(a) Referring to the metal rod in Figure 9.2 a (under tensile stress), show that Eq. 9.4 can be

(a) Referring to the metal rod in Figure 9.2 a (under...

(a) Referring to the metal rod in Figure $9.2 \mathrm{a}$ (under tensile stress), show that Eq. 9.4 can be rewritten to resemble a Hooke's law type of spring relationship for the rod. That is, show that it can be written as $F=k \Delta L,$ where $k$ is the "effective" spring constant for the rod. Express $k$ symbolically in terms of the rod's cross-sectional area $A$, its Young's modulus $Y,$ and its unstressed length $L_{\mathrm{o}}$ and show that it has the proper SI units. (b) Now consider a thin rod of iron that is subjected to a tensile force of $2.00 \times 10^{3} \mathrm{~N}$. If it has a cross-section of radius $1.00 \mathrm{~cm}$ and an unstressed length of $25.0 \mathrm{~cm},$ determine its effective spring constant. (c) By how much does this rod stretch when this force is applied? (d) How much work is done by this stretching force?

Key Concepts

Young's Modulus

Young's modulus is a fundamental material property that measures a material's stiffness. It is defined as the ratio of tensile stress (force per unit area) to tensile strain (relative deformation) in the linear elastic region of the material's stress-strain curve. It allows one to predict how much a material will stretch under a given load and is a key factor in determining the behavior of structures subject to mechanical stress.

Hooke's Law

Hooke's Law is a principle of mechanics that states that, for small deformations, the extension or compression of a spring is directly proportional to the applied force. Mathematically, it is expressed as F = k?L, where F is the force applied, ?L is the change in length, and k is the spring constant. This law is fundamental in understanding and modeling elastic behavior in materials and mechanical systems.

Effective Spring Constant

The effective spring constant is a parameter that characterizes the stiffness of an elastic solid (such as a rod) by analogizing it to a spring. It can be derived using the relationship between tensile stress, strain, and Young's modulus, resulting in an expression that relates the force applied to a rod, its cross-sectional area, its unstressed length, and its material's Young's modulus. This concept is essential for comparing and converting the elastic properties of different systems into a common framework.

Tensile Stress and Strain

Tensile stress is the force applied per unit area of a material, while tensile strain is the relative change in length due to the applied force. Their relationship, governed by Young's modulus, is fundamental in analyzing how materials respond to forces. This concept helps in determining the deformation of materials and is pivotal in the analysis of the elastic behavior of structures under load.

Work Done in Elastic Deformation

The work done in stretching or compressing an elastic material, such as a rod, is related to the energy stored in that material. For a system obeying Hooke's Law, this work can be calculated using the formula W = ½ k(?L)², where k is the effective spring constant and ?L is the extension. This concept is crucial in energy analysis and in understanding how mechanical energy is stored and released in elastic components.

Transcript

00:01 This is the equation for young's module and we want to show that this equation is similar to this equation for the spring, where this was the spring constant.

00:12 In order to show that this equation is similar to that one, what i'm going to do, i'm going to rearrange these terms here.

00:23 I keep y here.

00:25 I'm going to bring this delta l outside.

00:28 So i get one over l0, a, delta l.

00:31 Now i put everything else in the parentheses and i can write it like y times a divided by l not times delta l and if i compare these two equations right now i have delta l here delta l here f here so whatever is next to delta l they have it has to be equal to k so i can conclude that k is equal to y a over l not in order to make sure that this is correct, we need to double check that the unit matches.

01:06 The unit of k is neotone per meter.

01:11 The unit of yonge stap law is neotone per meter squared.

01:20 The unit of area is meter squared.

01:23 The unit of l is meter.

01:26 Now what happens? m squared cancels with the m squared and the right side give newton per meter.

01:32 So this was the unit of k.

01:34 This was the unit of y a over l -0 and both are equal to newton per meter.

01:44 That was part a.

01:46 In part b the question is asking us find the value of k which is the effective spring constant.

01:56 What we need to know the value of young's module for iron is given to be 15 times 10 to the 10 newton per meter squared...

Please give Ace some feedback