Physics
Equilibrium
Equilibrium in physics refers to a state where the net force and net torque acting on an object are both zero, resulting in no acceleration or rotation. It can occur in various forms, such as static equilibrium where an object is at rest, or dynamic equilibrium where it moves at a constant velocity.
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12 Key excerpts on "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)
- 2020(Publication Date)
- Wiley(Publisher)
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 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. 280 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.
- S Eskinazi(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
4 Static Equilibrium of the Environment 4.1 I N T R O D U C T I O N In the considerations of the atmosphere and the ocean, sooner or later we must look into the conditions that enable these masses to be in mechanical and thermal Equilibrium. The word static is derived from the Greek word statikos, meaning causing to stand still. The Latin word Equilibrium implies a state of balance among the forces acting on a system. Thermodynamic Equilibrium implies mechanical, thermal, and chemical equilibria. On the basis of Newtonian mechanics, the sum of all external forces and moments on a given mass system must balance out to zero for mechanical equi-librium. In case this fails to be true, an accelerating or decelerating motion will develop in which the net resulting force will be the product of the mass of the system and the acceleration, and the direction of the force will be in the direction of acceleration, according to Newton's law of inertia [Eq. (2.2)]. This is Newton's second law. The third law deals with the balancing of internal forces, and the first law states that when the resultant external force is zero, the system is either at rest or moves at a constant velocity (constant direction and magnitude). Since external and internal forces play an important role in the state of the environment, it is important that we describe them here. External forces are imposed on a mass system from its environment. These forces can be imposed by contact with the surface of the system, or from afar, as with gravitational force (see Fig. 4.1). Frictional and pressure forces act through contact on the surface, while body or volumic forces such as gravity and Coriolis act on the whole of the volume. 83 84 4 Static Equilibrium of the Environment g Fig. 4.1 Fluid parcel under surface and body forces. 4.2 B O D Y A N D S U R F A C E F O R C E S Body forces act on the extent of the mass. The gravitational force is a body force, and according to Eq.
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.
- 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 - PDFEngineering Mechanics
Statics
- James L. Meriam, L. G. Kraige, J. N. Bolton(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Bialobrzeski/laif/Redux Pictures In many applications of mechanics, the sum of the forces acting on a body is zero or near zero, and a state of Equilibrium is assumed to exist. This apparatus is designed to hold a car body in Equilibrium for a considerable range of orienta- tions during vehicle production. Even though there is motion, it is slow and steady with minimal acceleration, so that the assumption of Equilibrium is justified during the design of the mechanism. CHAPTER 3 Equilibrium CHAPTER OUTLINE 3/1 Introduction SECTION A Equilibrium in Two Dimensions 3/2 System Isolation and the Free-Body Diagram 3/3 Equilibrium Conditions SECTION B Equilibrium in Three Dimensions 3/4 Equilibrium Conditions 3/5 Chapter Review 3/1 Introduction Statics deals primarily with the description of the force conditions necessary and sufficient to maintain the Equilibrium of engineering structures. This chapter on Equilibrium, therefore, constitutes the most important part of statics, and the proce- dures developed here form the basis for solving problems in both statics and dynam- ics. We will make continual use of the concepts developed in Chapter 2 involving forces, moments, couples, and resultants as we apply the principles of Equilibrium. When a body is in Equilibrium, the resultant of all forces acting on it is zero. Thus, the resultant force R and the resultant couple M are both zero, and we have the Equilibrium equations R = ΣF = 0 M = ΣM = 0 (3/1) These requirements are both necessary and sufficient conditions for Equilibrium. All physical bodies are three-dimensional, but we can treat many of them as two-dimensional when the forces to which they are subjected act in a single plane or can be projected onto a single plane. When this simplification is not possible, the problem must be treated as three-dimensional.
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.
eBook - ePub- Robert DeHoff(Author)
- 2006(Publication Date)
- CRC Press(Publisher)
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.The vector sum of the forces acting on the point P is zero:FIGURE 5.1 A simple mechanical system in Equilibrium. The vector sum of the forces acting at P is zero.which implies:(5.1)= 0∑ iF i(5.2)= 0 ;∑ iFi x= 0 ;∑ iFi y= 0∑ iFi zwhere 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. Furthermore, if the system is displaced from its Equilibrium position, e.g., if the body is moved to any position in the neighborhood of O, it will return to O.In thermodynamics, the influences that operate to modify the condition of a system are more general than mechanical forces. Nonetheless, the intuitive notion of an Equilibrium condition also applies to such systems. This idea of an Equilibrium state has two components:- It is a state of rest.
- It is a state of balance.
The first component means that the condition of the system, no matter what it might be, is time independent. The system has achieved a stationary state. No changes can occur in a system that has come to Equilibrium except by the action of influences that originate outside the system/surroundings complex. The second component to the concept means that if the system is perturbed from its Equilibrium condition by some outside influence, it will return again to the same condition when it again comes to rest.
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.
eBook - PDF- J D Renton(Author)
- 2002(Publication Date)
- Woodhead Publishing(Publisher)
Static Equilibrium is achieved if no work is done by the set of forces F, under any rigid motion of the body. In (2.6), W, must be zero for any arbitrary small deflexions u and 8 This is 2.4 Statics [Ch. 2 then an example ofa virttral work equation. It is true if both of the expressions in (2.7) are zero. More generally, if a set of point moments M, ( j = 1 to m ) is also applied to the body, the conditions for static Equilibrium become In terms of components of the above vectors, there are six equations of Equilibrium for a single rigid body. These are given by (2.9a) n n m If another origin 0’, at a position relative to 0 given by r,, is chosen and the positions of the forces F, with respect to 0’ are given by r; ’, then a new set of equations is given by replacing ri with r,‘ in (2.9). This of course only affects the second part of these equations. This now takes the form (It may help to consider P, at a position r, relative to 0 in Figure 2.4 to be the point 0‘ at a position ro.) However, the sum of the forces F, is already known to be zero and thus no independent equations arise by choosing a new position for 0. Suppose that all the forces are parallel. For example, the body might consist of a set of point masses m, under the influence of the acceleration due to gravity, g. In vector terms, this means that each force F, is a scalar multiple F, of a unit vector, U say. Equations (2.7) then become 2 F~ = (2 F ~ > U = F , 2 (ri X F ~ ) = (2 ~ ~ r ~ ) x u = r x F n n (2.10) t = I 1 . 1 1.1 1 . 1 = r X ( C F , ) U = ( C I ; ; ) r x U 1 = 1 : = I It is possible to find a point through which the resultant Facts regardless of the direction of the forces. The position of this point is given by the vector r , where r = ( e F l r i ) / $ F l for the second of the conditions in (2.10) is satisfied, regardless of the value of U. Suppose that the above collection of masses has a total mass m. Then : = I 1.1 n (2.11) and the point given by the position vector r is known as the centre of mass. For problems confined to two dimensions, only three of equations
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