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14.4 Elasticity of Materials 449

What pressure must you exert on a sample of water if you want
EXAMPLE 9
to compress its volume by 0.10%?
SOLUTION: For volume compression, the relevant elastic modulus is the bulk
modulus B. By Eq. (14.20), the pressure, or the force per unit area, is
F ¢V
B
A V
For 0.10% compression, we want to achieve a fractional change of volume of
10 2
V V 0.0010. Since the bulk modulus of water is 0.22 10 N/m , the
required pressure is
F 10 2 6 2
0.22 10
/m 0.0010 2.2 10
/m
A

The simple uniform deformations of elongation, shear, and compression described
above require a rather special arrangement of forces. In general, the forces applied to
a solid body will produce nonuniform elongation, shear, and compression. For instance,
a beam supported at its ends and sagging in the middle because of its own weight or
the weight of a load placed on it will elongate along its lower edge, and compress along
its upper edge.
Finally, note that the formulas (14.18)–(14.20) are valid only as long as the defor-
mation is reasonably small—a fraction of a percent or so. If the deformation is exces-
sive, the material will be deformed beyond its elastic limit; that is, the material will
suffer a permanent deformation and will not return to its original size and shape when
the force ceases. If the deformation is even larger, the material will break apart or crum-
ble. For instance, steel will break apart (see Fig. 14.29) if the tensile stress exceeds
8 2 8 2
5 10 N/m , or if the shearing stress exceeds 2.5 10 N/m , and it will crumble if FIGURE 14.29 These rods of steel broke
8 2
the compressive stress exceeds 5 10 N/m . apart when a large tension was applied.

✔ Checkup 14.4

QUESTION 1: When a tension of 70 N is applied to a piano wire of length 1.8 m, it
stretches by 2.0 mm. If the same tension is applied to a similar piano wire of length
3.6 m, by how much will it stretch?
QUESTION 2: Is it conceivable that a long cable hanging vertically might snap under
its own weight? If so, does the critical length of the cable depend on its diameter?
QUESTION 3: The bulk modulus of copper is about twice that of aluminum. Suppose
that a copper and an aluminum sphere have exactly equal volumes at normal atmo-
spheric pressure. Suppose that when subjected to a high pressure, the volume of the alu-
minum sphere shrinks by 0.01%. By what percentage will the copper sphere shrink at
the same pressure?
QUESTION 4: While lifting a load, the steel cable of a crane stretches by 1 cm. If you
want the cable to stretch by only 0.5 cm, by what factor must you increase its diameter?

(A) 22 (B) 2 (C) 222 (D) 4