Physics
Object in Equilibrium
An object in equilibrium is a state in which the net force and net torque acting on the object are both zero. This means that the object is either at rest or moving with a constant velocity, with no acceleration. In this state, the forces and torques acting on the object are balanced, resulting in a stable and unchanging condition.
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9 Key excerpts on "Object in Equilibrium"
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Key Ideas 327 ● A rigid body at rest is said to be in static equilibrium. For such a body, the vector sum of the external forces acting on it is zero: F → net = 0 (balance of forces). If all the forces lie in the xy plane, this vector equation is equivalent to two component equations: F net, x = 0 and F net, y = 0 (balance of forces). ● Static equilibrium also implies that the vector sum of the external torques acting on the body about any point is zero, or τ → net = 0 (balance of torques). If the forces lie in the xy plane, all torque vectors are par- allel to the z axis, and the balance-of-torques equation is equivalent to the single component equation net, z = 0 (balance of torques). ● The gravitational force acts individually on each ele- ment of a body. The net effect of all individual actions may be found by imagining an equivalent total gravita- tional force F → g acting at the center of gravity. If the gravi- tational acceleration g → is the same for all the elements of the body, the center of gravity is at the center of mass. 1. The linear momentum P → of its center of mass is constant. 2. Its angular momentum L → about its center of mass, or about any other point, is also constant. We say that such objects are in equilibrium. The two requirements for equilibrium are then P → = a constant and L → = a constant. (12-1) Our concern in this chapter is with situations in which the constants in Eq. 12-1 are zero; that is, we are concerned largely with objects that are not mov- ing in any way — either in translation or in rotation — in the reference frame from which we observe them. Such objects are in static equilibrium. Of the four objects mentioned near the beginning of this module, only one — the book resting on the table — is in static equilibrium. The balancing rock of Fig. 12-1 is another example of an object that, for the present at least, is in static equilibrium.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
344 Equilibrium and Elasticity 12.1 EQUILIBRIUM Learning Objectives After reading this module, you should be able to . . . 12.1.1 Distinguish between equilibrium and static equilibrium. 12.1.2 Specify the four conditions for static equilibrium. 12.1.3 Explain center of gravity and how it relates to center of mass. 12.1.4 For a given distribution of particles, calculate the coordinates of the center of gravity and the center of mass. Key Ideas ● A rigid body at rest is said to be in static equilibrium. For such a body, the vector sum of the external forces acting on it is zero: F → net = 0 (balance of forces). If all the forces lie in the xy plane, this vector equation is equivalent to two component equations: F net, x = 0 and F net, y = 0 (balance of forces). ● Static equilibrium also implies that the vector sum of the external torques acting on the body about any point is zero, or τ → net = 0 (balance of torques). If the forces lie in the xy plane, all torque vectors are parallel to the z axis, and the balance-of-torques equa- tion is equivalent to the single component equation net, z = 0 (balance of torques). ● The gravitational force acts individually on each ele- ment of a body. The net effect of all individual actions may be found by imagining an equivalent total gravi- tational force F → g acting at the center of gravity. If the gravitational acceleration g → is the same for all the ele- ments of the body, the center of gravity is at the center of mass. What Is Physics? Human constructions are supposed to be stable in spite of the forces that act on them. A building, for example, should be stable in spite of the gravitational force and wind forces on it, and a bridge should be stable in spite of the gravitational force pulling it downward and the repeated jolting it receives from cars and trucks. One focus of physics is on what allows an object to be stable in spite of any forces acting on it.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
A building, for example, should be stable in spite of the gravitational force and wind forces on it, and a bridge should be stable in spite of the gravitational force pulling it downward and the repeated jolting it receives from cars and trucks. One focus of physics is on what allows an object to be stable in spite of any forces acting on it. In this chapter we examine the two main aspects of stability: the equilibrium of the forces and torques acting on rigid objects and the elasticity of nonrigid objects, a property that governs how such objects can deform. When this physics is done correctly, it is the subject of countless articles in physics and engineering journals; when it is done incorrectly, it is the subject of countless articles in newspapers and legal journals. Equilibrium and Elasticity 326 CHAPTER 12 Equilibrium and Elasticity Equilibrium Consider these objects: (1) a book resting on a table, (2) a hockey puck sliding with constant velocity across a frictionless surface, (3) the rotating blades of a ceiling fan, and (4) the wheel of a bicycle that is traveling along a straight path at constant speed. For each of these four objects, 1. The linear momentum P → of its center of mass is constant. 2. Its angular momentum L → about its center of mass, or about any other point, is also constant. We say that such objects are in equilibrium. The two requirements for equilibrium are then P → = a constant and L → = a constant. (12.1.1) Our concern in this chapter is with situations in which the constants in Eq. 12.1.1 are zero; that is, we are concerned largely with objects that are not moving in any way—either in translation or in rotation—in the reference frame from which we observe them. Such objects are in static equilibrium. Of the four objects mentioned near the beginning of this module, only one—the book resting on the table—is in static equilibrium.
eBook - PDF- David Agmon, Paul Gluck;;;(Authors)
- 2009(Publication Date)
- WSPC(Publisher)
The type of equilibrium may be determined as follows. We displace the body from equilibrium by a small amount Ax —► 0, (or give it a tiny initial velocity) release it from rest in the displaced position, and consider all the forces acting on it. 1. If the resultant force in the displaced position is in the direction of the equilibrium position, we have stable equilibrium (one might say that the body 'tends' to return to its original position). 2. If the resultant force is directed away from the equilibrium position, equilibrium is unstable (the slightest disturbance drives the body away from equilibrium). 3. If the resultant force vanishes and the body stays in the position from which it was released, we have neutral equilibrium. Example 1 The three states of equilibrium. A small bead can slide on a smooth rigid wire set in a vertical plane. What kind of equilibrium pertains to each of the points A,B,C,D1 Solution Using the above criteria it is easy to see that A and C are stable, B is unstable and D is neutral equilibrium. 4.4 Conditions for equilibrium (a) Point-like objects. A particle in three-dimensional space has three degrees of freedom, one each along each of the three axes x,y 9 z. Therefore the necessary and sufficient condition for a particle to be in equilibrium is the vanishing of the vector sum of external forces acting on it, Z F = 0 (4-1) stable stable At the minima equilibrium is stable, at the maximum it is unstable, on the horizontal part it is neutral. Chapter 4 Statics 93 This equation is equivalent to three component equations 5X=°. !*,=«. 2X=° We show two examples for the vanishing vector sum of four forces acting on a particle in a plane. In this chapter, all the problems will be restricted to the xy plane.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Descriptions and Applications of Different Forces in Physics 1. Mechanical equilibrium A pendulum in a stable equilibrium (left) and unstable equilibrium (right) A standard definition of static equilibrium is: A system of particles is in static equilibrium when all the particles of the system are at rest and the total force on each particle is permanently zero. This is a strict definition, and often the term static equilibrium is used in a more relaxed manner interchangeably with mechanical equilibrium, as defined next. A standard definition of mechanical equilibrium for a particle is: The necessary and sufficient conditions for a particle to be in mechanical equilibrium is that the net force acting upon the particle is zero. The necessary conditions for mechanical equilibrium for a system of particles are: (i)The vector sum of all external forces is zero; (ii) The sum of the moments of all external forces about any line is zero. As applied to a rigid body, the necessary and sufficient conditions become: ________________________ WORLD TECHNOLOGIES ________________________ A rigid body is in mechanical equilibrium when the sum of all forces on all particles of the system is zero, and also the sum of all torques on all particles of the system is zero. A rigid body in mechanical equilibrium is undergoing neither linear nor rotational acceleration; however it could be translating or rotating at a constant velocity. However, this definition is of little use in continuum mechanics, for which the idea of a particle is foreign. In addition, this definition gives no information as to one of the most important and interesting aspects of equilibrium states – their stability.
- David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
CHAPTER 12 Equilibrium and elasticity 12.1 Equilibrium LEARNING OBJECTIVES After reading this module, you should be able to: 12.1.1 distinguish between equilibrium and static equilibrium 12.1.2 specify the four conditions for static equilibrium 12.1.3 explain centre of gravity and how it relates to centre of mass 12.1.4 for a given distribution of particles, calculate the coordinates of the centre of gravity and the centre of mass. KEY IDEAS • A rigid body at rest is said to be in static equilibrium. For such a body, the vector sum of the external forces acting on it is zero: F net = 0. If all the forces lie in the xy plane, this vector equation is equivalent to two component equations: F net,x = 0 and F net,y = 0. • Static equilibrium also implies that the vector sum of the external torques acting on the body about any point is zero: net = 0. If the forces lie in the xy plane, all torque vectors are parallel to the z axis, and the balance‐of‐torques equation is equivalent to the single component equation net,z = 0. • The gravitational force acts individually on each element of a body. The net effect of all individual actions may be found by imagining an equivalent total gravitational force F g acting at the centre of gravity. If the gravitational acceleration g is the same for all the elements of the body, the centre of gravity is at the centre of mass. Why study physics? Design considerations of elasticity and equilibrium make the world a safer place. In New Zealand, building codes calculate ‘equivalent static actions’ for a building’s foundations from static equilibrium, so buildings are designed to perform safely under specific earthquake conditions. Meanwhile, the Australian Airports Association uses a ‘layered elastic’ method when analysing flexible pavement design of runways.
eBook - PDF- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
Figure 9.2 This motionless person is in static equilibrium. The forces acting on him add up to zero. Both forces are vertical in this case. Figure 9.3 This car is in dynamic equilibrium because it is moving at constant velocity. There are horizontal and vertical forces, but the net external force in any direction is zero. The applied force F app between the tires and the road is balanced by air friction, and the weight of the car is supported by the normal forces, here shown to be equal for all four tires. 316 Chapter 9 | Statics and Torque This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 However, it is not sufficient for the net external force of a system to be zero for a system to be in equilibrium. Consider the two situations illustrated in Figure 9.4 and Figure 9.5 where forces are applied to an ice hockey stick lying flat on ice. The net external force is zero in both situations shown in the figure; but in one case, equilibrium is achieved, whereas in the other, it is not. In Figure 9.4, the ice hockey stick remains motionless. But in Figure 9.5, with the same forces applied in different places, the stick experiences accelerated rotation. Therefore, we know that the point at which a force is applied is another factor in determining whether or not equilibrium is achieved. This will be explored further in the next section. Figure 9.4 An ice hockey stick lying flat on ice with two equal and opposite horizontal forces applied to it. Friction is negligible, and the gravitational force is balanced by the support of the ice (a normal force). Thus, net F = 0 . Equilibrium is achieved, which is static equilibrium in this case. Figure 9.5 The same forces are applied at other points and the stick rotates—in fact, it experiences an accelerated rotation. Here net F = 0 but the system is not at equilibrium. Hence, the net F = 0 is a necessary—but not sufficient—condition for achieving equilibrium.
eBook - PDFIntroduction to Physics
Mechanics, Hydrodynamics Thermodynamics
- P. Frauenfelder, P. Huber(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
This equivalence is based on Newton's first law. We shall use the term particle to mean a body of finite, nonzero mass whose dimensions are extremely small compared with other lengths involved in a problem. A particle can be considered as a geometric point, and is also called a point mass. 23 24 I N T R O D U C T I O N T O P H Y S I C S The statics of rigid bodies deals with systems of particles. As a consequence, Newton's third law plays an especially important role in this subject. In contrast to a particle or a point mass whose motion can be one of translation only, an extended system of particles can also rotate about an axis. The statics of a rigid body deals with both translations and rotations, whereas the statics of a particle is concerned only with translations. This will be recognized from the definition of equilibrium, the basic concept of statics. DEFINITION. A body (or a system of particles) is (a) in translational equilibrium if it remains in a state of rest or of uniform rectilinear motion, and (b) is in rotational equilibrium if it remains in a state of rest or of uni-form rotation. Five concepts are necessary in treating the statics of rigid bodies: (1) the concept of force, (2) the concept of the moment of ¿ force with respect to an axis, (3) the concept of a rigid body, (4) the concept of equilibrium and (5) Newton's third law. 9-1 The Concept of Force Our muscular sensations give us an idea of the concept of force, but they are obviously not suited for a precise description or measurement of forces. A better basis for understanding this concept lies in the fact that a force deforms a solid body on which it acts. In certain cases there is a simple relation between a force and the deformation produced by it.
eBook - PDF- Bogdan Skalmierski(Author)
- 2013(Publication Date)
- Elsevier(Publisher)
CHAPTER 3 Statics 3.1 Equations of equilibrum We shall now examine the conditions which are satisfied by forces when a body remains at rest. If a body is at rest both the linear momentum p and the angular momentum K are simultaneously equal to zero. On this basis we can tell that its being at rest entails the disappearance of derivatives dp = 0 dt and dK dt = 0. By the principles of linear and angular momentum, the disappearance of the derivatives (1) and (2) brings about the disappearance of the sum of the forces and of the moments of external forces acting on the body. Thus in the considered case, the system of vector equations, called the equations of equilibrium, is satisfied. This system of equations is the basis for the solution of all statical problems: P i(1) = 0, ( 3 ) ri c Pi(1 ) = 0. (4) Equations (3) and (4) give the fundamental theorem on equilibrium of forces. A body will remain in equilibrium when such conditions of support are satified that ensure the standstill of the body. In this situation, the applied forces, together with the reactions, satisfy (1) (2) 96 STATICS Ch. 3 the conditions of equilibrium, i.e., satisfy equations (3) and (4). In other words: If a body remains in equilibrium, then the sum of external forces and the sum of moments of these forces about an arbitrary point in space are both simultaneously equal to zero. The converse is not true. Note should be taken of the linear character of equations (3) and (4), by virtue of which in solving them use can be made of superposition of solutions. 3.2 Couple Let us imagine two forces in two parallel straight lines l and m (Fig. 3.1). If the line 1 together with the force is translated until the two lines super-pose, then the lengths of the two vectors can be compared. When these lengths are identical but the vectors have opposite senses, we are dealing with what is called a couple.
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