Young's Modulus
Learning Goal: To understand the meaning of Young's modulus, to perform some real-life calculations related to stretching steel, a common construction material, and to introduce the concept of breaking stress.
Hooke's law states that for springs and other "elastic" objects

,

where is the magnitude of the stretching force, is the corresponding elongation of the spring from equilibrium, and is a constant that depends on the geometry and the material of the spring. If the deformations are small enough, most materials, in fact, behave like springs: Their deformation is directly proportional to the external force. Therefore, it may be useful to operate with an expression that is similar to Hooke's law but describes the properties of various materials, as opposed to objects such as springs. Such an expression does exist. Consider, for instance, a bar of initial length and cross-sectional area stressed by a force of magnitude . As a result, the bar stretches by .

Let us define two new terms:

• Tensile stress is the ratio of the stretching force to the cross-sectional area:

.

• Tensile strain is the ratio of the elongation of the rod to the initial length of the bar:

.

It turns out that the ratio of the tensile stress to the tensile strain is a constant as long as the tensile stress is not too large. That constant, which is an inherent property of a material, is called Young's modulus and is given by

Part A
What is the SI unit of Young's modulus?
Hint A.1 Look at the dimensions Hint not displayed
 Pa   pascal   Pascal   pascals   Pascals   kg/(m*s^2)   N/(m^2)
Part B
Consider a metal bar of initial length and cross-sectional area . The Young's modulus of the material of the bar is . Find the "spring constant" of such a bar for low values of tensile strain.
Hint B.1 Use the definition of Young's modulus Hint not displayed
 = Y*A/L
Part C
Ten identical steel wires have equal lengths and equal "spring constants" . The wires are connected end to end, so that the resultant wire has length . What is the "spring constant" of the resulting wire?
Hint C.1 The spring constant Hint not displayed
Part D
Ten identical steel wires have equal lengths and equal "spring constants" . The wires are slightly twisted together, so that the resultant wire has length and is ten times as thick as each individual wire. What is the "spring constant" of the resulting wire?
Hint D.1 Hint not displayed
Part E
Ten identical steel wires have equal lengths and equal "spring constants" . The Young's modulus of each wire is . The wires are connected end to end, so that the resultant wire has length . What is the Young's modulus of the resulting wire?
Part F
Ten identical steel wires have equal lengths and equal "spring constants" . The Young's modulus of each wire is . The wires are slightly twisted together, so that the resultant wire has length and is ten times as thick as the individual wire. What is the Young's modulus of the resulting wire?
By rearranging the wires, we create a new object with new mechanical properties. However, Young's modulus depends on the material, which remains unchanged. To change the Young's modulus, one would have to change the properties of the material itself, for instance by heating or cooling it.
Part G
Consider a steel guitar string of initial length meter and cross-sectional area square millimeters. The Young's modulus of the steel is pascals. How far ( ) would such a string stretch under a tension of 1500 newtons?