Modelling with Statics

Decide whether the model is for a knot, a suspended load, an individual beam or the whole connected system. Include only forces acting on that selected body. If the system contains both a string and a load, their mutual forces are internal to the system; they are not an extra pair of external forces to add on top of the weight.

Weight acts vertically downwards. A taut string pulls along its own direction, away from the particle or object to which it is attached. A normal contact force acts perpendicular to a surface. Friction acts parallel to the contact and opposes the tendency to slip; its direction must be reasoned from the situation.

For a particle, the concurrent force model has no independent rotational condition. For a rigid body such as a beam, zero resultant force must be combined with zero resultant moment. State which model you are using rather than choosing equations by habit.

A light string has negligible weight, so you do not include the weight of each string segment. An inextensible string does not stretch. A smooth contact has no friction. A light string passing over a smooth pulley supports the usual equal-tension idealisation along that single string; it does not make the tensions in two different strings equal.

A particle model neglects size and rotation. A rigid-body model retains geometry and turning effects. Treat these as purposeful approximations, then explain a limitation if the question asks how the real arrangement might differ. For example, an actual rough pulley can make the tensions on its two sides different.

At the knot below, two strings pull upwards at different angles to the horizontal. A 120 N load pulls downwards. The steeper right-hand string does not generally have the same tension as the left-hand string. Horizontal components must cancel, while the two upward components together support the load.

Scroll the diagram to see the full view.

For a body at rest on a rough plane, static friction adjusts up to a maximum magnitude $\mu R$. You may use $|F|\le\mu R$, with equality only when the contact is in limiting equilibrium. Setting friction equal to $\mu R$ in every stationary problem adds a condition that may be false.

Resolve along and perpendicular to the plane. First establish the tendency to move, then obtain the friction required for equilibrium. Compare that required value with the available limiting friction. If the required magnitude exceeds the limit, the proposed static model cannot hold.

Check units, directions and the assumptions that made each equation valid. If a calculated tension is negative, the assumed taut-string direction is inconsistent with that model: a string cannot push. Recheck the diagram and consider whether the string would become slack.

For a beam, check both force and moment balance. For a rough contact, check the friction inequality. A force diagram and a correct set of equations are valuable even if the numerical answer looks plausible; they make missing forces and unjustified assumptions visible.

When evaluating a model, connect the limitation to the result. Saying a pulley is not perfectly smooth is useful if you then explain why equal tensions may no longer follow. Saying a particle has a finite size matters when rotation or contact geometry affects the prediction.