Forces and equilibrium in a finite body

Much of this book is concerned with the response of solids and liquids in bulk to applied forces and is, therefore, concerned with the application of Newton s laws of motion to such bodies. Newton s laws of motion as usually stated (see, for example, Qassical Mechanics by B.P. Cowan in this series) describe the effect of forces on particles, i.e. on entities so small that their size plays no role in the description of their behaviour. Matter in bulk can be treated as an assembly of such particles and, as a first step in the study of the response of bulk matter to applied forces, some of the implications of Newton s laws for a finite body will be examined. 

As a specific example, consider a right circular cylinder of area of cross-section a and length , initially at rest, that is acted upon by a force applied in the direction of its axis, as indicated in Fig. 2.1. If p is the density of the material of the cylinder (so that its mass is paL) and u is its velocity (which here is in the same direction as Fa), the equation of motion of the body as a whole is 

Newton s laws can be applied to any part of the body. In particular, consider the behaviour of that part, shown shaded in Fig. 2.1, of length /. Since the acceleration of this part of the body is the same as that of the body as a whole, the force F acting 

According to Newton s third law of motion, that part of the body corresponding to the shaded region in Fig. 2.1 exerts a force —on the remainder of the body. Therefore, the equation of motion of that part of the body is 

This analysis shows that the force acting on a finite body must be considered as being transmitted through that body, and that to specify the value of that force the part of the body under consideration must be specified. The force F is the response of the body to the applied force F and arises from the interactions of the molecules making up the material. These forces, which resist the applied, or external, forces are called internal forces. 

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