Equilibrium of a particle

A crane hook holding a pallet, the ring where three slings meet above a load, a traffic light hanging from two cables, a pulley riding on a zip line: in each case, several forces act through a single point, and that point is not moving. These are particle equilibrium problems, the simplest kind in statics and the foundation for everything that follows.

This chapter sets out the equilibrium condition, then spends real time on the free-body diagram, the drawing that turns a physical situation into equations. You will model cables, pulleys, springs and smooth surfaces, see why tight cables carry surprisingly large forces, find the largest load a system can safely carry, and extend the method to three dimensions.

The equilibrium condition

Newton's first law says that a particle with no net force on it stays at rest (or moves at constant velocity). So a particle is in equilibrium when the vector sum of all forces on it is zero:

In two dimensions this is two scalar equations; in three dimensions, three:

Two equations can solve for two unknowns, three for three. Unknowns are typically cable tensions, an unknown angle, or the weight a system can carry. If a problem has more unknowns than equations, you need more information, such as a spring stiffness or a geometric relationship, before it can be solved.

The free-body diagram

The free-body diagram (FBD) is a sketch of the particle by itself, cut free from everything around it, with every external force drawn as an arrow. It is the single most important step in a statics problem. Almost every wrong answer in this course can be traced to a wrong free-body diagram, not a wrong equation.

Draw one in four steps:

• Isolate the particle. Imagine cutting every cable, rope and contact that touches it, and draw the particle on its own.
• Add the weight (if it is not negligible) acting straight down.
• At every cut, add the force that the removed object exerted. A cable becomes a tension along the cable; a smooth surface becomes a normal force perpendicular to it.
• Label each force with its known value or an unknown symbol, mark the angles and slopes, and choose x and y axes.

Forces between parts of the particle (internal forces) never appear. Forces the particle exerts on other things do not appear either: only forces acting on it.

Modeling the connections

Each kind of connection exerts a particular kind of force. Learning these is like learning vocabulary; you will use them for the rest of your career.

• Cables, ropes and chains can only pull. The tension acts along the cable, away from the particle. An ideal cable is weightless and inextensible.
• A frictionless pulley changes the direction of a cable without changing its tension. The tension is the same on both sides.
• A linear spring exerts a force proportional to its change in length: \(F = ks\), where \(k\) is the stiffness (N/m) and \(s = L - L_0\) is the stretch from the unstretched length \(L_0\). A stretched spring pulls; a compressed spring pushes.
• A smooth surface pushes perpendicular to itself (a normal force) and cannot pull. "Smooth" means no friction, so there is no force along the surface.
• Weight acts at the center of gravity, vertically downward, with magnitude \(mg\).

Solving two-dimensional problems

With the free-body diagram drawn, the rest is algebra. Resolve every force into x and y components, write the two equilibrium equations, and solve.

Two checks are worth building into your habits. First, both tensions are positive: a negative tension would mean a cable pushing, which is impossible, so the assumed geometry would be wrong. Second, the steeper cable carries more load, which matches intuition: the more nearly vertical a cable, the more directly it opposes the weight.

The force triangle and Lami's theorem

When exactly three forces hold a particle in equilibrium, they must form a closed triangle when drawn tip to tail. That gives an alternative to components. Lami's theorem states that each force is proportional to the sine of the angle between the other two:

where \(\alpha_1\) is the angle between \(F_2\) and \(F_3\), and so on. For the crate, the angle between \(T_{AC}\) and \(W\) is 135°, between \(W\) and \(T_{AB}\) is 120°, and between the two cables is 105°. Then \(T_{AB} = 2452.5 \sin 135^\circ / \sin 105^\circ = 1795\) N, matching the component method. Lami's theorem is quick for three forces but does not extend to four or more, so components remain the general tool.

Why tight cables are dangerous

Look at what happens as a cable flattens. A load \(W\) hangs from the middle of a cable with each half at angle \(\theta\) below horizontal. By symmetry the tensions are equal, and vertical equilibrium gives

As \(\theta\) goes to zero, \(\sin\theta\) goes to zero and the tension grows without limit. A perfectly straight horizontal cable cannot support any load at all.

This is why riggers never tighten a lifting sling until it is nearly flat, why a "slackline" anchored between two trees can pull them over, and why overhead power lines are strung with a deliberate sag. Rigging handbooks give sling capacity as a function of the sling angle, and capacity drops sharply below about 45°. The same geometry appears in the nearly flat members of a shallow truss, which carry large forces for the same reason.

Pulleys and springs

Smooth surfaces and inclines

On an incline, choose axes along and perpendicular to the surface. The normal force then lies along one axis, which removes it from the other equation.

Design: the largest safe load

In practice the question is often reversed. Each cable has a rated working load, and you need the largest mass the system can carry. Solve for every tension in terms of the load, find which member reaches its limit first, and size the load to that member.

Note the method: you cannot just set both tensions to 5 kN and solve, because then you would have three equations (two equilibrium plus a ratio) and only one free variable. Solve generally first, then let the governing member set the load.

Particles in three dimensions

In 3D, write each force as a Cartesian vector, usually using the position-vector method from the last lesson, and set the sum of each component to zero. Three equations solve for three unknowns, such as the tensions in three cables holding a load.

In 3D, solving by substitution is fine for three equations, but a calculator or a few lines of Python with a matrix solver is faster and avoids arithmetic slips. You will write exactly that program in ENCM-100. Whichever way you solve, substitute the answers back into all three equations as a check.