Physics 9.3 - 9.5 - Scale vectors and free-body diagrams

Scale drawings and resultants

Higher tier only

Forces along one straight line can be combined by choosing a positive direction. Forces in that direction are added; forces in the opposite direction are subtracted. For forces in different directions, use a scale vector drawing.

To add force vectors graphically:

1. Choose and write a convenient scale, such as 1 cm represents 10 N.
2. Draw the first vector to scale and in the correct direction.
3. Draw the second vector with its tail at the head of the first. Preserve its original length and direction.
4. Continue head-to-tail for any further forces.
5. Draw the resultant from the tail of the first vector to the head of the final vector.
6. Measure the resultant's length and angle, then use the scale to state its magnitude and direction.

Use a ruler and protractor to draw and measure these constructions on paper. The displayed image may be resized by the screen, so its centimetre lengths must not be measured from the screen. Use the stated drawing scale on your own paper.

The order of the force vectors can change without changing the resultant, provided every vector keeps its magnitude and direction.

Worked example

Two forces act at one point: $30\,\mathrm{N} \text{ east}$ and $40\,\mathrm{N} \text{ north}$. Determine their resultant by scale drawing.

Solution

1. Use the scale 1 cm represents 10 N.
2. Draw a $3.0\,\mathrm{cm}$ arrow east.
3. From its head, draw a $4.0\,\mathrm{cm}$ arrow north.
4. Join the first tail to the final head. The resultant measures $5.0\,\mathrm{cm}$.
5. Convert with the scale: $5.0\,\mathrm{cm} \times 10\,\mathrm{N\,cm^{-1}} = 50\,\mathrm{N}$.
6. Measure the angle from east towards north: about $53 ^{\circ}$.

The resultant is $50\,\mathrm{N}$, about $53^\circ$ north of east. This is sensible: it points between the two forces and is closer to north because the northward force is larger.

The equilibrant is the force that balances the resultant: it has the same magnitude and the opposite direction. Thus a $50\,\mathrm{N}$ force directed from the final head back to the initial tail balances the $30\,\mathrm{N}$ east and $40\,\mathrm{N}$ north pair.

If vectors drawn head-to-tail form a closed shape, the final head returns to the first tail. The resultant then has zero length, so the forces are in equilibrium.

Resolving a force

Higher tier only

Resolving a force is the reverse of finding a resultant. One force vector is replaced by component vectors, usually in two perpendicular directions such as horizontal and vertical. The components are not extra forces: together they represent the same original force.

To resolve a force by scale drawing:

1. Choose a scale and draw the original force vector accurately.
2. Draw perpendicular axes through the vector's tail.
3. From the vector's head, draw construction lines parallel to the axes to complete a rectangle.
4. Draw the two component vectors from the original tail along the axes.
5. Measure each component and convert its length using the scale.

Worked example

A $50\,\mathrm{N}$ force acts at $53 ^{\circ} \text{ north of east}$. Resolve it into eastward and northward components using a scale drawing.

Solution

1. Use 1 cm represents 10 N, so draw the original force $5.0\,\mathrm{cm}$ long at $53 ^{\circ} \text{ north of east}$.
2. Complete the horizontal-vertical rectangle from the force arrow's head.
3. The eastward component measures about $3.0\,\mathrm{cm}$, giving $30\,\mathrm{N} \text{ east}$.
4. The northward component measures about $4.0\,\mathrm{cm}$, giving $40\,\mathrm{N} \text{ north}$.