Torque From Force and Distance

Torque describes a force's tendency to turn an object about a selected point or axis. Force alone is not enough: where the force acts, the direction in which it acts and the axis about which the moment is calculated all matter. A modest force far from an axis can create more torque than a larger force whose line of action passes close to that axis.

Read the force-and-distance geometry correctly

Open full image

Diagram description

A shaft carries a lever extending to the right. A downward force acts at the lever end, perpendicular to the radius from the shaft axis. A curved arrow shows the resulting turning direction. The formula panel gives T = Fr sin θ = Fd and identifies the SI unit N·m.

General moment of a force

T⃗ = r⃗ × F⃗; |T⃗| = rF sin θ = Fd

The cross product preserves both magnitude and direction. In a two-dimensional sketch, the same calculation can be performed as a signed scalar moment about the axis perpendicular to the page. In three dimensions, calculate the moment vector and then use its component along the actual shaft axis when the question is shaft-twisting torque.

T⃗

moment or torque vector about the chosen origin, N·m

r⃗

position vector from that origin to the force application point, m

F⃗

applied force vector, N

θ

angle between r⃗ and F⃗

d

perpendicular distance to the force's line of action, m

Example

If F is perpendicular to r, sin 90° = 1 and T = Fr. If F points directly toward or away from the axis, θ is 0° or 180°, the line of action crosses the axis and T is zero.

Why the line of action is more reliable than the visible handle length

Extend the force arrow into an infinite straight line, then draw the shortest line from the axis to it. That shortest line must meet the force line at 90 degrees; its length is d. This construction works even when the force is applied obliquely or when the application point is not directly above or beside the pivot.

For an ideal rigid body, sliding a force along its own line of action leaves its moment about a given point unchanged because F, d and the turning direction are unchanged. Moving the force to a parallel but different line changes d and therefore changes the moment.

Worked example: a perpendicular force on a lever

Open full image

Diagram description

A lever extends rightward from a rotation axis. A downward red arrow at its end is labelled 100 newtons, and the lever arm is labelled 0.20 metres. A calculation panel multiplies the force by the lever arm and reports 20 newton metres.

Perpendicular-force result

|T| = Fd = (100 N)(0.20 m) = 20 N·m

The 0.20 m distance is perpendicular to the 100 N force line, so no trigonometric correction is needed. Preserve the units during multiplication: newtons times metres produce newton metres. This result is a moment magnitude; its sign still depends on the declared positive direction.

|T|

magnitude of the moment about the shaft axis, N·m

F

perpendicular applied force, 100 N

d

perpendicular moment arm, 0.20 m

Example

If counterclockwise is declared positive, a downward force acting to the right of the axis produces a clockwise moment and would be written T = −20 N·m. Reversing the force reverses the sign but not the magnitude.

Use a repeatable moment-calculation method

From a physical load to an auditable torque value

1. 1. Isolate the body.Draw a free-body diagram of the lever, shaft, gear, pulley or assembly. Include every relevant external force, applied couple and support reaction rather than calculating from one convenient load only.
2. 2. Choose the point or axis.State exactly where the moment is required. A force has different moments about different points; in three dimensions, shaft torque is the moment component about the shaft's centerline.
3. 3. Draw each force line of action.Mark the force magnitude, application point and direction. Resolve an unknown angled force into components or determine the perpendicular distance geometrically.
4. 4. Convert to one unit system.For N·m, use force in newtons and every distance in metres. Record conversions next to the source values so that millimetres, centimetres and metres cannot be mixed silently.
5. 5. Calculate each moment separately.Use Fd, rF sin θ or r⃗ × F⃗ as appropriate. Assign the sign or vector direction immediately instead of trying to recover it after summing magnitudes.
6. 6. Sum to net torque.Add all signed moments and any directly applied couples about the same point or axis. Do not add moment magnitudes when their directions oppose.
7. 7. Check the result physically.Confirm that moving the force farther out increases its moment, a force through the axis gives zero moment and the final sign matches the expected turning tendency.

Direction determines the net torque

For a planar problem, a convenient convention is counterclockwise positive and clockwise negative. This matches the positive out-of-page direction from the right-hand rule for a standard x–y drawing. Another convention can be used, but it must be stated once and applied to every force. In a spatial mechanism, retain vectors or explicitly project each moment onto the axis being checked.

Signed net moment about one axis

T_net = ΣT_i = T₁ + T₂ + … + T_n

Each T_i must refer to the same point or axis and use the same positive direction. A support force whose line of action crosses the selected pivot contributes zero moment about that pivot, although it may still be essential to force balance and structural loading.

T_net

net torque about the chosen axis, N·m

T_i

signed moment of force or applied couple i about that axis, N·m

Example

With counterclockwise positive, suppose the example force contributes −20 N·m and a second force contributes +9 N·m. The net is −11 N·m, so the remaining turning tendency is 11 N·m clockwise. Equal 20 N·m moments in opposite directions would give zero net torque.

Torque about a shaft is not every moment acting on the assembly

A three-dimensional force generally creates a moment vector. Its component parallel to the shaft axis twists the shaft. Components perpendicular to the shaft axis tend to bend or reorient the assembly and must be carried by bearings and structure. Reporting only the vector magnitude as “shaft torque” can therefore overstate twisting demand while hiding other important loads.

Likewise, two equal and opposite parallel forces separated by a distance form a couple. Their net force is zero but their moments add to a nonzero free moment. Include directly applied motor, brake or spring couples as moments rather than inventing a single force without geometry.

Applied torque is not automatically available acceleration

Fixed-axis rotational dynamics

T_net = Iα

The force-distance calculation supplies one applied moment. To predict acceleration, sum driving and resisting moments and use the total moment of inertia about the same axis. A static balance has zero angular acceleration, while constant speed also requires zero net torque even though the motor may supply torque to offset friction or process load.

T_net

signed net external torque about the fixed axis, N·m

I

total moment of inertia about that axis, kg·m²

α

angular acceleration, rad/s²

Example

A measured 20 N·m applied moment does not imply 20 N·m of acceleration torque. If 12 N·m is required by an opposing process load and 3 N·m by friction, the simplified remaining net torque is 5 N·m before other effects are considered.

Catch common torque mistakes before they reach hardware

Review questions for every force-to-torque calculation

• AxisIs the moment calculated about the actual pivot or shaft centerline, and are all terms referred to that same axis?
• Line of actionIs the distance perpendicular to the force line, or has a slanted physical length been used incorrectly?
• AngleIs θ really the included angle between r and F, and is the calculator using the intended degree or radian mode?
• DirectionHas clockwise/counterclockwise sign or the right-hand-rule vector been assigned before summing?
• UnitsAre newtons and metres used consistently, with mass converted to force only through a valid load model?
• CompletenessDo gravity, friction, process force, inertia, transmission effects and directly applied couples appear where relevant?
• Load pathHave non-torsional force and moment components been passed into the bearing, guide, fastener and structure checks?
• EvidenceCan every input be traced to a drawing, measurement, requirement or reviewed manufacturer source?

Continue with Gear Ratio, Speed and Torque in Real Machines to move torque through a transmission, then use Sizing a Motor and Screw as One System to combine load, motion, inertia, peak and RMS checks. For forces and moments that the supporting structure must carry, revisit Linear Guides and Their Load Ratings.

One-minute questions

Questions to answer before you apply this lesson

What should I select or check first for Torque From Force and Distance?

1. Isolate the body. Draw a free-body diagram of the lever, shaft, gear, pulley or assembly. Include every relevant external force, applied couple and support reaction rather than calculating from one convenient load only.

Which assumption can change the conclusion?

Load, motion profile, geometry, alignment, duty, lubrication and environment match the selected mechanism and ratings.

What can go wrong if I copy the example directly?

Convert millimetres before calculating newton metres. A 200 mm moment arm is 0.200 m. Substituting 200 as though it were metres would produce a result one thousand times too large.

How do I verify the result in the real system?

7. Check the result physically. Confirm that moving the force farther out increases its moment, a force through the axis gives zero moment and the final sign matches the expected turning tendency.

Sources and further reading

1. 10.6 Torque — University Physics Volume 1OpenStax, Rice University · 2026-10-12Authoritative university textbook section reviewed for r × F, rF sin θ, perpendicular lever arm, torque direction and signed addition of moments.
2. Calculating TorqueMIT OpenCourseWare, Massachusetts Institute of Technology · 2026-10-12University course material reviewed for the cross-product definition and the equivalent perpendicular-distance and perpendicular-force interpretations.
3. 10.7 Newton's Second Law for Rotation — University Physics Volume 1OpenStax, Rice University · 2026-10-12Authoritative university textbook section reviewed for signed net torque and the fixed-axis relationship T_net = Iα.
4. NIST Guide to the SI, Chapter 4: The Two Classes of SI Units and the SI PrefixesNational Institute of Standards and Technology · 2026-10-12National metrology guidance reviewed for expressing moment of force in newton metres rather than joules despite identical base dimensions.
5. Machine Guarding — Hazardous Motions and ActionsOccupational Safety and Health Administration · 2026-10-12Government safety guidance reviewed for hazards associated with rotating shafts, couplings, projections and in-running nip points.
6. Maintenance of Work EquipmentUK Health and Safety Executive · 2026-10-12Government safety guidance reviewed for isolation, lock-off, preventing unintended restart and releasing or restraining stored energy during maintenance.