How this instrument works
Force pushes; moment twists. Multiply how hard you push by how far out you push it, and you have whatever decides if a bolt turns, a door swings, or a crankshaft picks up speed. That sin θ factor carries every bit of subtlety: only whatever component acts across your lever arm twists anything at all. Shove straight down a spanner handle, directly at your bolt, and θ is zero — arithmetic gives nothing back, and neither does that bolt.
Read r·sin θ as one thing and it becomes your moment arm: perpendicular distance from pivot out to whatever line your force acts along. Slide that force anywhere on its own line and its moment never budges, which is why a cranked or offset wrench delivers what straight-line geometry says rather than what handle length hints at. Archimedes proved this underlying law of levers geometrically around 250 BC in On the Equilibrium of Planes, centuries before anyone possessed a word for force; Pierre Varignon recast it in 1687 as force times perpendicular distance. Our English name arrived very late — James Thomson, elder brother of Lord Kelvin, proposed torque in 1884, borrowing Latin torquere, to twist.
Everything here assumes one rigid body, one force, and one clearly chosen pivot inside a single plane. Add further forces and you add their moments, minding signs. Full torque is a vector, r × F, aimed along its rotation axis by right-hand rule; this sheet hands back magnitude alone. Nor does one line of algebra say what happens next — that wants τ = I·α, where an object's moment of inertia settles how briskly it truly spins up. For scale: a spectacle hinge screw asks about 0.1 N·m, a bicycle stem bolt 5, a car wheel nut near 110, a loaded truck wheel nut 600, and main shafts on large wind turbines several million.
- Enter your push into Force — newtons, kilonewtons or pounds-force, whichever your gauge reads out.
- Put distance from pivot centre out to where that push lands into Lever arm length.
- Set Angle to the lever arm. Ninety degrees is a clean perpendicular shove and yields as much twist as any given force can produce.
- Read Torque underneath, flipping its unit menu to inch-pounds or foot-pounds if that is how your wrench happens to be marked.
- Unsure of your angle? Measure perpendicular distance from pivot out to your force's line of action, enter that as Lever arm length, and leave Angle to the lever arm at 90.
Worked example — a 50 N·m bolt and a half-metre bar
A bolt is specced at 50 N·m and your bar measures 500 mm from bolt centre to grip. Enter 100 into Force, 0.5 into Lever arm length, and 90 into Angle to the lever arm. Torque reads 50 N·m: 100 × 0.5 × sin 90° = 100 × 0.5 × 1 = 50, exactly. One hundred newtons is roughly what ten kilograms weigh, so this is firm leaning rather than heaving.
Now let that bar foul a bracket and oblige you to pull at 30° instead. Change Angle to the lever arm to 30 and your answer halves to 25 N·m, since sin 30° is precisely one half — identical effort, half as much job done. Losses stay gentle up near perpendicular, though: at 45° you still keep 70.7%, at 80° some 98.5%. Only that last stretch toward zero collapses.
This bargain runs both ways. Swap to 200 into Force with 0.25 into Lever arm length and Torque returns an identical 50 N·m — twice as much shove across half as much reach. Such trades are levers in one line, and why a longer bar makes a seized fastener feel almost reasonable.
Questions
Which distance do I measure for the lever arm?
From your axis of rotation out to where that force lands — not to a tool's tip, and not traced along a bent handle. If your pivot is a bolt, that means bolt centre. Fit a crowfoot adapter in line with your wrench handle, or reach for a cranked spanner, and effective reach grows: your fastener then receives more twist than its wrench was ever set to deliver. Rotate that adapter 90° to your handle and any extra reach disappears.
Why does the angle appear as a sine rather than a cosine?
Because only that share of force perpendicular to your arm turns anything. Resolve your push into two parts: one running lengthwise, one crossing it. Whatever runs lengthwise merely stretches or compresses your lever and tugs its pivot in its seat; whatever crosses, F·sin θ, generates moment. A cosine would belong here had θ been measured from perpendicular instead of from your arm itself — precisely how people end up with maximum and zero swapped.
Can a moment be negative?
Yes, and the sign carries real information: it names the direction of rotation. Convention follows the right-hand rule — curl your fingers the way the body turns and your thumb points along the vector, with counter-clockwise in a standard xy plane counted positive. Between 0 and 180° this sheet stays positive; past 180° the sine goes negative by itself, flagging the opposite sense. Where several moments act on one body, signs are what let you add them honestly.
How does torque differ from power?
Twist is one thing, the rate at which twist does work is another: P = τ·ω, with ω in radians per second. An engine can make a large moment at low revs while making little power, which is how a tractor and a motorcycle can share a peak figure yet behave nothing alike. Gearing follows from the same relation — a reduction gear multiplies the moment and divides the speed, leaving power (bar friction) alone.
Will hitting the specified value give me the clamping force I want?
Only approximately. On a typical dry steel fastener something like 85–90% of what you apply is eaten overcoming friction beneath the bolt head and inside the threads; barely a tenth arrives as tension in the bolt. That is why lubricated threads reach far higher preload at an unchanged wrench setting, and why critical joints get tightened by turn-of-nut angle, by measured bolt stretch, or by ultrasonic gauging rather than by twist alone.
Where does this formula stop being true?
It assumes a rigid body, a single force, a fixed and clearly named pivot, and everything living in one plane. Flexible shafts, forces pointing out of plane, and pivots that are themselves accelerating each break at least one assumption. The result also only sets motion up; how quickly the body actually spins follows from τ = I·α, with I the moment of inertia about that same axis. Pick a different axis and both quantities change together.