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Chapter 11: Problem 14

Shear forces are applied to a rectangular solid. The same forces are applied to another rectangular solid of the same material, but with three times each edge length. In each case, the forces are small enough that Hooke's law is obeyed. What is the ratio of the shear strain for the larger object to that of the smaller object?

Short Answer

Expert verified
The shear strain ratio is \( \frac{1}{9} \).

Step by step solution

01

Understand Shear Strain

Shear strain (\( \gamma \)) is defined as the deformation per unit length, which changes the shape of a body due to shear forces.
02

Determine Volume and Surface Area

The volume of the smaller object is \( V = L^3 \) and its surface area over which the force is applied is \( A = L^2 \). The larger object has volume \( V' = (3L)^3 = 27L^3 \) and area \( A' = (3L)^2 = 9L^2 \).
03

Apply Shear Stress Equation

Shear stress (\( \tau \)) is force (\( F \)) divided by area (\( A \)), so for the smaller object: \( \tau = \frac{F}{L^2} \). For the larger object: \( \tau' = \frac{F}{9L^2} \).
04

Apply Hooke's Law for Shear

Shear strain is related to shear stress by Hooke's Law, given by \( \gamma = \frac{\tau}{G} \), where \( G \) is the shear modulus of the material. So for the smaller object, \( \gamma = \frac{F}{GL^2} \), and for the larger object, \( \gamma' = \frac{F}{9GL^2} \).
05

Calculate the Ratio of Shear Strains

The ratio of the shear strain for the larger object (\( \gamma' \)) to that of the smaller object (\( \gamma \)) is \( \frac{\gamma'}{\gamma} = \frac{\frac{F}{9GL^2}}{\frac{F}{GL^2}} = \frac{1}{9} \). This indicates the shear strain decreases with the increase of object size.

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Key Concepts

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

Hooke's law
Hooke's law is a principle of physics that describes how materials deform in response to applied forces. When forces are applied to a material, it stretches or compresses. According to Hooke's law, the change in shape is directly proportional to the force applied, assuming the material's elastic limit is not exceeded.

This can be expressed mathematically as:

  • The formula: \( F = kx \)
  • Where \( F \) is the force applied, \( k \) is the spring constant or stiffness of the material, and \( x \) is the deformation experienced by the material.
In the case of shear forces, Hooke's law relates shear stress to shear strain through the shear modulus. This is a measure of how rigid a material is when deformed by shear forces.
shear stress
Shear stress is the component of stress coplanar with a material cross-section. It arises when forces are applied in parallel but opposite directions on a surface, causing deformation.

To calculate shear stress \( \tau \), use the formula:

  • \( \tau = \frac{F}{A} \)
  • Where \( F \) is the force applied, and \( A \) is the area over which the force is applied.
Shear stress essentially measures how much force is affecting an area in a parallel manner. Understanding shear stress is crucial for studying the deformation of structures, as it determines how forces can alter their shape without breaking them. In our exercise, a larger area meant a reduced shear stress given the same force.
shear modulus
Shear modulus \( G \), also known as the modulus of rigidity, is a material's elasticity coefficient under shear stress. It plays an essential role in Hooke's law for shear, expressing the relationship between shear stress and shear strain.

The relationship is given by:

  • \( \gamma = \frac{\tau}{G} \)
  • Where \( \gamma \) is the shear strain, \( \tau \) is the shear stress, and \( G \) is the shear modulus.
A higher shear modulus indicates a more rigid material that deforms less under shear stress. In the context of the exercise, both the smaller and larger objects are made of the same material, so their shear modulus \( G \) remains constant across both scenarios.
rectangular solid deformation
Rectangular solid deformation refers to the change in shape experienced by a rectangular object when subjected to external forces. In the exercise, we analyzed how shear forces affect such objects.

Key factors that influence deformation include:

  • The force applied and its distribution over the object's surface.
  • The material's properties, including the shear modulus.
  • The dimensions of the object, including its surface area and volume.
When scaling the size of a rectangular solid while applying identical forces, the extent of deformation changes according to the object's surface area ratio. As demonstrated in the solution, when the dimensions of a rectangular solid are increased uniformly, the shear strain shifts, reducing by a factor calculated in relation to the increase in size.

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Most popular questions from this chapter

(a) If a vibrating system has total energy \(E_{0},\) what will its total energy be (in terms of \(E_{0}\) ) if you double the amplitude of vibration? (b) If you want to triple the total energy of a vibrating system with amplitude \(A_{0}\), what should its new amplitude be (in terms of \(A_{0}\) )?

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