Physics · Mechanics

Torque

Torque is the product of force and lever arm; it describes the turning and tilting effect of a force about an axis.

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Formula

M = F \cdot r \cdot sin (θ)

LaTeX: M = F \cdot r \cdot \sin(\theta)

M in N·m · F in N · r in m · θ in degrees [°]

Variables & units – Torque

Symbol	Meaning	Unit
M	Torque about the axis of rotation	N·m
F	Applied force	N
r	Distance from axis to point of application	m
θ	Angle between r and F	°

Derivation & background – Torque

The law of the lever goes back to Archimedes: force times force arm equals load times load arm (F₁·r₁ = F₂·r₂). Vectorially, torque is the cross product M⃗ = r⃗ × F⃗; only the force component perpendicular to the lever arm turns, hence the factor sin θ. A body is in rotational equilibrium when the sum of all torques is zero. For rotational motion torque plays the role of force: M = J·α with the moment of inertia J.

Exam blueprint

Validity range

Applies to rigid bodies with a fixed axis of rotation. Only the force component perpendicular to the lever arm counts; r is the distance from the axis to the line of action of the force.

Derivation steps

Only the force component perpendicular to the lever produces a turning effect.

1Decompose F: the component F·sin θ is perpendicular to the lever arm, F·cos θ only pulls on the axis.
2Turning effect = lever arm times perpendicular component: M = r·F·sin θ; in equilibrium ΣM = 0 (law of the lever).

Rearrangements

Required force

F = \frac{M}{r \cdot s i n ( θ )}

The longer the lever, the smaller the required force.

Law of the lever

F_{1} \cdot r_{1} = F_{2} \cdot r_{2}

Balance of two torques about the same axis.

Task variant

A bolt needs 40 N·m. What force is required on a 20 cm lever (perpendicular)?

F = M/r = 40/0.2 = 200 N. A wrench twice as long halves the force to 100 N.

A child (30 kg) sits 2 m from a seesaw pivot. Where must a 40 kg child sit?

Law of the lever: m₁·r₁ = m₂·r₂ (g cancels). r₂ = 30 × 2/40 = 1.5 m.

Common mistakes

Dropping the sin θ factor for an obliquely applied force.

Only the perpendicular component turns; at θ = 90° sin θ = 1, otherwise smaller.

Measuring r to the point of application instead of the line of action.

The lever arm is the perpendicular distance from the axis to the line of action.

Equating N·m with joules.

Formally the same unit, but torque is not energy: the quantities have different meanings.

Ignoring the sense of rotation and adding all torques.

Balance counterclockwise and clockwise torques with opposite signs.

Exam context

Tasks on levers, seesaws and torque wrenches as well as equilibrium conditions (ΣF = 0 and ΣM = 0) for beams and cranes.

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Formula cluster

Statics and rotation

Torque plays the role of force for rotational motion.

a \times b = a_{2} b_{3} - a_{3} b_{2} a_{3} b_{1} - a_{1} b_{3} a_{1} b_{2} - a_{2} b_{1}

Cross Product (Vector Product)

F = m \cdot a

Newton's Second Law

Worked example

Wrench: F = 80 N perpendicular at lever arm r = 0.25 m: M = 80 × 0.25 = 20 N·m. If the force acts at 60°, only M = 80 × 0.25 × sin(60°) ≈ 17.3 N·m is effective.

Applications

Bolted joints (torque wrench), engines (Nm rating), levers and pulleys, seesaw, statics of beams and cranes

Quanta exam set

Curated exam set for "Torque":

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Which formula describes Torque?

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Question (front)

How do you rearrange M = F·r·sin(θ) for Required force?

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Question (front)

Which common mistake happens with Torque?

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Scientific sources

[1]Tipler, P.A., & Mosca, G. (2015). Physik für Wissenschaftler und Ingenieure. Springer Spektrum.

Common notations & search queries

M=F*rM = F·r·sinθDrehmoment FormelHebelgesetz Formeltorque formulaKraftarm LastarmNewtonmeter berechnenDrehmoment Schraubenschlüssel

Related formulas

F = m \cdot a

Newton's Second Law

a \times b = a_{2} b_{3} - a_{3} b_{2} a_{3} b_{1} - a_{1} b_{3} a_{1} b_{2} - a_{2} b_{1}

Cross Product (Vector Product)

W = F \cdot s \cdot cos (α)

Mechanical Work

More Physics formulas

F = m \cdot a

Newton's Second Law

F_{G} = G \cdot \frac{m _{1} \cdot m _{2}}{r ^{2}}

Newton's Law of Universal Gravitation

E_{kin} = \frac{1}{2} m v^{2}

Kinetic Energy

E_{p o t} = m \cdot g \cdot h

Potential Energy (Gravitational)

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Frequently asked questions about Torque

How do you calculate torque?+

Multiply the force by the lever arm: M = F·r when the force acts perpendicular to the lever. Example: 80 N perpendicular at the end of a 25 cm wrench gives M = 80 × 0.25 = 20 N·m. If the force acts at an oblique angle θ, only its perpendicular component counts: M = F·r·sin(θ). At 60° you get only 20 × sin(60°) ≈ 17.3 N·m instead of 20 N·m. Watch the units: r in metres, F in newtons, the result in newton metres. The lever arm is the distance from the axis to the line of action of the force, not just any distance on the tool.

What does the law of the lever state and how is it related to torque?+

The law of the lever, force times force arm equals load times load arm (F₁·r₁ = F₂·r₂), is nothing but the torque balance: a lever does not rotate when the counterclockwise and clockwise moments are equal. Seesaw example: a 30 kg child 2 m from the pivot creates the moment 30 × 9.81 × 2 ≈ 589 N·m. A 40 kg child balances it at r = 589/(40 × 9.81) = 1.5 m. Because g appears on both sides it cancels, and you can directly compute m₁·r₁ = m₂·r₂. The beam balance, crowbar, wheelbarrow and all pliers rest on this principle.

Why does a longer lever save force?+

For a required torque M the relation F = M/r holds: the necessary force is inversely proportional to the lever arm. If a stuck bolt needs 40 N·m, a 20 cm wrench requires 200 N, a 40 cm wrench only 100 N. Extending the wrench with a pipe lowers the force further. The price is a longer path: the handle must travel a larger arc, and the work done W = F·s stays the same; there is no free energy (the golden rule of mechanics). This is exactly why wheel-nut wrenches have long arms and door handles sit far from the hinge.

Is torque the same as energy, since both are measured in N·m?+

No, the matching unit is a coincidence of dimensions; the quantities are fundamentally different. In work W = F·s force and path point in the same direction, and the scalar product yields an amount of energy in joules. In torque M = r × F lever arm and force are perpendicular, and the cross product yields a directed turning action. A torque of 20 N·m does not mean 20 J of stored energy; energy only flows when something actually rotates: W = M·φ with the rotation angle φ in radians. That is why torques are deliberately written N·m and energies J, although the units are formally identical.

What is a torque wrench for?+

Bolted joints need a defined preload: bolts tightened too loosely work free under vibration, bolts tightened too hard overstretch the thread or snap. Since the preload is directly linked to the tightening torque, manufacturers specify torques, for instance 110 N·m for car wheel bolts or only 5 N·m for delicate carbon bicycle parts. The torque wrench measures the applied moment via a calibrated spring and audibly breaks over or clicks once the set value is reached. The physics behind it is M = F·r: the mechanism limits the product of hand force and lever length, no matter where you grip.

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How do you calculate with Torque?

Here is how to work through a typical Torque (M = F·r·sin(θ)) task step by step:

1
Task
A bolt needs 40 N·m. What force is required on a 20 cm lever (perpendicular)?
Solution path
F = M/r = 40/0.2 = 200 N. A wrench twice as long halves the force to 100 N.
2
Task
A child (30 kg) sits 2 m from a seesaw pivot. Where must a 40 kg child sit?
Solution path
Law of the lever: m₁·r₁ = m₂·r₂ (g cancels). r₂ = 30 × 2/40 = 1.5 m.