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\(\mathrm{A}\) 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 \(7.2 \times 10^{-2} \mathrm{kg} .\) 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 Maximum Static Friction Force

The maximum force before the plate slips is determined by static friction. This force is given by \( F_{ ext{max}} = \mu_s \times F_n \), where \( \mu_s = 0.91 \) (coefficient of static friction) and \( F_n = mg \) (normal force, equal to weight). The weight \( mg = 7.2 \times 10^{-2} \times 9.8 = 0.7056 \) N. Therefore, \( F_{\text{max}} = 0.91 \times 0.7056 = 0.642096 \) N.

02

Calculate Maximum Shear Stress

Shear stress \( \tau \) is defined as the force per unit area. The area \( A \) of the square face is \((3.0 \times 10^{-2})^2 = 9.0 \times 10^{-4} \) m². Therefore, the maximum shear stress is \( \tau = \frac{F_{\text{max}}}{A} = \frac{0.642096}{9.0 \times 10^{-4}} = 713.44 \) Pa.

03

Calculate Maximum Shear Strain

Shear strain \( \gamma \) is related to shear stress \( \tau \) and shear modulus \( G \) by the formula \( \gamma = \frac{\tau}{G} \). Given \( G = 2.0 \times 10^{10} \) N/m², the shear strain is \( \gamma = \frac{713.44}{2.0 \times 10^{10}} = 3.5672 \times 10^{-8} \).

04

Calculate Maximum Shear Deformation

Shear deformation \( \Delta X \) is related to shear strain by the formula \( \gamma = \frac{\Delta X}{t} \), where \( t = 1.0 \times 10^{-2} \) m is the thickness. So, \( \Delta X = \gamma \times t = 3.5672 \times 10^{-8} \times 1.0 \times 10^{-2} = 3.5672 \times 10^{-10} \) m.

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Key Concepts

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

Shear Strain

When a force is applied parallel to the surface of an object, the material experiences shear strain. Imagine it as a shift between layers in the material without any change in its volume. It's like sliding a deck of cards sideways - the top card shifts relative to the bottom one. In mathematical terms, shear strain \( \gamma \) is defined as the displacement (deformation) across the thickness of a material divided by its thickness.

It's calculated using the formula:

• \[ \gamma = \frac{\tau}{G} \]

where \( \tau \) is the shear stress and \( G \) is the shear modulus. This relationship helps illustrate how the material's response (strain) to an applied force (stress) depends on its properties (shear modulus).

Shear Deformation

Shear deformation in a material is the actual change or distortion that occurs when a force is applied. It reflects the physical shift of particles from their initial alignment. Consider a block of gelatin - when you push the top surface slightly, it causes the block to tilt and deform, akin to the shear deformation concept.

• The extent of this deformation can be quantified by the formula:
• \[ \Delta X = \gamma \cdot t \]

Here, \( \Delta X \) is the deformation, \( \gamma \) is the shear strain, and \( t \) is the thickness of the material. This formula highlights that the amount of deformation depends on both the strain and the thickness.

Static Friction

Static friction is the force that keeps an object at rest when a force is applied, preventing it from sliding. Think of trying to push a heavy box on the floor; the resistance you feel before it starts moving is static friction. This frictional force must be overcome to initiate motion. It is dependent on two factors: the coefficient of static friction and the normal force.

• Mathematically, it's expressed as:
• \[ F_{\text{max}} = \mu_s \times F_n \]

where \( \mu_s \) is the coefficient of static friction and \( F_n \) is the normal force (often the weight of the object if on a horizontal surface). This formula helps calculate the maximum force needed to move an object before it starts to slide.

Shear Modulus

The shear modulus, also known as the modulus of rigidity, describes how a material reacts to shear stress. It measures the material's stiffness when subject to shear forces, and it is an intrinsic property of materials. Think of it as how difficult it is to twist or shift the material. The higher the shear modulus, the stiffer the material, meaning it will deform less under the same applied shear stress.

• It's calculated through the relationship:
• \[ G = \frac{\tau}{\gamma} \]

Here, \( G \) is the shear modulus, \( \tau \) is the shear stress, and \( \gamma \) is the shear strain. This formula showcases a direct link between stress and strain controlled by the material's stiffness.