Moment

A moment is the turning effect of a force about a point. Learn M = F·d, sign conventions, Varignon's theorem, couples, and how levers and spanners multiply force.

Push a door near its hinges and it barely moves; push at the handle and it swings easily. The same force produces a bigger turning effect — a bigger moment — when applied further from the pivot.

Moments explain levers, spanners, beams, cranes tipping over and shafts transmitting torque. Along with forces, they are the second half of every equilibrium calculation.

An analogy

A see-saw: a light child far from the pivot can balance a heavy adult close to it, because what matters is force × distance, not force alone.

Force times perpendicular distance

Only the perpendicular distance from the pivot to the force's line of action counts. A force pointing straight at the pivot has zero moment however large it is. Convention: counter-clockwise positive, clockwise negative (or state your choice).

Varignon's theorem

The moment of a force equals the sum of the moments of its components. It is often easier to split an angled force into x and y components and take each one's moment than to find the perpendicular distance geometrically.

Couples

Two equal, opposite, parallel forces form a couple. It produces pure rotation with no net force, and its moment is the same about every point. Turning a steering wheel with both hands or a tap spanner applies a couple.

Moments in 3D and force–couple systems

In 3D, M = r × F is a vector along the axis of rotation. Any force can be moved to another point by adding a couple equal to its moment about that point (a force–couple system), which is how loads are transferred through structures and into supports.

Formulas

• Moment: M = F · d⊥ = F · r · sin θ
• Varignon's theorem: M_O = F_x · y − F_y · x (sum of component moments)
• Couple: C = F · d (same about every point)
• Lever mechanical advantage: MA = load / effort = effort arm / load arm

Worked example: Spanner torque

Given: F = 200 N; Handle length 0.3 m; Force at 60° to the handle

Find: Moment on the bolt when pushing at 90° and at 60°.

• At 90°: M = 200 × 0.3 = 60 N·m
• At 60°: M = 200 × 0.3 × sin 60° = 52 N·m

Answer: 60 N·m (perpendicular), 52 N·m (at 60°)

Worked example: Balancing a see-saw

Given: Child A: 60 kg at 1.5 m from pivot; Child B: 40 kg

Find: Where must child B sit?

• ΣM = 0: 60 g × 1.5 = 40 g × d
• d = 90/40 = 2.25 m

Answer: 2.25 m on the other side

Common mistakes

• Wrong: Using the straight-line distance to the force's point of application. — Right: Use the perpendicular distance to the force's line of action.
• Wrong: Mixing clockwise and counter-clockwise signs. — Right: Choose a sign convention and apply it to every moment in the equation.

Applications

• Bolt tightening: Torque wrenches apply specified moments so bolts reach the right preload.
• Crane stability: Load moment indicators compare the tipping moment of the load with the restoring moment of the counterweight.
• Human biomechanics: Muscles act at short lever arms, so the elbow joint and biceps carry forces many times the weight held.

Key takeaways

• M = F × perpendicular distance.
• Varignon: sum the moments of components.
• A couple gives pure rotation, same moment about any point.
• Levers trade force for distance: MA = effort arm / load arm.

Going further

• Torque vs moment: Engineers often say 'torque' for moments that twist a shaft about its own axis and 'bending moment' for moments that bend a member — the physics is the same.
• Stability and tipping: An object tips when the moment of applied loads about an edge exceeds the restoring moment of its weight; vehicle rollover thresholds and crane load charts come from this.
• Screw threads as levers: A bolt converts applied torque to axial preload via the thread helix and friction: T ≈ K·F·d, with nut factor K ≈ 0.2 for dry steel.