Resultant forces

Force is a vector: it has a magnitude, measured in newtons (N), and a direction. This means that forces cannot always be combined by simply adding their numerical values. Their directions matter too.

Reading and drawing a free body diagram

A free body diagram isolates one object from its surroundings. Represent the object by a simple box or dot, then draw a labelled arrow for every external force acting on it:

• the arrow points in the direction of the force;
• the arrow's length represents the force's magnitude when relative sizes are known;
• the label identifies the force, such as weight, friction, drag, thrust or normal contact force.

The arrows show forces, not the object's velocity. A car could be moving to the right while its resultant force points left—for example, while braking.

Before drawing, decide exactly what the chosen object or system is. For a pushed crate, include the push on the crate, friction from the floor, its weight and the floor's normal contact force. Do not include the force that the crate exerts on the person or floor, because those forces act on other objects.

Finding the resultant force

The resultant force, or net force, is the single force that would have the same effect as all the forces together. For forces acting along the same straight line:

• forces acting in the same direction are added;
• forces acting in opposite directions are subtracted;
• the resultant points in the direction of the larger total force.

For example, suppose a trolley has a driving force of 12 N to the right and friction of 5 N to the left. The resultant is $12-5=7\text{ N}$ to the right. It is incomplete to give only “7 N”: direction is part of a vector answer.

In the falling-object example, weight and drag oppose one another. If weight is 10 N downwards and drag is 4 N upwards, the resultant is 6 N downwards. The object therefore accelerates downwards. This does not by itself tell us whether it is moving up or down at that instant: resultant force determines the direction in which velocity changes, not necessarily the direction of the existing velocity.

Resultant force and changing motion

A non-zero resultant force causes acceleration in the direction of the resultant. Acceleration means a change in velocity, so the object may speed up, slow down or change direction.

Consider a car travelling to the right:

• if its resultant force is to the right, its velocity changes towards the right, so it speeds up;
• if its resultant force is to the left, its velocity changes towards the left, so it slows down while it is still travelling right;
• a sideways resultant changes the direction of its velocity.

The resultant is found from the forces, not from how fast the object is moving. An object does not require a forward resultant force merely to keep moving. The special case in which the forces cancel and the resultant is zero is called equilibrium and is considered separately.

Forces in more than one direction

A complete free body diagram must show every relevant force even if some do not contribute to the resultant in the direction being discussed. For a crate accelerating horizontally along a level floor, its weight downwards and the normal contact force upwards may cancel vertically, while the push and friction leave a horizontal resultant. The overall resultant is then horizontal.

If forces act at angles to one another, the resultant is found by vector addition rather than ordinary arithmetic. The same principle still applies: replace all the force vectors by one vector with the same combined effect.

For OCR Physics A, this learning outcome is shown in bold in the specification and is therefore assessed on Higher Tier papers.