Equilibrium: when the forces cancel out

A particle is an object treated as a single point, so we don't worry about its size or about it spinning. A particle is in equilibrium when all the forces on it cancel out completely. The total force, called the resultant, is zero.

This does not mean that no forces act. A book resting on a table has its weight pulling down and the table pushing up. Both forces are real; they just cancel.

Why does this matter? Newton's first law says that with no resultant force, an object doesn't speed up, slow down or change direction. So a particle in equilibrium is either at rest or moving at a constant velocity. Whenever a question says a particle is "at rest", "hangs", "is held" or "moves with constant speed in a straight line", you know the forces cancel, and that gives you equations.

Forces can point in any direction, so "they cancel" is hard to use directly. The trick is to split every force into two pieces at right angles, usually horizontal and vertical. These pieces are called components, and splitting a force up is called resolving it.

Forces cancel overall only if they cancel in every direction. So equilibrium gives two separate equations:

• the horizontal components add to zero (everything pushing right balances everything pushing left),
• the vertical components add to zero (everything up balances everything down).

Two equations means you can find two unknowns.

A force of size $F$ at an angle $\theta$ to a direction has these components:

Why? Draw the force as the long side (hypotenuse) of a right-angled triangle whose other two sides are the components. The side next to the angle is $F\cos\theta$ and the side opposite the angle is $F\sin\theta$, because $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$ and $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$ (SOH CAH TOA), with the hypotenuse equal to $F$.

The easy way to remember it: cos goes with the side touching the angle. If the angle is measured from the horizontal, the horizontal component uses $\cos$. If the angle is measured from the vertical, the vertical component uses $\cos$ instead.

1. Draw a force diagram. Mark every force with an arrow and a label, including the weight.
2. Choose directions. Usually right is positive and up is positive.
3. Resolve every slanted force into horizontal and vertical components.
4. Write two equations: horizontal components total zero, and vertical components total zero.
5. Solve them, then check the answers are sensible (tensions in strings must come out positive).