Object in Equilibrium

3 Key excerpts on "Object in Equilibrium"

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  eBook - ePub

  Einstein's Theory of Relativity

  • Max Born(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  CHAPTER II THE FUNDAMENTAL LAWS OF CLASSICAL MECHANICS 1. Equilibrium and the Concept of Force Historically, mechanics took its start from the doctrine of equilibrium, or statics; the development from this point is also the most natural one logically. The fundamental concept of statics is force. It is derived from the subjective feeling of exertion experienced when we perform work with our bodies. The stronger of two men, we say, is the one who can lift the heavier stone or stretch the stiffer bow. This measure of force, with which Ulysses established his right among the suitors, and which indeed plays a great part in the stories of ancient heroes, already contains the germ of the objectivization of the subjective feeling of exertion. The next step was the choice of a unit of force and the measurement of all forces in terms of their ratios to this unit, that is, the relativization of the concept of force. Weight, being the most evident manifestation of force, and making all things tend downwards, offered the unit of force in a convenient form—a piece of metal which was chosen as the unit of weight through some decree of the state or of the Church. Nowadays it is an international congress that fixes the units. In technical matters, the unit of weight today is the weight of a definite piece of platinum which is maintained in Paris. This unit, called the pond (p) will be used in our discussion till otherwise stated. The instrument used to compare the weights of different bodies is the balance. Two bodies have the same weight, or are equally heavy, when on being placed in the two scales of the balance they do not disturb its equilibrium. If we place these two bodies in one pan of the balance, and in the other a body such that the equilibrium is again not disturbed, this new body has twice the weight of either of the other two

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  eBook - ePub

  Foundations of Mechanical Engineering

  • A. D. Johnson(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  Fig. 4.31 Types of contact force.

  4.8 Equilibrium Synopsis

  Equilibrium was introduced and explained in section 4.4. It is essential to understand the importance of equilibrium since it is the basis for all the calculations performed in this chapter. Indeed it is also at the heart of analysis of many other branches of science, such as thermodynamics, fluid dynamics, kinematics, stress, electronics, chemistry, etc.

  The general assumption made in this chapter is that if a system is in equilibrium it must be static. However, there are many situations where the system does not have to be stationary for equilibrium to apply, but the motion must be uniform

  Within the scope of this chapter the use of the principles of equilibrium allows the balancing of action forces and moments with reaction forces and moments. For this to be true, force systems have to be considered as static. If these systems were unbalanced they would move, that is, they would be ‘dynamic systems’ and other, appropriate theory would be required.

  Generally, the following ‘rules of equilibrium’ may be -applied to any force/moment system:

  • The vector sum of the forces must equal zero, i.e.
    ∑ F = 0
    where F represents vector forces.
  • The algebraic sum of all the moments must equal zero, i.e.
    ∑ M = 0
  • The algebraic sum of the vertical forces equals zero, i.e.
    ∑

    F v

    = 0
    where F v represents vertical forces.
  • The algebraic sum of the horizontal forces equals zero, i.e.
    ∑

    F H

    = 0
    where F H represents horizontal forces.

  4.9 Summary

  4.9.1 Polygon of forces

  When four or more forces, acting through a single point, are in equilibrium, the magnitudes and directions of these forces can be represented on a vector diagram which forms the sides of a polygon. All the forces must lie in one plane and must be considered in cyclic order.

  Note: this rule also covers, in a general way, the triangle of forces and the parallelogram of forces.

  For any vector diagram, for equilibrium, the vector diagram must close. For equilibrium the vector sum of the forces must equal zero, or

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  eBook - ePub

  Analysis of Structures

  An Introduction Including Numerical Methods

  • Joe G. Eisley, Antony M. Waas(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  2 Static Equilibrium 2.1 Introduction

  While all solid bodies deform to some extent under loads these deformations are often so small that the bodies may be considered to be rigid for certain purposes. A rigid body is an idealization that considers a body to have no deformation when subject to loads. When all forces, both known applied forces and unknown restraint forces, are identified and we attempt to sum the forces and moments we can have two possible situations:

  1. The number of unknown forces is less than the number of independent equations of motion. Rigid body motion may result. This is the subject of rigid body dynamics . We shall not study this case here.

  2. The number of unknown forces is equal to or greater than the number of independent equations of motion. There is no rigid body motion and we have static equilibrium . There are two subcases:

  a. The body is statically determinate when the number of unknown forces is equal to the number of independent equations of static equilibrium. All the unknown forces can be determined by the summation of forces and moments without regard to what the body is made of or to its deformation.

  b. The body is statically indeterminate when the number of unknown forces is greater than the number of independent equations of static equilibrium. The summation of forces and moments is not sufficient to find the unknown forces. We need to consider the deformation of the body and the physical properties that contribute to resisting deformation.

  In the case of a statically indeterminate body we must introduce additional equations to obtain a solution. In this chapter we shall be interested primarily in statically determinate bodies. The subsequent chapters will contend with both statically determinate and statically indeterminate problems.

  2.2 Free Body Diagrams

  We shall illustrate each subcase with a simple example in two dimensions using concentrated forces only. In these examples it is assumed that the applied forces initially are known and the restraint forces are unknown.

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