Page 177 - Schaum's Outline of Theory and Problems of Applied Physics

162 ELASTICITY [CHAP. 13

BULK MODULUS
When compressive forces act over the entire surface of a body, its volume decreases. If the compressive force
per unit area F/A is uniform, the bulk modulus is given by

F/A
B =−
V/V 0
volume stress
Bulk modulus =−
volume strain
The minus sign is included because an increase in the volume stress leads to a decrease in the volume.
Volume stresses occur when objects are immersed in liquids, since a liquid exerts a uniform force perpen-
dicular to any surface in its interior. As will be discussed in Chapter 16, the stress F/A exerted by a liquid is
called pressure p, so we can also write

p
B =−
V/V 0

SOLVED PROBLEM 13.10
The pressure at a depth of 300 m in the ocean exceeds sea-level atmospheric pressure by 3.03 kPa. By
3
how much does the volume of a 0.2-m aluminum object contract when lowered to this depth in the
ocean? The bulk modulus of aluminum is 7 × 10 10 Pa.

3
3
pV 0 (3.03 × 10 Pa)(0.2m ) −9 3 3
V =− =− =−8.7 × 10 m =−8.7mm
10
B 7 × 10 Pa
The minus sign means that the volume of the object decreases.

SOLVED PROBLEM 13.11
The reciprocal of the bulk modulus B of a liquid is called its compressibility k, so k = 1/B. The bulk
9
modulus of water is 2.3×10 Pa. (a) Find its compressibility per atmosphere of pressure, where 1 atm =
5
1.013 × 10 Pa is the pressure exerted by the earth’s atmosphere at sea level. (b) How much pressure in
atmospheres is needed to compress a sample of water by 0.1 percent?
(a) In terms of atmospheres, the bulk modulus of water is

9
2.3 × 10 Pa 4
B = = 2.27 × 10 atm
5
1.013 × 10 Pa/atm
and so its compressibility is

1 1 −5 −1
k = = = 4.4 × 10 atm
4
B 2.27 × 10 atm
(b) Here V/V 0 =−0.1 percent =−0.001. Hence the required pressure is

1 V 0.001
p =− = −5 −1 = 23 atm
k V 0 4.4 × 10 atm