Resolving forces

Splitting a force at an angle into two perpendicular components, F cosθ and F sinθ — to add forces in any directions and to solve equilibrium by resolving in two perpendicular directions.

Mechanics · Forces & Newton's laws

Resolving forces

Splitting a force at an angle into two perpendicular components, F cosθ and F sinθ — to add forces in any directions and to solve equilibrium by resolving in two perpendicular directions.

Why it works

A force pulling at an angle is awkward — but it has exactly the same effect as two perpendicular forces acting together: one along your chosen direction and one at right angles to it. Replacing a force by these is resolving it.

For a force

F

acting at angle

\theta

to a direction, the component along that direction is

F\cos\theta

and the component perpendicular to it is

F\sin\theta

F_{\parallel} = F\cos\theta, \qquad F_{\perp} = F\sin\theta.

The cosine goes with the angle you're measuring from, the sine with the perpendicular — a quick check is that when

\theta = 0

the force is entirely along (

F\cos 0 = F

) and nothing perpendicular (

F\sin 0 = 0

).

This is the master tool for forces in two dimensions. To combine several forces, pick two perpendicular axes (often horizontal and vertical), resolve every force onto them, add the components in each direction, and recombine if you need the resultant.

Equilibrium by resolving. A particle in equilibrium has zero resultant force, which means the components balance separately in each of two perpendicular directions:

\sum F_x = 0 \quad\text{and}\quad \sum F_y = 0.

These two equations let you find unknown forces or angles. The skill is choosing directions that make the algebra clean — resolving along and perpendicular to one of the unknown forces often makes it disappear from one equation.