Moments and resolution of forces
This lesson deals with forces acting on a body at rest. The difference between the particle of the last lecture and the body in this lecture is that all the forces on the particle act through the same point, which is not the case for forces on an extended body. The important concept, again, is the resolution of forces to obtain the equations determining equilibrium. The simplest examples involve essentially one-dimensional bodies such as ladders. Again, it is essential start with a good diagram showing all the forces.
2.2 Key concepts
• Resolution of forces into a single resultant force or a couple.
• The moment of a force about a fixed point.
• Condition for equilibrium: zero resultant force and zero total couple.
2.3 Resolving forces
The difference between forces acting on a particle and forces acting on an extended body is immediately obvious from the intuitive in equivalence of the two situations below: for an extended body, it matters through which points the forces act — i.e. on the position of the line of action of the force
Translate the body in the direction parallel to the line of action of the force; and a tendency to rotate the body.1 Clearly, for the body to be in equilibrium these effects must separately balance.For the translational effects to balance, we need (as in the case of a particle) the vector sum of the forces to be zero:
For the rotational effects about a point to balance, we need the sum of the effects to be zero, but what does this mean? Intuitively, we expect that a force whose line of action is a long way from to have more rotating effect than a force of the same magnitude that is nearer and it turns out (see below) that the effect is linear in distance. The rotation effect of a force is called the moment of the force.
2.4 The moment of a force
In two dimensions, or in three dimensions in the case of a planar body and forces acting in the same plane as the body, any force tends to rotate the body within the plane or, in other words, about an axis perpendicular to the plane. In this case, we define:
Moment of a force about a point P
magnitude of the force
the shortest distance between the line of action of the force and P
A couple is a pair of equal and opposite forces. We define the moment of a couple about any point in the obvious way, as the sum of the moments of the two forces about that point. The sum of the moments of two forces will in general depend on the point about which the moment of the individual forces is taken; but this is not the case for a couple. Let the two forces be and and let and be the position vectors of any fixed points on their respective lines of action, with respect to a point . Then
and this does not depend on the choice of . Choosing P on the line of action of one of the forces shows that the magnitude of the couple is just , where is the distance between the lines of action of the forces.