Moments, Torque & Equilibrium Calculator

Moments and Torque Guide

What's the key thing to understand about Moment of a Force?

Moment = Force × perpendicular distance from pivot. M = F × d. Units: N·m (newton-metres). The perpendicular distance is crucial — the line of action of the force must be perpendicular to the distance measured. If the force is at angle θ to the lever arm: M = F × d × sin θ. This is equivalent to torque: τ = r × F (cross product). Example: 50N force at 2.0m from pivot: M = 50 × 2 = 100 N·m. Clockwise convention: clockwise = positive moment (or negative, depending on convention — be consistent).

What do I need to know about Principle of Moments?

For a body in rotational equilibrium: sum of clockwise moments = sum of anticlockwise moments. Σ(F × d)_clockwise = Σ(F × d)_anticlockwise. Example: seesaw with 50N at 2m left and 80N at 1.5m right of pivot. Clockwise (right): 80 × 1.5 = 120 N·m. Anticlockwise (left): 50 × 2 = 100 N·m. Not in equilibrium — 20 N·m net clockwise. To balance: move the 80N force: 100/80 = 1.25m from pivot. Applications: levers, beams, bridges, human joints (the forearm is a class 3 lever — the fulcrum is the elbow, and the effort applied by the muscle is amplified into a larger load force at the hand via the lever arm of the forearm bones).

What's the key thing to understand about Couples and Pure Torque?

A couple: two equal and opposite forces separated by a perpendicular distance d. Torque of couple = F × d (no net force, only rotation). Examples: turning a steering wheel (two hands applying opposite forces). Using a spanner — the net torque equals F × width of spanner. A couple produces pure rotation with no translation — it is the same regardless of where you take moments. Unlike a single moment, the torque of a couple is the same about any point. This is why the torque output of an engine, a couple produced within the crankshaft, can be usefully quoted as a single figure without needing to specify exactly where on the engine it's measured.

What do I need to know about Centre of Mass?

Centre of mass position: x_cm = Σ(m_i × x_i) / Σm_i. Sum of (mass × distance from reference) divided by total mass. Example: uniform beam 4m long (mass 20kg, CM at 2m from left end). With 50kg person at 1m from left: x_cm = (20×2 + 50×1) / (20+50) = (40+50)/70 = 1.286m from left end. For uniform shapes: CM is at the geometric centroid. Composite objects: combine individual CMs weighted by mass. The object balances when supported at the centre of mass — tipping occurs if the CM moves outside the base of support.

What should I know about the Three Classes of Levers?

Levers are classified by where the fulcrum, load, and effort sit relative to one another. A first-class lever (seesaw, scissors) has the fulcrum between load and effort. A second-class lever (wheelbarrow, bottle opener) has the load between the fulcrum and the effort, always giving a mechanical advantage greater than one. A third-class lever (tweezers, the human forearm lifting with the biceps) has the effort between the fulcrum and the load, trading mechanical advantage for a larger range of motion at the load end — exactly why your hand can move much faster and farther than your biceps contracts.

What do I need to know about Full Equilibrium Conditions?

A rigid body is in complete static equilibrium only when two conditions hold simultaneously: the sum of all forces is zero (ΣF = 0, so there's no linear acceleration) and the sum of all moments about any point is zero (ΣM = 0, so there's no angular acceleration). Satisfying only the moment condition (principle of moments) while ignoring net force — or vice versa — is a classic error; a seesaw balanced in moments but with an unbalanced net vertical force would still accelerate downward even while 'balanced' rotationally. Structural engineers check both conditions for every beam, bracket, and support.

What should I know about Torque and Angular Acceleration?

The moment of a force (torque) is not only useful for analysing static equilibrium — it is also the rotational equivalent of force in Newton's second law, governing how objects spin up or slow down. Just as F = ma relates linear force to linear acceleration through mass, the rotational analogue τ = Iα relates net torque τ to angular acceleration α through the moment of inertia I, a measure of how an object's mass is distributed relative to its axis of rotation. A larger moment of inertia means more torque is needed to produce the same angular acceleration, which is why a long wrench extension (increasing the lever arm and torque) makes loosening a tight bolt easier, while an object with mass concentrated far from its spin axis (like a flywheel's heavy rim) resists changes in rotational speed more strongly than one with the same mass concentrated near the axis. This relationship explains why figure skaters spin faster when they pull their arms inward: reducing their moment of inertia, with angular momentum conserved, forces angular velocity to increase. Engineering applications from flywheel energy storage to motor gearing all depend on this same torque-moment of inertia-angular acceleration relationship.

Torque Reference Table for Common Tools and Actions

Torque (moment) = force × perpendicular distance from the pivot, so the same force produces very different torque depending on the lever arm:

This is exactly why a longer wrench or breaker bar makes a stuck bolt easier to turn without needing more muscular force — doubling the lever arm doubles the torque for the same applied force, which is also why door handles are placed far from the hinge rather than close to it.

Worked Example: Balancing a Seesaw Using the Principle of Moments

Two children sit on opposite sides of a seesaw pivoted exactly at its center. Child A, with mass 30 kg, sits 1.8 m from the pivot. How far from the pivot must Child B, with mass 27 kg, sit on the other side for the seesaw to balance? The principle of moments states that for balance, the clockwise moment must equal the counterclockwise moment: m_A × g × d_A = m_B × g × d_B. Since g appears on both sides, it cancels, leaving simply m_A × d_A = m_B × d_B. 30 kg × 1.8 m = 54 kg·m. Solving for d_B: d_B = 54 / 27 = 2.0 m. So Child B must sit exactly 2.0 m from the pivot — right at the far end if the seesaw is 4 m long. Check: moment of A = 30 × 1.8 = 54 kg·m (ignoring g, which cancels); moment of B = 27 × 2.0 = 54 kg·m — equal, confirming balance. This is the same principle behind every balance scale, crowbar, and see-saw: the product of mass (or force) and distance from the pivot must match on both sides, which is why a lighter person can balance a heavier one simply by sitting farther from the center.

Torque When the Force Isn't Perpendicular to the Lever Arm

The full torque formula is τ = rF sin(θ), where θ is the angle between the force vector and the lever arm (the line from the pivot to the point of application). Only the component of force perpendicular to the lever arm contributes to torque. A mechanic pulls on a 0.4 m wrench handle with 100 N of force, but pulls at an angle of 60° to the handle rather than straight perpendicular. Find the resulting torque. τ = rF sin(θ) = 0.4 × 100 × sin(60°) = 40 × 0.8660 = 34.64 N·m. Compare this to pulling fully perpendicular (θ = 90°, sin = 1), which would give the maximum possible torque: τ_max = 0.4 × 100 × 1 = 40 N·m. The angled pull at 60° only achieves 34.64/40 = 86.6% of the maximum available torque for the same applied force — exactly sin(60°) — which is why mechanics are taught to pull perpendicular to a wrench handle whenever possible: any angle away from 90° wastes some of the applied force on a component that pulls along the handle's length and contributes nothing to turning it.