Equilibrium of a Particle

4 Key excerpts on "Equilibrium of a Particle"

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  eBook - PDF

  Statics and Mechanics of Materials

  An Integrated Approach

  • William F. Riley, Leroy D. Sturges, Don H. Morris(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  EQUILIBRIUM: CONCURRENT FORCE SYSTEMS 3-1 INTRODUCTION Statics was defined in Chapter 1 as the branch of rigid-body mechanics concerned with bodies that are acted upon by a balanced system of forces (the resultant of all forces acting on the body is zero) and hence are at rest or moving with a con- stant velocity in a straight line. A body with negligible dimensions is commonly referred to as a particle. In mechanics, either large bodies or small bodies can be referred to as particles when the size and shape of the body have no effect on the response of the body to a system of forces. Under these conditions, the mass of the body can be as- sumed to be concentrated at a point. For example, the earth can be modeled as a particle for orbital motion studies because the size of the earth is insignificant when compared with the size of its orbit and the shape of the earth does not in- fluence the description of its position or the action of forces applied to it. Since it is assumed that the mass of a particle is concentrated at a point and that the size and shape of a particle can be neglected, a particle can be subjected only to a system of concurrent forces. Newton’s first law of motion states that “in the absence of external forces (R  0), a particle originally at rest or moving with a constant velocity will remain at rest or continue to move with a constant velocity along a straight line.” Thus, a necessary condition for Equilibrium of a Particle is R  F  0 (3-1) A particle in equilibrium must also satisfy Newton’s second law of motion, which can be expressed in equation form [Eq. (1-1)] as R  F  ma (1-1) In order to satisfy both Eqs. (1-1) and (3-1), ma  0 Since the mass of the particle is not zero, the acceleration of a particle in equi- librium is zero (a  0). Thus, a particle initially at rest will remain at rest and a particle moving with a constant velocity will maintain that velocity.

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  Thermodynamics in Materials Science

  • Robert DeHoff(Author)
  • 2006(Publication Date)
  • CRC Press

    (Publisher)

  These equations are the basis for making practical calculations about the final resting condition of complex systems. The chapter ends with a collection of alternate statements of the criterion for equilibrium, each of which is useful in the description of systems with specific external constraints. 5.1 INTUITIVE NOTIONS OF EQUILIBRIUM The idea that a system changes toward some final ultimate condition and, once there, remains in that condition unless acted upon by some external agent is a familiar one. Such a system is said to come to equilibrium or approach its equilibrium state. The specific nature of that state depends upon the chemical and energy content of the system and that of its surroundings. A system comes to equilibrium in given surroundings. It may only be disturbed from its equilibrium state by changing the state of its surroundings. A simple application of this idea is illustrated in Figure 5.1. The system acted upon by a set of mechanical forces shown comes to equilibrium when it arrays itself with respect to the forces so that they are in balance. The mathematical statement that describes this condition is particularly simple for this mechanical system. W 2 P W 1 W 3 F Æ 3 F Æ 1 F Æ 2 FIGURE 5.1 A simple mechanical system in equilibrium. The vector sum of the forces acting at P is zero. Thermodynamics in Materials Science 108 The vector sum of the forces acting on the point P is zero: X i F i ¼ 0 ð 5 : 1 Þ which implies: X i F ix ¼ 0 ; X i F iy ¼ 0 ; X i F iz ¼ 0 ð 5 : 2 Þ where the subscripts x , y , and z represent components of the force vectors resolved in some orthogonal coordinate system, and the summation over the index i is taken over all of the forces acting on the body. Once the system has found the position in space O which achieves this balance, it will remain fixed in that position until one of the weights is changed.

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  eBook - ePub

  Engineering Statics

  • M. Rashad Islam, M. Abdullah Al Faruque, Bahar Zoghi, Sylvester A. Kalevela(Authors)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  3 Equilibrium of Particle and Rigid Body

  3.1

  CONCEPT OF EQUILIBRIUM

  From Newton’s third law of motion, we know that every action has its own and opposite reaction. That means, if you apply a force to a wall it will react with an equal amount of force on you, if the wall is not moving (rigid). Therefore, every action is equal to its reaction in any direction. This is called the equilibrium condition. If we sum up this action and reaction in a direction, it will be zero as these two are numerically equal but the signs are opposite. Thus, we can write the following three equations for two-dimensional conditions:

  ∑

  F x

  = 0  

  (

  the   summation   of   forces   along   the   x -direction   (horizontal)   is   zero

  )

  ∑

  F y

  = 0  

  (

  the   summation   of   forces   along   the   y -direction   (vertical)   is   zero

  )

  ∑ M = 0  

  (

  the   summation   of   moment   at   any   point   about   any   axis   perpendicular   to   the   plane   is   zero

  )

  Figure 3.1a shows a body is experiencing forces from different sides. The summation of forces in any direction must be zero to be in equilibrium. Figure 3.1b shows an inclined structure which is in equilibrium.

  3.2

  PARTICLE VERSUS RIGID BODY

  A particle is a body of infinitely small volume and is considered to be concentrated at a point. Every point in a particle can be considered to have the same characteristics.

  A rigid body is one which does not deform under the action of the loads or the external forces. In the case of a rigid body, the distance between any two points of the body remains constant, when this body is subjected to loads. Examples of a particle and a rigid body are shown in Figure 3.2 .

  3.3

  IDEALIZATION OF STRUCTURES

  Real structures are complicated and have different features in terms of loading application, loading distribution, support restrictions, etc. These conditions are simplified for analyzing the structures using paper and pencil or a computer. A simplified version of the structures is known as the idealized structure.

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  Dynamics of Mechanical Systems

  • Harold Josephs, Ronald Huston(Authors)
  • 2002(Publication Date)
  • CRC Press

    (Publisher)

  Finally, the expression equilibrium position for a mechanical system means a constant or steady-state solution of the governing equations of motion of the system. To illustrate these concepts, consider again the simple pendulum of Figure 14.2.1, here consisting of the concentrated mass (or bob) P connected to a pin supported by a light (massless) rod, as opposed to a string. Recall from Eq. (8.4.4) that the governing differential equation for this system is: (14.2.1) where as before θ describes the orientation of the pendulum, is its length, and g is the gravity constant. ˙˙ / sin θ θ + ( ) = g l 0 480 Dynamics of Mechanical Systems Observe that there are two equilibrium, or “constant,” solutions to Eq. (14.2.1). That is, (14.2.2) These solutions represent static equilibrium positions shown graphically in Figure 14.2.2. Intuitively, we would expect the solution θ = 0 to be stable and the solution θ = π to be unstable. That is, if we displace (or “perturb”) the pendulum a small amount away from θ = 0 and then release it, we would expect the pendulum to simply oscillate about θ = 0. Alternatively, if we perturb the pendulum by a small amount away from θ = π and then release it, we would expect the pendulum to fall increasingly further away from the equilibrium position. We can show this mathematically through an analysis of the governing equation (Eq. (14.2.1)). Let us designate “small” quantities by a superscript star or asterisk ( * ). Specifi-cally, a small displacement of the pendulum will be represented by the angle θ * . Hence, a perturbation or disturbance away from the equilibrium position θ = 0 may be expressed as: (14.2.3) By substituting from Eq. (14.2.3) into Eq. (14.2.1), we obtain: (14.2.4) Because θ * is small, we can approximate and represent sin θ * by θ * leading to the equation: (14.2.5) The solution of Eq. (14.2.5) may be expressed in the form: (14.2.6) FIGURE 14.2.1 The simple pendulum.

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