Vector Diagrams

For example, with a scale of 1 cm representing 10 N, a 30 N force is drawn as a 3 cm arrow. The scale turns a physical force into a manageable length on paper. Labels should identify the force, and a direction can be stated using an angle, a compass direction or a direction relative to an object.

Combining forces to find the resultant

Several forces can be replaced by one resultant, or net force, with the same overall effect. Forces along one straight line are straightforward: forces in the same direction add, while forces in opposite directions subtract. The resultant points in the direction of the greater total force.

For forces in different directions, use a scale vector drawing. Draw the first force to scale, then move the second force without turning it so that its tail begins at the head of the first. This is called placing the vectors head-to-tail. Draw the resultant from the tail of the first vector to the head of the final vector. Its measured length gives the resultant magnitude after conversion using the scale, and a protractor gives its direction.

An equivalent construction for two forces is the parallelogram method. Draw both vectors from the same starting point, complete a parallelogram using lines parallel to the vectors, and draw the diagonal from the shared starting point. That diagonal is the resultant. Both constructions represent the same vector addition.

Suppose a scale drawing uses 1 cm to represent 2 N. If the resultant measures 4.3 cm, its magnitude is $4.3 \times 2 = 8.6\text{ N}$. This is a value obtained from the scale drawing, so sensible measuring precision matters.

Resolving one force into components

Resolution reverses the combining process. Instead of replacing several forces with one resultant, it replaces one force with components which together have the same effect. Components are usually chosen at right angles: one horizontal and one vertical.

To resolve an oblique force by scale drawing, draw the force vector to scale as the diagonal of a rectangle. Draw horizontal and vertical sides which meet at the arrowhead. The two directed sides are the horizontal and vertical components. Measure their lengths and use the original scale to find their magnitudes.

The component arrows must combine head-to-tail to reproduce the original force. A component is therefore not an extra force: it is one part of an alternative representation of the original force.

Components make a complicated diagram easier to interpret. Horizontal components can be combined with other horizontal forces, and vertical components with other vertical forces. The resultant horizontal and vertical effects can then be combined in a scale drawing to construct the final resultant.

Recognising equilibrium

An object is in force equilibrium when all its force vectors add to zero. In a head-to-tail diagram, this means the vector chain returns exactly to its starting point: the force polygon closes. There is then no resultant force and therefore no acceleration. The object may remain at rest, or it may continue moving at constant velocity.

For two forces alone to produce equilibrium, they must be equal in magnitude and opposite in direction. With several forces, it is not enough merely to find two equal arrows; the combined horizontal effect and the combined vertical effect must each be zero.

Force equilibrium concerns the resultant force. Preventing rotation also requires the turning effects of forces to balance, so a zero resultant force by itself does not always prove complete rotational equilibrium.

A reliable scale-drawing method

1. Choose a scale which makes the diagram large enough to measure accurately but small enough to fit the page.
2. Convert every force into a drawing length using that same scale.
3. Draw each vector accurately in its stated direction, using a ruler and protractor.
4. For addition, place vectors head-to-tail without changing their lengths or directions.
5. Draw the resultant from the initial tail to the final head, or check whether the chain closes for equilibrium.
6. Measure the required length and angle.
7. Convert the length back into newtons and state the resultant's direction.

Because the answer comes from measurement, a slightly different result may arise from normal drawing uncertainty. Accuracy improves when the chosen scale produces long, clear vectors.