Equilibrium and Elasticity

We’ve devoted a good deal of effort to understanding why and how bodies 
accelerate in response to the forces that act on them. But very often we’re 
interested in making sure that bodies don’t accelerate. Any building, 
from a multistory skyscraper to the humblest shed, must be designed so that it 
won’t topple over. Similar concerns arise with a suspension bridge, a ladder leaning 
against a wall, or a crane hoisting a bucket full of concrete.


A body that can be modeled as a particle is in equilibrium whenever the vector 
sum of the forces acting on it is zero. But for the situations we’ve just described, 
that condition isn’t enough. If forces act at different points on an extended body, 
an additional requirement must be satisfied to ensure that the body has no 
tendency to rotate: The sum of the torques about any point must be zero. This 
requirement is based on the principles of rotational dynamics developed in 
Chapter 10. We can compute the torque due to the weight of a body by using the 
concept of center of gravity, which we introduce in this chapter.


Idealized rigid bodies don’t bend, stretch, or squash when forces act on 
them. But all real materials are elastic and do deform to some extent. Elastic 
properties of materials are tremendously important. You want the wings of an 
airplane to be able to bend a little, but you’d rather not have them break off. 
Tendons in your limbs need to stretch when you exercise, but they must return 
to their relaxed lengths when you stop. Many of the necessities of everyday 
life, from rubber bands to suspension bridges, depend on the elastic proper-
ties of materials. In this chapter we’ll introduce the concepts of stress, strain,
and elastic modulus and a simple principle called Hooke’s law, which helps 
us predict what deformations will occur when forces are applied to a real (not 
perfectly rigid) body.

 

 

Frequently Asked Questions

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Ans: Some physical quantities, such as time, temperature, mass, and density, can be described completely by a single number with a unit. But many other important quantities in physics have a direction associated with them and cannot be described by a single number. view more..
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Ans: We have stressed the importance of knowing the accuracy of numbers that represent physical quantities. But even a very crude estimate of a quantity often gives us useful information. Sometimes we know how to calculate a certain quantity, but we have to guess at the data we need for the calculation. Or the calculation might be too complicated to carry out exactly, so we make rough approximations. view more..
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Ans: Measurements always have uncertainties. If you measure the thickness of the cover of a hardbound version of this book using an ordinary ruler, your measurement is reliable to only the nearest millimeter, and your result will be 3 mm. It would be wrong to state this result as 3.00 mm; given the limitations of the measuring device, you can’t tell whether the actual thickness is 3.00 mm, 2.85 mm, or 3.11 mm. view more..
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Ans: A body that can be modeled as a particle is in equilibrium whenever the vector sum of the forces acting on it is zero. But for the situations we’ve just described, that condition isn’t enough. If forces act at different points on an extended body, an additional requirement must be satisfied to ensure that the body has no tendency to rotate: The sum of the torques about any point must be zero. This requirement is based on the principles of rotational dynamics view more..
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Ans: In this chapter we’ll apply the first and second conditions for equilibrium to situations in which a rigid body is at rest (no translation or rotation). Such a body is said to be in static equilibrium view more..
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Ans: In most equilibrium problems, one of the forces acting on the body is its weight. We need to be able to calculate the torque of this force. The weight doesn’t act at a single point; it is distributed over the entire body. But we can always calculate the torque due to the body’s weight by assuming that the entire force of gravity (weight) is concentrated at a point called the center of gravity view more..
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Ans: We can often use symmetry considerations to locate the center of gravity of a body, just as we did for the center of mass. The center of gravity of a homoge-neous sphere, cube, or rectangular plate is at its geometric center. The center of gravity of a right circular cylinder or cone is on its axis of symmetry. view more..
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Ans: There are just two key conditions for rigid-body equilibrium: The vector sum of the forces on the body must be zero, and the sum of the torques about any point must be zero. To keep things simple, we’ll restrict our attention to situations in which we can treat all forces as acting in a single plane, which we’ll call the xy-plane view more..
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Ans: Here are some solved examples to help your concepts to be more clear. view more..
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Ans: The rigid body is a useful idealized model, but the stretching, squeezing, and twisting of real bodies when forces are applied are often too important to ignore. view more..
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Ans: The simplest elastic behavior to understand is the stretching of a bar, rod, or wire when its ends are pulled (Fig. 11.12a). Figure 11.14 shows an object that initially has uniform cross-sectional area A and length l0. We then apply forces of equal magnitude F# but opposite directions at the ends (this ensures that the object has no tendency to move left or right). We say that the object is in tension. view more..
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Ans: When a scuba diver plunges deep into the ocean, the water exerts nearly uniform pressure everywhere on his surface and squeezes him to a slightly smaller volume. This is a different situation from the tensile and compressive stresses and strains we have discussed. view more..
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Ans: The third kind of stress-strain situation is called shear. The ribbon in Fig. 11.12c is under shear stress: One part of the ribbon is being pushed up while an adjacent part is being pushed down, producing a deformation of the ribbon. view more..
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Ans: Hooke’s law—the proportionality of stress and strain in elastic deformations— has a limited range of validity. In the preceding section we used phrases such as “if the forces are small enough that Hooke’s law is obeyed.” Just what are the limitations of Hooke’s law? What’s more, if you pull, squeeze, or twist anything hard enough, it will bend or break view more..
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Ans: summary of equilibrium and elasticity view more..
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Ans: Fluids play a vital role in many aspects of everyday life. We drink them, breathe them, swim in them. They circulate through our bodies and control our weather. The physics of fluids is therefore crucial to our understanding of both nature and technology view more..
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Ans: A fluid is any substance that can flow and change the shape of the volume that it occupies. (By contrast, a solid tends to maintain its shape.) We use the term “fluid” for both gases and liquids. The key difference between them is that a liquid has cohesion, while a gas does not. The molecules in a liquid are close to one another, so they can exert attractive forces on each other and thus tend to stay together (that is, to cohere). That’s why a quantity of liquid maintains the same volume as it flows: If you pour 500 mL of water into a pan, the water will still occupy a volume of 500 mL. The molecules of a gas, by contrast, are separated on average by distances far larger than the size of a molecule. Hence the forces between molecules are weak, there is little or no cohesion, and a gas can easily change in volume. If you open the valve on a tank of compressed oxygen that has a volume of 500 mL, the oxygen will expand to a far greater volume. view more..
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Ans: A fluid exerts a force perpendicular to any surface in contact with it, such as a container wall or a body immersed in the fluid. This is the force that you feel pressing on your legs when you dangle them in a swimming pool. Even when a fluid as a whole is at rest, the molecules that make up the fluid are in motion; the force exerted by the fluid is due to molecules colliding with their surroundings view more..
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