowbar, scissors |
| Second-class lever (load in the middle) | VM = a / b > 1 | F = Q · b / a | wheelbarrow, nutcracker |
| Third-class lever (effort in the middle) | VM = a / b < 1 | F = Q · b / a | tweezers, the human arm: gains travel, not force |
| Fixed pulley | VM = 1 | F = Q | only changes the direction |
| Movable pulley | VM = 2 | F = Q / 2 | the load hangs from two parts of rope |
| Block and tackle, n parts on the moving block | VM = n | F = Q / n | real: F ≈ Q / (n · η), with overall η of 0.8 to 0.95: it drops with each sheave |
| Windlass (drum r, crank R) | VM = R / r | F = Q · r / R | – |
| Inclined plane (length L, height h, angle α) | VM = L / h | F = Q · (sen α + μ · cos α) | without friction F = Q · h / L |
| Wedge (length L, thickness e) | VM = L / e | F = Q · e / L | ideal; friction makes it self-locking |
| Screw jack (lead p, lever r) | VM = 2 · π · r / p | F = Q · p / (2 · π · r · η) | screw η 0.2 to 0.4: most of the effort goes into friction |
Two vertical legs split the load in half. As the angle between them opens, each leg carries more than half, and at 120° each carries the whole load. The factor is 1/cos(θ/2), with θ the included angle: 1.41 at 90°. That is why slings are rated by angle and why a short sling on a wide part is a problem. How lifting lugs and eye bolts are chosen is in the lifting guide.
T: tension in each leg; Q: weight of the load (same unit); θ: included angle between the two legs; WLL: working load limit of one leg. With unequal legs, the more vertical one carries more.
| Included angle θ | Angle to vertical ° | Tension per leg × Q | Factor 1 / cos(θ/2) | Capacity of the pair × WLL of one leg |
|---|---|---|---|---|
| 0° | 0 | 0.500 | 1.000 | 2.000 |
| 30° | 15 | 0.518 | 1.035 | 1.932 |
| 45° | 23 | 0.541 | 1.082 | 1.848 |
| 60° | 30 | 0.577 | 1.155 | 1.732 |
| 90° | 45 | 0.707 | 1.414 | 1.414 |
| 120° | 60 | 1.000 | 2.000 | 1.000 |
WLL: working load limit of one vertical leg. Included angles over 120° are not used; many standards and makers stop at 90°.
M = 9550 · P / n with P in kW and n in rpm, or T = 5252 · HP / n in lbf·ft. A 5.5 kW motor at 1450 rpm gives 36 N·m; after a 20:1 gearbox at 94 % efficiency, 681 N·m at 72.5 rpm.
Pressure times piston area: F [kN] = p [bar] × A [cm²] / 100, or lbf = psi × in². A 100 mm bore (78.5 cm²) at 210 bar pushes 165 kN, about 16.8 metric tons; retracting with a 70 mm rod, 84 kN.
About 13 A at full load by the formula (7.5 kW with 0.88 efficiency and 0.84 power factor); the NEC table lists 14 A. At 380 V it is about 15 A. The nameplate value is the one to use for the overload relay.
Sources: ISO 80000-3:2019 and ISO 80000-4:2019 (quantities of space, time and mechanics); IEC 60034-1 (rotating electrical machines: rating and performance); IEC 60072-1 (dimensions and output series of IEC frames 56 to 400) and EN 50347 (power-to-frame assignment); ISO 6020-2 (160 bar compact hydraulic cylinders); ISO 15552 (pneumatic cylinders); ISO 3448 (ISO VG viscosity grades); DIN 15003 (lifting appliances: loads and forces); the lifting guides of the Escuela del Acero. Drive efficiencies and friction coefficients: indicative values from mechanical engineering handbooks and gearbox catalogs.
Belt profiles, belt length and chain ratings for the drive downstream of the motor.
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