Draw free-body diagrams showing all forces acting on an object, resolve forces into perpendicular components, and apply the equilibrium condition that the net force is zero in each direction.

What this topic is asking

A free-body diagram is the starting point of almost every Regents force problem, and equilibrium is the condition when the net force is zero. The Physical Setting/Physics course asks you to draw all the forces acting on an object as labelled arrows, resolve them into perpendicular components when needed, and apply the equilibrium condition (net force zero in each direction) to objects at rest or moving at constant velocity. Part B-2 often asks you to complete or draw a free-body diagram and read a force from it.

Drawing a free-body diagram

The standard forces to look for are: the weight $F_g = mg$ always pointing straight down; the normal force $F_N$ perpendicular to any surface the object rests on; friction $F_f$ along that surface, opposing sliding; tension $F_T$ along any rope or cable, pulling away from the object; and any applied force or spring force. Including the right forces, and no extras, is what the Regents rewards.

Equilibrium: the net force is zero

Equilibrium is the most common situation tested with free-body diagrams. A hanging sign, a book on a table, a box pushed at constant velocity, and a mass on a slope held by a rope are all equilibrium problems. In each, you set the forces in each direction equal and opposite. For the hanging sign, the upward tension equals the downward weight; for the box at constant velocity, the forward push equals the friction.

Resolving forces into components

When forces act at angles (a rope pulling diagonally, a weight on an incline), resolve each into perpendicular components before applying the equilibrium conditions. With an angle $\theta$ from the horizontal, a force $F$ has components $F_x = F\cos\theta$ and $F_y = F\sin\theta$. Then sum the components on each axis and set each sum to zero. On an incline it is usually easiest to tilt the axes along and perpendicular to the slope, so the weight resolves into $mg\sin\theta$ along the slope and $mg\cos\theta$ perpendicular to it.

Reference Tables note

Free-body diagrams and the equilibrium conditions are methods, not formulas, so they are not printed on the Reference Tables. The forces you place on the diagram do come from table equations: weight $F_g = mg$ and friction $F_f = \mu F_N$ are both printed, and the component resolution uses the right-triangle trigonometry given on the tables. Equilibrium is the $a = 0$ case of Newton's second law, $F_{net} = ma$.

Try this

Q1. State what a free-body diagram should never include. [1 point]

• Cue. Forces the object exerts on other objects, and invented forces (such as a "force of motion").

Q2. A $40.$ N picture hangs at rest from a single vertical cord. State the tension in the cord. [1 point]

• Cue. In equilibrium the tension balances the weight, so $F_T = 40.$ N.