ELASTIC PROPERTIES OF SOLIDS
We shall discuss the deformation of solids in terms of the concepts of stress and strain.
Stress is the external force acting on an object per unit cross-sectional area.
Strain is proportional to stress; the constant of proportionality depends on the material
being deformed and on the nature of the deformation. Strain is a measure of the
degree of deformation
strain is proportional to stress
Prestressed Concrete
The tension in the cable was 940 N. What diameter should a 10-m-long steel wire have
if we do not want it to stretch more than 0.5 cm under these conditions?
Solution: From the definition of Young’s modulus, we can solve for the required crosssectional area. Assuming that the cross section is circular, we can determine the diameter
of the wire. From Equation of Young Module, we have
The radius of the wire can be found from
To provide a large margin of safety, we would probably use a flexible cable made up of
many smaller wires having a total cross-sectional area substantially greater than our
calculated value.
Squeezing a Brass Sphere
A solid brass sphere is initially surrounded by air, and the air
pressure exerted on it is 1.0 105 N/m2 (normal atmospheric
pressure). The sphere is lowered into the ocean to a
depth at which the pressure is 2.0 107 N/m2. The volume
of the sphere in air is 0.50 m3. By how much does this volume
change once the sphere is submerged?
Solution From the definition of bulk modulus, we have
Because the final pressure is so much greater than the initial pressure, we can neglect the
initial pressure and state that
The negative sign indicates a decrease in volume.
TEMPERATURE AND THE ZEROTH LAW
OF THERMODYNAMICS
Heat is the transfer of energy from one object to another object as a result of a difference in
temperature between the two
Thermal equilibrium is a situation in which two objects in thermal contact with each other
cease to exchange energy by the process of heat.
Thermal equilibrium is a situation in which two objects in thermal contact with each
other cease to exchange energy by the process of heat.
If objects A and B are separately in thermal equilibrium with a third object C,
then objects A and B are in thermal equilibrium with each other.
Celsius temperature scale, this mixture is defined to have a temperature
of zero degrees Celsius, which is written as 0°C; this temperature is called
the ice point of water. Another commonly used system is a mixture of water and
steam in thermal equilibrium at atmospheric pressure; its temperature is 100°C,
which is the steam point of water.
THE CONSTANT-VOLUME GAS THERMOMETER AND
THE ABSOLUTE TEMPERATURE SCALE
Fahrenheit scale. This scale sets the temperature of the ice point at 32°F and the
temperature of the steam point at 212°F. The relationship between the Celsius and
Fahrenheit temperature scales is
THERMAL EXPANSION OF SOLIDS AND LIQUIDS
At ordinary temperatures, the atoms in a solid oscillate about their
equilibrium positions with an amplitude of approximately 10-11 [m] and
a frequency of approximately 1013 [Hz].
The average spacing between the atoms is about 10-10 [m].
As the temperature of the solid increases, the atoms oscillate with
greater amplitudes; as a result, the average separation between them
increases.
The average coefficient of linear expansion:
Expansion of a Railroad Track
A steel railroad track has a length of 30.000 m when the temperature is 0.0°C. (a)
What is its length when the temperature is 40.0°C?
Solution Making use of Table 19.2 and noting that the change in temperature is
40.0°C, we find that the increase in length is
If the track is 30.000 m long at 0.0°C, its length at 40.0°C is:
30.013 m.
(b) Suppose that the ends of the rail are rigidly clamped at 0.0°C so that expansion is
prevented. What is the thermal stress set up in the rail if its temperature is raised to
40.0°C?
From the definition of Young’s modulus for a solid, we have
Because Y for steel is 20 1010 N/m2 (see Table 12.1), we have
If the rail has a cross-sectional area of 30.0 cm2, what is the force of compression in the rail?
One mole of any substance is that amount of the substance that contains Avogadro’s number
The number of moles n of a substance is related to its mass m through the expression
where M is the molar mass of the substance (see Section 1.3), which is usually expressed
in units of grams per mole (g/mol). For example, the molar mass of oxygen (O2) is 32.0
g/mol. Therefore, the mass of one mole of oxygen is 32.0 g.
One mole (mol) of a substance is that amount of the substance that contains as many
particles (atoms, molecules, or other particles) as there are atoms in 12 g of the carbon-12
isotope. One mole of substance A contains the same number of particles as there are in 1 mol
of any other substance B. For example, 1 mol of aluminum contains the same number of
atoms as 1 mol of lead.
Size of each atom:
A solid cube of aluminum (density 2.7 g/cm3) has a volume of 0.20 cm3. How many
aluminum atoms are contained in the cube?
Solution Since density equals mass per unit volume, the mass m of the cube is
To find the number of atoms N in this mass of aluminum, we can set up a proportion
using the fact that one mole of aluminum (27 g) contains 6.02x1023 atoms:
What is the size of an aluminum atom: mAl=13 amu, 13x1.66x10-27 =2.158x10-26 kg
Size of each atom:
You are designing apparatus to support an actor of mass
65 kg who is to “fly” down to the stage during the performance
of a play. You decide to attach the actor’s harness to a
130-kg sandbag by means of a lightweight steel cable running
smoothly over two frictionless pulleys, as shown in Figure. You need 3.0 m of cable
between the harness and the nearest pulley so that the pulley can be hidden behind a
curtain.
For the apparatus to work successfully, the sandbag must never
lift above the floor as the actor swings from above the stage to the floor. Let us call the
angle that the actor’s cable makes with the vertical . What is the maximum value can have
before the sandbag lifts off the floor?
Sol.: