Moment of Force (Torque) Formula
Calculate torque using τ = F × d.
Learn how force and distance from a pivot create rotational motion in mechanical systems.
The Formula
Torque (or moment of force) measures the turning effect of a force about a pivot point.
The greater the force or the longer the distance from the pivot, the larger the torque.
Variables
| Symbol | Meaning |
|---|---|
| τ | Torque / moment of force (Newton-metres, N·m) |
| F | Applied force (Newtons, N) |
| d | Perpendicular distance from the pivot to the line of action of the force (metres, m) |
Example 1
A mechanic applies 80 N of force to a 0.3 m wrench. What is the torque?
τ = F × d
τ = 80 N × 0.3 m
τ = 24 N·m
Example 2
A 50 kg child sits 2 m from the pivot of a seesaw. What torque does gravity create?
First, find the force due to gravity: F = m × g = 50 × 9.81 = 490.5 N
τ = F × d = 490.5 N × 2 m
τ = 981 N·m
When to Use It
Use the torque formula when you need to:
- Calculate the turning force on bolts, nuts, or shafts
- Analyse balance in levers, seesaws, and beams
- Design motors, gears, and rotational machinery
- Determine the force needed to open or close a valve
Remember that d must be the perpendicular distance from the pivot to the force's line of action.
If the force is applied at an angle θ, use τ = F × d × sin(θ).
Key Notes
- Formula: M = F × d: The moment (torque) equals the force multiplied by the perpendicular distance from the line of action to the pivot point. Also called the moment arm or lever arm. SI unit: N·m.
- N·m is not the same as J (Joule): Both are dimensionally N·m but are conceptually different. Joules measure energy (force times displacement in the direction of motion). Torque is a rotating tendency — it does not move linearly. Never convert between them.
- Vector form: M = r × F: The cross product of the position vector r (from pivot to force application point) and force F. The magnitude is |r||F|sinθ where θ is the angle between them. Direction (axis of rotation) is given by the right-hand rule.
- Moment equilibrium: ΣM = 0: For a static object, the sum of all moments about any point must be zero. Combined with ΣF = 0, these two conditions fully define static equilibrium. Used to find unknown reaction forces at supports.
- Applications: Moment calculations are used in structural analysis (beam reactions, frame forces), mechanical design (bolt preload, gear torque), seesaws and levers, wrench force requirements, and determining the overturning moment on walls and foundations.