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Mechanics · Forces & Newton's laws

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Force diagrams, resultant force & equilibrium

What each force on a free-body diagram is and which way it points (weight always vertically down, normal reaction perpendicular to the surface, tension pulling along a string), why forces add as vectors so a resultant is found component-by-component, and what "in equilibrium" means — the resultant is zero, so the components balance.

Mechanics · Forces & Newton's laws

Force diagrams, resultant force & equilibrium

What each force on a free-body diagram is and which way it points (weight always vertically down, normal reaction perpendicular to the surface, tension pulling along a string), why forces add as vectors so a resultant is found component-by-component, and what "in equilibrium" means — the resultant is zero, so the components balance.

Why it works

Every mechanics problem starts the same way: draw the free-body diagram — the object on its own, with an arrow for every force acting on it. Get this picture right and the equations write themselves; get it wrong and no amount of algebra saves you. So the first skill is knowing the forces and which way each one points.
  • Weight W=mgW = mg. The pull of gravity, acting at the object's centre and
always vertically downwards — never along a slope, never tilted. Its size is mass ×g\times g, with g≈9.8 m s−2g \approx 9.8\ \text{m s}^{-2}.
  • Normal reaction RR (or NN). The push of a surface on the object, always
perpendicular to that surface. On flat ground it points straight up; it is not automatically equal to the weight (see below).
  • Tension TT. The pull of a taut string or rope, acting **along the string,
away from the object*. A string can only pull* — never push.
  • Thrust / compression. The push in a rigid rod that is being squashed, acting
along the rod. Unlike a string, a rod can push.
  • Friction / resistance. A force from a surface or the air that acts **along
the surface, opposing motion** (or the tendency to move). Air resistance and a vehicle's "resistance to motion" are the same idea.
  • Driving force / applied force. Whatever is pushing or pulling the object — an
engine's drive, a hand, a rope you pull.RWFFr*A block on the ground being pulled by a force FF (not accurately drawn). Weight WW down, normal reaction RR up, the pull FF, and a resistance FrFr opposing the motion. These four are all the forces on the block — nothing the block does to anything else belongs here.*

Normal reaction is not always the weight. R=mgR = mg only when the only other vertical force is gravity. Push down on the block with an extra vertical force and RR grows; pull up on it and RR shrinks. RR is whatever it has to be to make the vertical forces balance — work it out, don't assume it.

Forces add as vectors — that's the whole game. The single force that has the same effect as all of them together is the resultant: add them tip-to-tail, or (far easier) add their components. Writing forces in i,j\mathbf{i}, \mathbf{j} form, the resultant is just R=∑F=(∑Fx)i+(∑Fy)j,\mathbf{R} = \sum \mathbf{F} = \left(\textstyle\sum F_x\right)\mathbf{i} + \left(\textstyle\sum F_y\right)\mathbf{j}, and its magnitude is (∑Fx)2+(∑Fy)2\sqrt{(\sum F_x)^2 + (\sum F_y)^2} with direction tan⁡−1 ⁣(∑Fy∑Fx)\tan^{-1}\!\big(\tfrac{\sum F_y}{\sum F_x}\big). Two perpendicular forces of 3 N3\,\text{N} and 4 N4\,\text{N} give a resultant of 32+42=5 N\sqrt{3^2+4^2}=5\,\text{N}, not 7 N7\,\text{N} — you never add the sizes of forces that point different ways.

Equilibrium means the resultant is zero. If an object is at rest or moving at constant velocity, the forces on it are balanced: ∑F=0\sum \mathbf{F} = \mathbf{0}. In components that is two equations at once, ∑Fx=0and∑Fy=0,\sum F_x = 0 \quad\text{and}\quad \sum F_y = 0, which is exactly enough to pin down unknown forces. (This is Newton's first law, and the a=0a = 0 special case of F=maF = ma in [[forces.newtons-laws]].) The art is choosing two convenient directions — usually horizontal and vertical — and resolving every force onto them.