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A square plate is \(1.0 \times 10^{-2} \mathrm{m}\) thick, measures \(3.0 \times 10^{-2} \mathrm{m}\) on a side, and has a mass of \(7.2 \times 10^{-2}\) kg. The shear modulus of the material is \(2.0 \times 10^{10} \mathrm{N} / \mathrm{m}^{2} .\) One of the square faces rests on a flat horizontal surface, and the coefficient of static friction between the plate and the surface is 0.91 . A force is applied to the top of the plate, as in Figure \(10.29 a .\) Determine (a) the maximum possible amount of shear stress, (b) the maximum possible amount of shear strain, and (c) the maximum possible amount of shear deformation \(\Delta X\) (see Figure \(10.29 b\) ) that can be created by the applied force just before the plate begins to move.

Step by step solution

01

Calculate the Maximum Static Frictional Force

The maximum force of static friction can be found using \( F_{\text{friction}} = \mu_s \cdot F_{\text{normal}} \). Here, \( \mu_s = 0.91 \) is the coefficient of static friction. The normal force for an object on a horizontal surface is equal to its weight: \( F_{\text{normal}} = mg \), where \( m = 7.2 \times 10^{-2} \) kg and \( g = 9.8 \) m/s虏 is the acceleration due to gravity. Thus, \( F_{\text{normal}} = 7.2 \times 10^{-2} \times 9.8 = 0.7056 \) N. Therefore, \( F_{\text{friction}}_{\max} = 0.91 \times 0.7056 = 0.642096 \) N.

02

Calculate the Maximum Possible Shear Stress

Shear stress \( \tau \) is defined as the force applied parallel to the surface divided by the area over which it acts. The area of the top of the square plate is \( A = (3.0 \times 10^{-2})^2 = 9.0 \times 10^{-4} \) m虏. Thus, the maximum possible shear stress is \( \tau_{\max} = \frac{F_{\text{friction}_{\max}}}{A} = \frac{0.642096}{9.0 \times 10^{-4}} = 713.44 \) N/m虏.

03

Calculate the Maximum Possible Shear Strain

Shear strain \( \gamma \) is related to shear stress via the shear modulus \( G \): \( \tau = G \cdot \gamma \). Given \( \tau_{\max} = 713.44 \) N/m虏 and \( G = 2.0 \times 10^{10} \) N/m虏, solve for shear strain: \( \gamma = \frac{\tau_{\max}}{G} = \frac{713.44}{2.0 \times 10^{10}} = 3.5672 \times 10^{-8} \).

04

Calculate the Maximum Possible Shear Deformation

Shear deformation \( \Delta X \) is given by \( \Delta X = \gamma \cdot L \), where \( L = 1.0 \times 10^{-2} \) m is the thickness of the plate. Thus, \( \Delta X = 3.5672 \times 10^{-8} \times 1.0 \times 10^{-2} = 3.5672 \times 10^{-10} \) m.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Shear Strain

Shear strain describes the change in shape of a material when it's subjected to a shear force. Unlike normal strain, which involves changes in length, shear strain involves angular distortion. It tells us how much the material is deformed in response to shear stress.
The formula for shear strain is \( \gamma = \frac{\Delta X}{L} \), where \( \Delta X \) is the horizontal displacement and \( L \) is the original length perpendicular to the applied force. In simpler terms, shear strain is a way of knowing how skewed a material becomes. Think of it as how the material "slides" over itself due to applied force.

Shear Deformation

Shear deformation refers to the actual change in shape that occurs when shear stress is applied to a material. It results in the sliding of layers within the material. Imagine a stack of cards, and pushing the top card to the side: that's shear deformation in action.
Shear deformation is quantified by the displacement, \( \Delta X \), and can be measured visually or calculated by understanding the shear strain and thickness of the material. It plays a crucial role in material science and engineering, as excessive shear deformation can lead to structural failures.

Shear Modulus

The shear modulus, also known as modulus of rigidity, is a property that quantifies a material's ability to resist shear deformation. It's denoted by \( G \) and measures the relationship between shear stress and shear strain.
Mathematically, it's expressed as \( G = \frac{\tau}{\gamma} \), where \( \tau \) is the shear stress and \( \gamma \) is the shear strain. A high shear modulus means the material is very rigid and less likely to deform under shear stress. For our exercise, the shear modulus of the plate indicates how easily it can be distorted or stay robust against the applied force.

Static Friction

Static friction is the force that keeps an object at rest when a force is applied. It needs to be overcome for the object to start moving. It plays a critical role in preventing slipping and is represented as \( F_{\text{friction}} = \mu_s \times F_{\text{normal}} \), where \( \mu_s \) is the coefficient of static friction and \( F_{\text{normal}} \) is the normal force.
In our context, the static frictional force ensures the plate remains on the surface without sliding, until a certain threshold is surpassed. Understanding this helps in predicting when an object will begin to slide, based on the applied force.

Physics Education

Physics education aims to provide students with a comprehensive understanding of the fundamental principles governing physical phenomena. It's not just about solving equations but understanding concepts like shear stress and strain, and how they apply to real-world situations.
Effective physics education combines theoretical knowledge with practical applications, helping students visualize concepts through experiments and problem-solving exercises. By exploring topics like shear deformation, students can better understand the material properties and how they react under various forces.

Shear Force

Shear force is the component of force that acts parallel to the surface of an object. It causes layers within the material to slide past each other. This force is central to understanding shear stress and deformation in materials.
In structural engineering, shear force is crucial in analyzing beams and structural elements, ensuring they can safely resist applied loads without excessive deformation or failure. It helps in designing materials and structures that are safe and efficient.

Material Properties

Material properties describe how a substance reacts under various conditions and forces. These include density, stiffness, ductility, and shear modulus, among others. Understanding these properties allows engineers to select the right materials for specific applications.
For instance, a high shear modulus indicates a material can withstand high shear stress without deforming significantly. Such properties inform decisions in building, manufacturing, and more, ensuring that materials meet the demands of their intended use.