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Newton-metre

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Newton-metre
One newton-metre is the torque resulting from a force of one newton applied perpendicularly to the end of a moment arm that is one metre long.
General information
Unit system	SI
Unit of	torque
Symbol	N⋅m, N m
Conversions
1 N⋅m in ...	... is equal to ...

FPS system	0.73756215 lbf.ft
inch⋅pound-force	8.8507 in lbf
inch⋅ounce-force	141.6 in oz

The newton-metre or newton-meter (also non-hyphenated, newton metre or newton meter; symbol N⋅m^[1] or N m^[1])^[a] is the unit of torque (also called moment) in the International System of Units (SI). One newton-metre is equal to the torque resulting from a force of one newton applied perpendicularly to the end of a moment arm that is one metre long.

The unit is also used less commonly as a unit of work, or energy, in which case it is equivalent to the more common and standard SI unit of energy, the joule.^[2] In this usage the metre term represents the distance travelled or displacement in the direction of the force, and not the perpendicular distance from a fulcrum (i.e. the lever arm length) as it does when used to express torque. This usage is generally discouraged,^[3] since it can lead to confusion as to whether a given quantity expressed in newton-metres is a torque or a quantity of energy.^[4] "Even though torque has the same dimension as energy (SI unit joule), the joule is never used for expressing torque".^[4]

Newton-metres and joules are dimensionally equivalent in the sense that they have the same expression in SI base units,

1\,{\text{N}}{\cdot }\mathrm {m} =1\,{\frac {{\text{kg}}{\cdot }{\text{m}}^{2}}{{\text{s}}^{2}}}\quad ,\quad 1\,\mathrm {J} =1\,{\frac {\mathrm {kg} {\cdot }\mathrm {m} ^{2}}{\mathrm {s} ^{2}}}

but are distinguished in terms of applicable kind of quantity, to avoid misunderstandings when a torque is mistaken for an energy or vice versa. Similar examples of dimensionally equivalent units include Pa versus J/m³, Bq versus Hz, and ohm versus ohm per square.

Conversion factors

[edit]

1 kilogram-force metre = 9.80665 N⋅m^[5]^[6]
1 newton-metre ≈ 0.73756215 pound-force-feet
1 pound-foot ≡ 1 pound-force-foot ≈ 1.35581795 N⋅m
1 ounce-inch ≡ 1 ounce-force-inch ≈ 7.06155181 mN⋅m (millinewton-metres)
1 dyne-centimetre = 10⁻⁷ N⋅m

Notes

[edit]

^ The nonstandard notation "Nm" occurs in some fields.

References

[edit]

^ ^a ^b "BIPM – unit symbols". Archived from the original on 2017-10-01. Retrieved 2016-10-18.
^ For example: Eshbach's handbook of engineering fundamentals - 10.4 Engineering Thermodynamics and Heat Transfer "In SI units the basic unit of energy is newton-metre".
^ Fundamentals of Physics, 9th edition by Halliday Resnick Walker, p. 309. "The SI unit of torque is the newton-meter. In our discussion of energy we called this combination the joule. But torque is not work and torque should be expressed in newton-meters, not joules. google books link
^ ^a ^b "BIPM - special names". Archived from the original on 2019-03-21. Retrieved 2015-09-27.
^ Mechanical Engineering Formulas Pocket Guide, p6
^ Concise encyclopedia of plastics, by Donald V. Rosato, Marlene G. Rosato, Dominick V. Rosato, p621

v t e SI units
Base units	ampere candela kelvin kilogram metre mole second
Derived units with special names	becquerel coulomb degree Celsius farad gray henry hertz joule katal lumen lux newton ohm pascal radian siemens sievert steradian tesla volt watt weber
Other accepted units	astronomical unit dalton day decibel degree of arc electronvolt hectare hour litre minute minute and second of arc neper tonne
See also	Conversion of units Metric prefixes Historical definitions of the SI base units 2019 revision of the SI System of units of measurement
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The newton-metre (symbol: N⋅m) is the SI derived unit of torque, also known as moment or moment of force.^[1] It represents the torque produced by a force of one newton applied perpendicularly at a distance of one metre from the axis of rotation.^[1] Although dimensionally equivalent to the joule (J), the SI unit of energy and work, the newton-metre is used exclusively for torque to distinguish it from scalar quantities like energy, preventing confusion in physical applications.^[2] The unit derives its name from the newton (N), the SI unit of force named after Sir Isaac Newton and adopted in 1948 by the 9th Conférence Générale des Poids et Mesures (CGPM), combined with the metre (m), the SI base unit of length established in 1791 and refined through international prototypes until 1983.^[3]^[4] In engineering, physics, and everyday applications such as automotive specifications and tool calibration, the newton-metre quantifies rotational effects, with common multiples like the kilonewton-metre (kN⋅m) used for larger scales in machinery and structures. The symbol N⋅m employs a middle dot to clearly denote multiplication, avoiding ambiguity with unrelated units like the millinewton-metre (mN⋅m).^[1]

Fundamentals

Definition

The newton-metre (symbol: N⋅m or N m with a space) is the SI derived unit of torque, representing the rotational equivalent of force in the International System of Units (SI).^[1] Torque quantifies the tendency of a force to cause rotation about an axis, distinguishing it from linear force by incorporating the perpendicular distance from the axis of rotation.^[2] In vector form, torque

\tau

is defined as the cross product

\tau = \mathbf{r} \times \mathbf{F}

, where

\mathbf{r}

is the position vector from the axis of rotation to the point of force application, and

\mathbf{F}

is the force vector.^[5] The magnitude of this torque is given by

|\tau| = r F \sin \theta,

where

r

is the magnitude of the position vector,

F

is the magnitude of the force, and

\theta

is the angle between

\mathbf{r}

and

\mathbf{F}

.^[5] This formulation arises from the fundamental principles of mechanics, emphasizing the role of leverage in rotational motion. When the force is applied perpendicular to the position vector (

\theta = 90^\circ

, so

\sin \theta = 1

), one newton-metre equals the torque produced by a force of one newton acting at a distance of one metre from the pivot point.^[1] This builds on the base SI units: the newton (N), defined as the force required to accelerate a one-kilogram mass by one metre per second squared, and the metre (m), the base unit of length defined by the distance light travels in vacuum in

1/299792458

of a second.^[1]

Relation to Force and Distance

The newton-metre quantifies torque, defined as the rotational equivalent of linear force that tends to produce angular acceleration about an axis.^[5] The torque

\vec{\tau}

emerges from the vector interaction between force and position, given by the cross product $\vec{\tau} = \vec{r} \times \vec{F}$

×F

, where $\vec{r}$

is the position vector from the axis of rotation to the point where the force $\vec{F}$

is applied.^[5] This operation yields a vector perpendicular to both $\vec{r}$

and $\vec{F}$

, with direction determined by the right-hand rule, emphasizing the three-dimensional nature of rotational dynamics.^[5] The magnitude of this torque is $\tau = r F \sin \theta$ , where $r = |\vec{r}|$

∣ is the distance from the axis, $F = |\vec{F}|$

∣ is the force magnitude, and $\theta$ is the angle between $\vec{r}$

and $\vec{F}$

. The $\sin \theta$ factor captures the effective component of the force that is perpendicular to the position vector, as only this component contributes to rotation; when $\theta = 0^\circ$ or $180^\circ$ , $\sin \theta = 0$ and no torque is produced.^[5] The perpendicular distance from the axis to the line of action of the force, termed the moment arm $d = r \sin \theta$ , serves as the effective lever length in torque calculations. This reduces the magnitude formula to $\tau = F d$ when the force is applied perpendicular to the lever arm ( $\theta = 90^\circ$ ), maximizing the rotational effect for a given force and distance. For example, twisting a doorknob applies a force at the handle, where the moment arm is the radial distance from the hinge axis to the point of force application, producing torque proportional to this lever length. Similarly, tightening a bolt with a wrench involves applying force perpendicular to the handle, with the torque magnitude equaling the force times the wrench's length as the moment arm.^[6] Dimensional analysis confirms the derived nature of the newton-metre: torque dimensions are $[\tau] = [F] \times [L] = (\mathrm{kg \cdot m \cdot s^{-2}}) \times \mathrm{m} = \mathrm{kg \cdot m^2 \cdot s^{-2}}$ , directly yielding the unit newton-metre in the SI system.^[1]

Measurement and Units

SI Designation

The newton metre is the official name of the SI unit for torque, expressed in lowercase letters without a hyphen to distinguish it from the unit newton per metre, though a hyphenated form "newton-meter" may be used in some national variants to avoid ambiguity.^[1] The plural form is newton metres.^[1] The symbol for the newton metre is N⋅m or N m, where either a middle dot (⋅) or a space indicates multiplication between the newton (N) and metre (m) symbols; the multiplication sign (×) must not be used, in accordance with BIPM and NIST guidelines for compound unit symbols.^[7] Unit symbols are always written in roman (upright) typeface, never italicized, and remain unchanged in the plural.^[7] As a coherent derived SI unit, the newton metre requires no numerical factor other than unity when formed from base units and does not need prefixes for standard values, ensuring consistency within the SI system.^[1] It is derived from the base SI units as the product of the newton (kg⋅m⋅s⁻²) and the metre (m), yielding the dimensional expression kg⋅m²⋅s⁻².

Conversions and Equivalents

The newton metre (N⋅m) is converted to the imperial foot-pound (lb⋅ft) using the approximate factor 1 N⋅m ≈ 0.737562 lb⋅ft, with the precise relation derived from 1 lb⋅ft = 1.3558179483314004 N⋅m exactly based on defined standards for force and length.^[8] Additional equivalents include 1 N⋅m = 8.8507457916 inch-pounds (in⋅lbf), calculated from 1 in⋅lbf = 0.1129848 N⋅m, and 1 N⋅m = 0.1019716213 kilogram-force metres (kgf⋅m), stemming from 1 kgf⋅m = 9.80665 N⋅m.^[8]^[9] For practical applications, the following table summarizes key conversions from 1 N⋅m to common torque units, using values aligned with SI and customary definitions:

Unit	1 N⋅m equals
lb⋅ft	0.7375621493
in⋅lbf	8.8507457916
kgf⋅m	0.1019716213
dyne⋅cm	10,000,000
N⋅mm	1000

These factors facilitate interoperability in engineering contexts, such as tool calibration or specification comparisons.^[8]^[9] Additionally, 1 N⋅m = 1000 newton-millimetres (N⋅mm), since torque scales linearly with the distance unit and 1 m = 1000 mm. The unit N⋅mm is equivalent to the millinewton-metre (mN⋅m). SI prefixes scale the newton metre for varying magnitudes, with kilonewton-metre (kN⋅m = 1,000 N⋅m) common in high-torque scenarios like automotive or industrial machinery, and millinewton-metre (mN⋅m = 0.001 N⋅m), equivalent to approximately 10.197 gf⋅cm based on 1 gf = 9.80665 mN and 1 m = 100 cm, for precision instruments.^[10] For instance, passenger vehicle engine torques typically range from 150 to 500 N⋅m, while heavy-duty trucks may exceed 1 kN⋅m.

Distinctions and Misconceptions

Difference from Joule

The newton-metre (N⋅m) and the joule (J) are dimensionally equivalent in the International System of Units (SI), both expressed as

\mathrm{kg \cdot m^2 \cdot s^{-2}}

.^[1] However, they quantify fundamentally different physical concepts: the joule measures energy or work, defined as the scalar product of a force and the displacement along its line of action, while the newton-metre measures torque, defined as the tendency of a force to cause rotation about an axis, given by the magnitude of the force multiplied by the perpendicular distance from the axis to the line of action of the force.^[1]^[9]^[1] The SI intentionally distinguishes these units to prevent confusion between rotational and translational quantities, recommending the explicit form "newton-metre" for torque rather than the joule, which is reserved exclusively for energy and work as scalar quantities.^[1]^[11] This separation emphasizes that torque is a pseudovector describing rotational effect without implying energy transfer.^[9] For instance, applying a 1 N force to lift a 1 N weight vertically through 1 m performs 1 J of work, transferring energy to increase the gravitational potential energy.^[9] In contrast, applying the same 1 N force tangentially at a 1 m radius on a balanced wheel in static equilibrium produces 1 N⋅m of torque, tending to rotate the wheel but performing no net work, as there is no displacement of the point of application.^[1]^[12] The International Bureau of Weights and Measures (BIPM) explicitly advises using "joule" only for energy-related quantities, even when numerical values match those of torque in newton-metres, to maintain conceptual clarity in scientific and technical contexts.^[1]

Common Confusions

A frequent source of confusion arises from the notation for the newton-metre, where writing it as "Nm" without a dot or space can lead to ambiguity with the nanometre (nm), a unit of length equal to 10^{-9} metres. To prevent this, the International System of Units (SI) recommends denoting the torque unit as N⋅m or N m, explicitly distinguishing it from the millinewton (mN), which is 10^{-3} newtons.^[1]^[13] Another common mix-up involves distinguishing the newton-metre (N⋅m) from the newton per metre (N/m), the latter being the SI unit for quantities like surface tension or spring stiffness, representing force divided by length rather than multiplied. The non-standard order "metre-newton" (mN) is occasionally encountered but equates to the same unit as N⋅m since multiplication is commutative; however, it risks confusion with the millinewton symbol mN.^[1]^[9] In energy contexts, the newton-metre is sometimes mistakenly equated directly to work or energy without considering angular displacement, as torque (N⋅m) measures rotational force application, whereas work requires multiplication by the angle in radians to yield joules (J). Although both share the same dimensional formula (kg⋅m²⋅s⁻²), the SI explicitly prohibits using the joule for torque to maintain clarity.^[1]^[9] An educational illustration of this distinction appears in automotive specifications, where engine torque is rated in N⋅m (e.g., 300 N⋅m for twisting force at the crankshaft), contrasting with fuel efficiency metrics expressed in energy equivalents like megajoules (MJ) per 100 km or kilowatt-hours (kWh), which quantify total work done over distance rather than instantaneous rotational capability.^[1]

Historical Development

Origin in SI System

The newton-metre (N·m) emerged as a derived unit within the International System of Units (SI), which was formally established in 1960 by Resolution 12 of the 11th General Conference on Weights and Measures (CGPM). This system built upon the metre-kilogram-second (MKS) framework, integrating the newton as the unit of force—named in 1948 by the 9th CGPM after Sir Isaac Newton for his contributions to mechanics—and the metre, originally defined in the 1790s by the French Academy of Sciences as one ten-millionth of the distance from the equator to the North Pole along a meridian through Paris.^[14]^[4] The combination of these base units yielded the newton-metre as the coherent SI unit for torque, or moment of force, explicitly listed in the inaugural SI brochure for its role in quantifying rotational effects.^[1] Although the mathematical concept of torque traces back to ancient times, its formal unitization in the modern metric context followed resolutions from the 1948 CGPM, which advanced the MKS system toward international standardization. The foundational idea of torque as the product of force and perpendicular distance originated with Archimedes in the 3rd century BCE, who articulated the lever principle in his treatise On the Equilibrium of Planes, enabling the balancing of weights through proportional moments.^[15] This principle provided the theoretical basis for torque long before standardized units, influencing subsequent developments in mechanics. Before the SI's adoption, torque measurements relied on non-coherent systems prevalent in scientific and engineering practice. In the centimetre-gram-second (CGS) system, the dyne-centimetre served as a common unit, where one dyne represents the force accelerating a 1-gram mass by 1 cm/s².^[9] Engineering contexts often used the kilogram-force metre (kgf·m), defined by the force exerted by standard gravity on a 1-kilogram mass, reflecting practical metric traditions outside the absolute MKS framework.^[9] These units were gradually supplanted by the newton-metre to ensure coherence within the SI, where 1 N·m equals 10^7 dyne·cm or approximately 0.10197 kgf·m.^[9]

Evolution and Standardization

Following the initial establishment of the International System of Units (SI) in 1960, refinements to the notation and usage of derived units like the newton-metre (N⋅m) emerged in the 1970s and 1980s. The 4th edition of the SI Brochure, published in 1981 and updated through subsequent clarifications by the International Bureau of Weights and Measures (BIPM), specified the use of a middle dot (⋅) as the preferred multiplication sign for unit symbols to avoid ambiguity, particularly for products like N⋅m, distinguishing it from inline juxtaposition or other operators.^[16]^[1] This convention was further reinforced in the 5th edition of 1985, promoting consistent typesetting in scientific and technical literature to enhance readability and precision.^[16] In the 1990s, the BIPM emphasized the conceptual distinction between torque and energy despite their dimensional equivalence, noting in the 6th edition of the SI Brochure (1991) that the joule (J) should not be used for torque measurements, reserving N⋅m exclusively for rotational quantities to prevent misinterpretation in engineering contexts.^[16] This guidance addressed growing concerns in applied physics and mechanics, where conflating the units could lead to errors in calculations involving moment of force versus work or energy transfer.^[1] The 2019 revision of the SI, adopted by the 26th General Conference on Weights and Measures (CGPM), redefined several base units including the kilogram but left the newton-metre unchanged as a coherent derived unit, reaffirming its status within the revised framework where all units are now expressed in terms of fixed constants like the speed of light and Planck's constant. This update ensured continuity for practical applications while enhancing the system's invariance to experimental artifacts.^[1] International adoption of the newton-metre as the standard torque unit advanced through ISO 80000-4:2006, which formally defines mechanics quantities including torque (symbol τ) with N⋅m as the SI unit, providing conversion factors and guidelines for global consistency in scientific and industrial documentation.^[17] In the European Union, Council Directive 80/181/EEC, as amended, mandates the use of SI units like N⋅m in engineering and commercial contexts to facilitate trade and safety, with updates in 2019 aligning it to the revised SI for sectors such as manufacturing and construction. As of 2025, no major revisions to the newton-metre have occurred, but its application has expanded in renewable energy specifications, particularly for wind turbine drivetrains where torque ratings in N⋅m (often in the megawatt-scale range) are critical for efficiency and load assessments in designs exceeding 10 MW capacity.^[18]

Applications

In Mechanics and Physics

In rigid body dynamics, the newton-metre (N·m) quantifies torque

\tau

in the rotational analog of Newton's second law,

\tau = I \alpha

, where

I

is the moment of inertia (in kg·m²) and

\alpha

is the angular acceleration (in rad·s⁻²). This relation ensures dimensional consistency, as the product of torque and time yields angular impulse in kg·m²·s⁻¹, matching changes in angular momentum. The unit's derivation from force (newton) and lever arm (metre) underscores its role in describing rotational effects perpendicular to linear motion. In planetary motion, gravitational torques—arising from non-central perturbations or tidal interactions—are calculated in N·m to assess rotational influences on orbits. For instance, in spacecraft near planetary bodies, such torques can reach scales of approximately

10^{-6}

N·m, affecting attitude control. Similarly, in geophysical models of planetary interiors like Earth's core, gravitational torques from surface-inner core interactions operate at magnitudes around

10^{20}

N·m, driving precession and dynamo effects. For gyroscopes, N·m measures the applied torque that induces precession, with the rate

\Omega = \tau / L

(where

L

is angular momentum) determining stability in rotational systems. In quantum and relativistic physics, torque concepts extend to particle interactions like spin-orbit coupling, where the effective torque arises from relativistic corrections to electron motion in atomic fields. However, due to the subatomic scales, these torques are rarely expressed in N·m—instead, they are quantified in units of

\hbar

(reduced Planck's constant, approximately

1.05 \times 10^{-34}

J·s) or energy equivalents, highlighting the unit's primary utility in macroscopic regimes. Laboratory measurements of torque in N·m often employ torsion balances, instruments suspended by thin fibers to detect minute rotational twists from gravitational attractions. Calibration involves relating the angular deflection

\theta

to torque via

\tau = \kappa \theta

, where

\kappa

is the torsional constant (in N·m/rad), enabling precise quantification in experiments probing weak forces analogous to gravitational interactions.

In Engineering and Technology

In automotive engineering, the newton-metre (N⋅m) quantifies engine torque, a critical parameter for vehicle performance and acceleration, related to rotational dynamics via the equation

\tau = I \alpha

, where

\tau

is torque,

I

is the moment of inertia, and

\alpha

is angular acceleration. Torque curves, plotted as torque versus engine speed, guide design and tuning; for mid-size sedans, peak values typically range from 150 to 250 N⋅m, enabling responsive acceleration in everyday driving conditions without excessive fuel consumption. In mechanical systems, N⋅m specifications ensure structural integrity during assembly and operation. Bolt tightening, such as wheel lug nuts on mid-size vehicles, commonly requires 100 to 140 N⋅m to secure wheels against vibration and loads while avoiding over-tightening that could strip threads or warp components. In robotics, joint torques measured in N⋅m dictate manipulator capabilities; for instance, high-performance underwater robotic arms feature joints rated up to 36 N⋅m to handle payloads at speeds of 0.7 rad/s, balancing precision and energy efficiency in tasks like subsea inspection. Renewable energy applications leverage N⋅m and its multiples for scaling power generation. Wind turbines, rated in megawatts (MW), achieve output through drivetrain torques in meganewton-metres (MN⋅m); the International Energy Agency's 22 MW offshore reference turbine delivers a rated torque of 31,465 kN⋅m (31.465 MN⋅m) at low rotor speeds, maximizing energy capture while minimizing mechanical stress. Electric motors in these systems prioritize high torque density for efficiency, often exceeding 10 N⋅m/kg in direct-drive configurations to reduce reliance on gearboxes and improve overall system reliability in variable wind conditions. Precision tools like torque wrenches are calibrated directly in N⋅m to meet exacting assembly requirements in manufacturing. These devices apply controlled force to fasteners, with calibration verifying accuracy to within ±4% as per ISO 6789 standards, preventing failures in aerospace or automotive components where deviations could compromise safety. Regular recalibration, typically every 5,000 cycles or annually, maintains performance across ranges from 10 to 200 N⋅m, supporting standardized processes in high-volume production.

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Newton-metre

Fundamentals

Definition

Relation to Force and Distance

Measurement and Units

SI Designation

Conversions and Equivalents

Distinctions and Misconceptions

Difference from Joule

Common Confusions

Historical Development

Origin in SI System

Evolution and Standardization

Applications

In Mechanics and Physics

In Engineering and Technology

References

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