Moments

When looking at forces acting on a rigid body, as opposed to objects modelled as particles, it is possible to study the turning effects of that rigid body as well as forces acting at different points on the rigid body. Moments measure the turning effect of a given force and so we must specify the point at which we are assessing that turning effect. The moment about a given point is calculated as follows:

\text{ Moment }=\text{ Force }\times\text{ Perpendicular Distance}

This can also be thought of as the direct distance to the point where the force is acting multiplied by the component of the force acting perpendicularly to that distance. For a force acting completely at right angles to the distance from the point, the formula simplifies to \text{Moment }=\text{ Force }\times \text{ Distance} – this is because in this case \sin(\theta)=\sin(90^\circ)=1.

Since we are talking about turning effect we must specify whether the moment is a clockwise moment or an anti-clockwise moment. We also give the units of a moment which is Newton metres (Nm) since they are calculated from the product of a force and a distance.

Equilibrium

As with a resultant force, there is also a resultant moment at a given point which is the sum of all moments acting at that point. In order to calculate the resultant moment of a rigid body at a given point, the forces acting on the body must be coplanar – see Example 1. This means that the forces are all acting in the same plane. When a body is in equilibrium, the resultant moments are 0 around any given point as well as the resultant forces – this means that, as well as the total upwards forces being equal to the total downwards forces, all clockwise moments balance anticlockwise moments:

\text{Equilibrium }\Leftrightarrow \text{ Total Clockwise Moments = Total Anti-clockwise Moments}

In many equilibrium cases, a careful selection of the point around which to calculate moments can simplify the problem vastly – see Example 2.

Centres of Mass and Tilting

For rigid bodies that are uniform, their mass is spread evenly throughout the body and the centre of mass (the point at which the weight of the body acts) is at the centre. For non-uniform rigid bodies, the mass is spread unevenly and the centre of mass does not act at the centre. Calculating moments around certain points in a rigid body can reveal the location of the body’s centre of mass – see Example 3. Questions like this may also include bodies that are at the point of tilting about a pivot. In these cases, the reaction force at any other support point that isn’t the pivot is equal to 0 – see Example 4.

Examples

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AS Mechanics Moments