Resultant forces

The resultant force is a single force that has the same effect as all the forces acting on an object combined. It tells you what the object will actually do.

Why resultant matters

Newton's first law: an object stays at rest or moves with constant velocity unless acted on by a resultant force. Newton's second law: $F = ma$ where $F$ is the resultant force.

So if the resultant is zero, the object's velocity doesn't change. If non-zero, the object accelerates in the direction of the resultant.

Adding co-linear forces

Forces along the same line just add or subtract.

• Two forces in the same direction: add magnitudes.
• Two forces in opposite directions: subtract — the result points in the direction of the larger.

Example: 30 N right and 12 N left → resultant 18 N right.

Equilibrium

If forces in all directions sum to zero, the object is in equilibrium. Up forces = down forces; left = right.

• Book on a table: weight (down) = normal (up); horizontal forces both zero. Equilibrium.

Adding perpendicular forces (Higher Tier)

Use Pythagoras and trig:

• Magnitude: $|R| = \sqrt{F_x^2 + F_y^2}$.
• Direction: $\theta = \tan^{-1}(F_y/F_x)$.

Higher Tier — resolving 2D forces

If a force acts at angle $\theta$ to the horizontal, its components are:

• $F_x = F\cos\theta$ (horizontal).
• $F_y = F\sin\theta$ (vertical).

Add up all $F_x$ values to get total horizontal; same for vertical. Combine with Pythagoras.