Statics_of_Rigid_Bodies_03_Rigid_Bodies - Section 2.3 2.3...

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Section 2.3 Solid Mechanics Part I Kelly 17 2.3 The Statics of Rigid Bodies A material body can be considered to consist of a very large number of particles. A rigid body is one which does not deform, in other words the distance between the individual particles making up the rigid body remains unchanged under the action of external forces. A new aspect of mechanics to be considered here is that a rigid body under the action of a force has a tendency to rotate about some axis. Thus, in order that a body be at rest, one not only needs to ensure that the resultant force is zero, but one must now also ensure that the forces acting on a body do not tend to make it rotate. This issue is addressed in what follows. 2.3.1 Moments, Couples and Equivalent Forces When one swings a door on its hinges, it will move more easily if (i) one pushes hard, i.e. if the force is large, and (ii) if one pushes furthest from the hinges, near the edge of the door. It makes sense therefore to measure the rotational effect of a force on an object as follows: The tendency of a force to make a rigid body rotate is measured by the moment of that force about an axis. The moment of a force F about an axis through a point o is defined as the product of the magnitude of F times the perpendicular distance d from the line of action of F and the axis o. This is illustrated in Fig. 2.3.1. Figure 2.3.1: The moment of a force F about an axis o (the axis goes “into” the page) The moment o M of a force F can be written as Fd M = 0 (2.3.1) Not only must the axis be specified (by the subscript o ) when evaluating a moment, but the sense of that moment must be given; by convention, a tendency to rotate counterclockwise is taken to be a positive moment. Thus the moment in Fig. 2.3.1 is positive. The units of moment are the Newton metre (Nm) Note that when the line of action of a force goes through the axis, the moment is zero. d Rigid body o F line of action of force axis point of application of force
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Section 2.3 Solid Mechanics Part I Kelly 18 It should be emphasized that there is not actually a physical axis, such as a rod, at the point o of Fig. 2.3.1. In this discussion, it is imagined that an axis is there. Two forces of equal magnitude and acting along the same line of action have not only the same components y x F F , , but have equal moments about any axis. They are called equivalent forces since they have the same effect on a rigid body. This is illustrated in Fig. 2.3.2. Figure 2.3.2: Two equivalent forces Consider next the case of two forces of equal magnitude, parallel lines of action separated by distance d , and opposite sense. Any two such forces are said to form a couple . The only motion that a couple can impart is a rotation; unlike the forces of Fig. 2.3.2, the couple has no tendency to translate a rigid body. The moment of the couple of Fig. 2.3.3 about o is Fd Fd Fd M = = 1 2 o (2.3.2) Figure 2.3.3: A couple The sign convention which will be followed in most of what follows is that a couple is positive when it acts in a counterclockwise sense, as in Fig. 2.3.3.
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 1 taught by Professor Staff during the Fall '11 term at Auckland.

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Statics_of_Rigid_Bodies_03_Rigid_Bodies - Section 2.3 2.3...

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