Equilibrium of a point mass

A body is in static equilibrium if it is at rest and remains at rest over time. For a point mass — an object so small that its dimensions and shape can be neglected — this condition translates into a very simple requirement on the forces acting on it: their resultant must be zero. If even a single direction were left “uncovered”, the point would begin to accelerate in that sense and the equilibrium would be broken.

Principle — Equilibrium condition for a point mass

$\ev{\sum_i \vv{F}_i = \vv{0}}$ The vector sum of all forces acting on the point must be zero.

Being an equality between vectors, it is equivalent to several scalar equalities, one for each direction in space. Working in the plane, the condition $\sum_i \vv{F}_i = \vv{0}$ splits into two independent equations:

$\begin{aligned} \sum_i F_{i,x} = 0 \qquad \sum_i F_{i,y} = 0 \end{aligned}$

that is, the sum of all horizontal components is zero and the sum of all vertical components is zero. Each force must first be resolved along the chosen axes, then the $x$ components and the $y$ components are summed separately. Choosing convenient axes — often one along the direction of a known force — greatly simplifies the working.

Rule of $N$ unknowns

If a problem contains $N$ unknowns (forces, angles, spring lengths, normal reactions…), determining all of them requires at least $N$ distinct, non-trivial equations: a relation of the form $0 = 0$ does not count. When the two conditions $\sum \vv{F} = \vv{0}$ are not enough, the third equation almost always comes from a geometric relation in the diagram: spring lengths that add up to a fixed distance, complementary angles, ropes running over pulleys. In many problems it is literally impossible to reach the solution without reading the geometry of the figure.

Collegamenti

Argomenti: Statica ed equilibrio Concetti: Equilibrio statico Oggetti: Molla

Esercizi collegati: Esercizio svolto — Blocco appeso a due funi · Esercizio svolto — Il Gorilla tira una cassa · Esercizio svolto — Scatola con molla: zona di equilibrio