Revision summary

Choose the body, draw its external forces, then decide whether you need force balance, moment balance or both.

Force models

• Name the source and recipient. Include forces on the chosen body, not forces it exerts on other bodies.
• Weight acts along the gravitational field. Electric force is along the field for a positive charge and opposite it for a negative charge.
• Magnetic force on a current-carrying conductor is perpendicular to current and field. In the given view, current right and field into the page give force up. Parallel current and field give zero magnetic force.
• Normal force is perpendicular to the contact surface. Tension pulls along a taut string.
• Friction opposes relative sliding or its tendency at the contact. Drag opposes motion relative to the fluid. Do not infer a universal drag-speed law.
• Components represent the original force; they are not extra forces to count alongside it.

Springs

Extension

x = loaded length - unloaded length. Convert to metres when k is in N/m.

Hooke's law

F = kx within the limit of proportionality. Show the restoring direction separately.

Force-extension graph

Force vertically and extension horizontally: gradient k. Reversed axes: gradient 1/k.

Practical reference

Include the hanger in total force and use a genuinely unloaded reference, or state a consistent incremental model. Read a settled pointer perpendicular to a fixed ruler.

Proportionality tests the linear force-extension relationship; elasticity tests recovery after unloading. Repeats, calibration, alignment and loading/unloading checks address different limitations.

For a scattered force-extension graph, choose a line that follows the whole trend. In y = mx + c, the gradient gives force per extension and the intercept is the reading at zero extension. At the intersection of two candidate lines, equate their force expressions; one crossing does not establish a good fit.

Moments and couples

Moment of a force

Fd about a stated point, with d perpendicular to the force's line of action. A line through the pivot gives zero moment.

Couple

Equal opposite parallel forces on the same body, with different lines of action. Resultant force zero; torque τ = Fd, where d is their full perpendicular separation.

Centre of gravity

Replace distributed weight by one equivalent force at the centre of gravity. A uniform straight beam in a uniform field has it at the midpoint.

Use a consistent clockwise/anticlockwise sign. A third-law pair acts on different bodies and is not a couple on one body.

Distinguish tension T from torque T by context and units. Although torque has the same dimensions as energy, report it in N m, not J.

Equilibrium

1. Draw the free-body diagram, including the body's own weight where relevant.
2. Use ΣF_x = 0 and ΣF_y = 0 for force balance.
3. Use equal clockwise and anticlockwise moments. Choosing a point on an unknown force's line removes its moment from that equation.
4. Check another balance and interpret any unexpected sign against the actual support or string model.

A closed force triangle means zero resultant force. For an extended body, also check moments using actual lines of action. In a non-right triangle, use opposite side-angle pairs for the sine rule and the interior included angle for the cosine rule:

The angle between two vectors drawn from a common origin need not be their head-to-tail triangle's interior angle.