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# lecture3 - Mechanics Lecture Notes 1 1.1 Lecture 3...

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Mechanics Lecture Notes 1 Lecture 3: Equilibrium of a solid body 1.1 Introduction This lecture 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. 1.2 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. 1.3 Resolving forces The difference between forces acting on a particle and forces acting on an extended body is imme- diately obvious from the intuitive inequivalence 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. F F In general, each force acting on a body can be thought of as having two effects: a tendency to 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: X i F i = 0 . (1) For the rotational effects about a point P 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 P 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. 1.4 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 . 1 Imagine that one point, not on the line of action of the force, is fixed. 1

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with account taken of the direction of the effect: either clockwise or anticlockwise. 2 In general (in three dimensions when the body and forces are not coplanar), the effect of different forces will be to tend to rotate the body about different axes. In this case, the ‘force times distance from line of action to the point’ definition of the moment of a force is not adequate. We have to represent the moment as a vector. The important thing to understand is that the direction of vector representing the moment of the force not
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