Introduction
Statics is the branch of mechanics dealing with bodies in equilibrium — where all forces and moments balance so there is no acceleration (linear or rotational). It requires the vector algebra of Chapter I applied specifically to systems of forces, and forms the analytical basis for structural and mechanical engineering problems.
1. Forces and resultants
Multiple forces acting on a body can be combined into a single equivalent resultant force by vector addition. Solving statics problems typically starts by resolving every force into convenient perpendicular components (usually horizontal and vertical) so that the resultant’s components can simply be summed algebraically rather than through repeated geometric constructions.
2. Moments and couples
A moment (or torque) of a force about a point measures its tendency to cause rotation about that point, calculated as the force multiplied by the perpendicular distance from the point to the force’s line of action (moment = F × d). A couple is a pair of equal, opposite, and non-collinear forces, which together produce a pure rotational effect (a moment) with zero net force — couples are important because their moment is the same about every point in the plane, simplifying certain calculations.
3. Conditions of equilibrium
A body is in static equilibrium if and only if both conditions hold simultaneously: the sum of all forces is zero (no net translational tendency), and the sum of all moments about any point is zero (no net rotational tendency). In two dimensions, this gives three independent scalar equations (ΣFx = 0, ΣFy = 0, ΣM = 0) — exactly enough to solve for three unknowns in a typical statically determinate problem.
4. Friction
Friction is the force resisting relative sliding between two surfaces in contact. In statics, the key relation is that the maximum available static friction is F ≤ μsN (μs is the coefficient of static friction, N is the normal force) — a body remains in equilibrium as long as the friction required to prevent sliding does not exceed this maximum; once it would, the body is on the point of slipping (limiting equilibrium), a condition frequently used to find critical angles or maximum loads in problems like ladders leaning against a wall.
5. Centres of mass and gravity
The centre of mass (or, in a uniform gravitational field, centre of gravity) of a body or system is the single point at which its entire weight can be considered to act for the purposes of moment calculations — for a system of discrete masses, x̄ = Σ(mᵢxᵢ)/Σmᵢ, and for continuous bodies the equivalent integral form is used. Finding the centre of mass of composite shapes (built from simpler standard shapes, such as rectangles and triangles) by combining known centres of mass, weighted by area or mass, is a standard and frequently tested technique.
Key Takeaways
- Resultant forces are found by resolving into components and summing algebraically, avoiding repeated geometric vector addition.
- A moment measures rotational tendency (force × perpendicular distance); a couple produces a pure moment with zero net force, independent of the point about which it is taken.
- Equilibrium requires both zero net force and zero net moment simultaneously — two conditions giving three independent equations in two dimensions.
- Static friction has a maximum value (μsN); limiting equilibrium (friction at its maximum) is the key condition for problems asking for critical angles or maximum loads.
- The centre of mass/gravity is the effective point of action of weight for moment calculations, and composite-shape centres of mass are found by combining known simple-shape results.