Page 240 - Schaum's Outline of Theory and Problems of Applied Physics

CHAP. 19] EXPANSION OF SOLIDS, LIQUIDS, AND GASES 225

SOLVED PROBLEM 19.2
A wooden wagon wheel has an outside diameter of 3750 mm. The iron tire for this wheel is deliberately
made smaller so that it can be shrunk in place to be a tight fit. If the tire’s inside diameter is 3737 mm
at 20 C, find the temperature to which it must be heated to fit over the wheel. The coefficient of linear
◦
expansion of steel is 1.2 × 10 / C.
−5 ◦
Here L 0 = 3737 mm and L = 13 mm. Hence
L 13 mm
◦
T = = = 290 C
−5 ◦
aL 0 (1.2 × 10 / C)(3737 mm)
The required temperature is therefore 20 C + 290 C = 310 C.
◦
◦
◦
SOLVED PROBLEM 19.3
Find the force associated with the expansion of a steel beam whose cross-sectional area is 200 cm 2
2
11
when its temperature increases from 5 to 30 C. Young’s modulus for steel is Y = 2.0 × 10 N/m , and
◦
a = 1.2 × 10 / C.
−5 ◦
The change in the beam’s length L due to thermal expansion is given by
L = aL 0 T
To obtain the same change in length by mechanical means would involve applying a tension force F such that
YA L
F =
L 0
according to Chapter 13. Thus the amount of force associated with the expansion is
YA L YA(aL 0 T )
F = = = YaA T
L 0 L 0
The force depends on the cross section of the beam but not on its length. For the beam under consideration here
11
2
6
2
◦
F = YaA T = (2.0 × 10 N/m )(1.2 × 10 / C)(0.02 m )(25 C) = 1.2 × 10 N = 1.2MN
−5 ◦
which is equivalent to nearly 135 tons.
SOLVED PROBLEM 19.4
3
How much water overflows when a Pyrex vessel filled to the brim with 1 L (1000 cm ) of water at
◦
◦
−6 ◦
20 C is heated to 90 C? The coefficients of volume expansion of Pyrex and water are 9 × 10 / C and
−4 ◦
2.1 × 10 / C, respectively.
A cavity in an object expands or contracts by the same amount as a solid body with the composition of the
object and the same volume as the cavity. Hence the volume of water that overflows is
V w − V p = b w V 0 T − b P V 0 T = (b w − b P )v 0 T
3
◦
= (210 × 10 / C − 9 × 10 / C)(1000 cm )(70 C)
−6 ◦
−6 ◦
= 14.1cm 3
SOLVED PROBLEM 19.5
3
Lead has a coefficient of linear expansion of 3 × 10 / C, and its density at 20 C is 11.0 g/cm . Find the
−5 ◦
◦
density of lead at 200 C.
◦