Physics › Mechanics › Moments and equilibrium
Moments and equilibrium
A force can turn as well as push, and the turning effect depends on where the force acts as much as how big it is. The principle of moments settles every balancing problem in the course, provided the distance you use is the perpendicular one.
Builds on Scalars and vectors.
IN THIS TOPIC
- Calculate the moment of a force as force times perpendicular distance from the point to the line of action.
- Recognise a couple and calculate its moment as force times the separation of the lines of action.
- Apply the principle of moments to balanced systems, using the centre of mass to place an object's weight.
WHAT YOU PROBABLY THINK
If the forces are equal, it balances.
The moment of a force
The turning effect of a force about a point is its moment:
where d is the perpendicular distance from the point to the line of action of the force, and the unit is the newton metre, N m. Every word of that definition earns marks, and “perpendicular” is the word that separates definitions that score from definitions that do not.
When a force is applied at an angle, extend its line of action as a construction line and drop a perpendicular from the pivot onto it. That perpendicular length, equal to L sin θ for a force at θ to the beam, is the d in Fd. Pushing along a line that passes through the pivot gives a moment of zero, however hard you push.
Couples
A couple is a pair of equal and opposite coplanar forces whose lines of action do not coincide, the grip of two hands on a steering wheel. The forces cancel, so a couple produces no resultant force and no acceleration of the centre of mass; it produces turning and nothing else.
Its moment is force times the perpendicular distance between the two lines of action, F × s, using one of the forces, not both. Doubling by counting each force separately is a standard slip.
The principle of moments
An object in equilibrium satisfies the principle of moments: about any point, the sum of clockwise moments equals the sum of anticlockwise moments. Equal forces are neither necessary nor sufficient; what balances is the products Fd.
The freedom to choose the pivot is the technique. Moments may be taken about any point, so take them about the point where an unknown force acts: that force then has zero distance, zero moment, and vanishes from the equation, leaving one unknown instead of two.
Centre of mass
An object's weight is spread through it, but for moments it behaves as if the whole weight acts at one point, the centre of mass. For a uniform regular solid, the centre of mass is at its geometric centre, so a uniform 4.0 m beam carries its weight at the 2.0 m mark. Placing the weight anywhere else is the commonest way a beam calculation goes wrong before it starts.
THE EXAM BIT
- Define the moment with all the furniture: force multiplied by the perpendicular distance from the point to the line of action. Definitions missing either phrase drop the mark.
- Take moments about the point where an unknown force acts, and that unknown disappears from the equation. It is the single most useful trick in the topic.
- A uniform beam's weight acts at its centre. Forgetting the beam's own weight, or placing it at an end, wrecks otherwise sound working.
- A couple's moment is one force times the full separation, not both forces. And a couple cannot be balanced by a single force, only by another couple.
- Check both conditions when asked whether a body is in equilibrium: resultant force zero and resultant moment zero. Each can hold without the other.
CHECK YOURSELF
A uniform 4.0 m beam of weight 200 N rests on supports at each end. A 600 N load sits 1.0 m from the left support. Find the force from the left support.
Show a hint
Take moments about the right support, so its unknown force vanishes.
Show the answer
About the right support: anticlockwise, Rleft × 4.0. Clockwise: the beam's weight, 200 N at its centre, 2.0 m away, gives 400 N m; the load, 600 N at 3.0 m from the right, gives 1800 N m.
So 4.0 Rleft = 400 + 1800 = 2200, giving Rleft = 550 N.
The pivot choice did the work: the right support's force never appeared, and one equation held one unknown.
Moments balance, not forces.
Choose the pivot that kills the unknown.
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