Conditions for equilibrium
We learned in Sections 4.2 and 5.1 that a particle is in equilibrium—that is, the
particle does not accelerate—in an inertial frame of reference if the vector sum
of all the forces acting on the particle is zero, ΣFS= 0. For an extended body,
the equivalent statement is that the center of mass of the body has zero acceleration
if the vector sum of all external forces acting on the body is zero, as discussed in
Section 8.5. This is often called the first condition for equilibrium:
A second condition for an extended body to be in equilibrium is that the body
must have no tendency to rotate. A rigid body that, in an inertial frame, is not
rotating about a certain point has zero angular momentum about that point. If it
is not to start rotating about that point, the rate of change of angular momentum
must also be zero. From the discussion in Section 10.5, particularly Eq. (10.29),
this means that the sum of torques due to all the external forces acting on the
body must be zero. A rigid body in equilibrium can’t have any tendency to start
rotating about any point, so the sum of external torques must be zero about any
point. This is the second condition for equilibrium:
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 (Fig. 11.1).
But the same conditions apply to a rigid body in uniform translational motion (without rotation),
such as an airplane in flight with constant speed, direction, and altitude. Such a body is in
equilibrium but is not static.
Frequently Asked Questions
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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: 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: 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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Ans: If the weight of the fluid can be ignored, the pressure in a fluid is the same throughout its volume. We used that approximation in our discussion of bulk stress and strain in Section 11.4. But often the fluid’s weight is not negligible, and pressure variations are important. Atmospheric pressure is less at high altitude than at sea level, which is why airliner cabins have to be pressurized. When you dive into deep water, you can feel the increased pressure on your ears. view more..
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