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Summary

  • An object or material is elastic if it comes back to its original shape and size when the stress vanishes. In elastic deformations with stress values lower than the proportionality limit, stress is proportional to strain. When stress goes beyond the proportionality limit, the deformation is still elastic but nonlinear up to the elasticity limit.
  • An object or material has plastic behavior when stress is larger than the elastic limit. In the plastic region, the object or material does not come back to its original size or shape when stress vanishes but acquires a permanent deformation. Plastic behavior ends at the breaking point.

Key equations

First Equilibrium Condition k F k = 0
Second Equilibrium Condition k τ k = 0
Linear relation between
stress and strain
stress = (elastic modulus) × strain
Young’s modulus Y = tensile stress tensile strain = F A L 0 Δ L
Bulk modulus B = bulk stress bulk strain = Δ p V 0 Δ V
Shear modulus S = shear stress shear strain = F A L 0 Δ x

Conceptual questions

Note: Unless stated otherwise, the weights of the wires, rods, and other elements are assumed to be negligible. Elastic moduli of selected materials are given in [link] .

What is meant when a fishing line is designated as “a 10-lb test?”

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Steel rods are commonly placed in concrete before it sets. What is the purpose of these rods?

It acts as “reinforcement,” increasing a range of strain values before the structure reaches its breaking point.

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Problems

A uniform rope of cross-sectional area 0.50 cm 2 breaks when the tensile stress in it reaches 6.00 × 10 6 N/m 2 . (a) What is the maximum load that can be lifted slowly at a constant speed by the rope? (b) What is the maximum load that can be lifted by the rope with an acceleration of 4.00 m/s 2 ?

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One end of a vertical metallic wire of length 2.0 m and diameter 1.0 mm is attached to a ceiling, and the other end is attached to a 5.0-N weight pan, as shown below. The position of the pointer before the pan is 4.000 cm. Different weights are then added to the pan area, and the position of the pointer is recorded in the table shown. Plot stress versus strain for this wire, then use the resulting curve to determine Young’s modulus and the proportionality limit of the metal. What metal is this most likely to be?

Added load (including pan)
(N)
Scale reading
(cm)
0 4.000
15 4.036
25 4.073
35 4.109
45 4.146
55 4.181
65 4.221
75 4.266
85 4.316
Figure shows vertical wire attached to a ceiling with the other end is attached to a weight pan.
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An aluminum ( ρ = 2.7 g/cm 3 ) wire is suspended from the ceiling and hangs vertically. How long must the wire be before the stress at its upper end reaches the proportionality limit, which is 8.0 × 10 7 N/m 2 ?

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Additional problems

The coefficient of static friction between the rubber eraser of the pencil and the tabletop is μ s = 0.80 . If the force F is applied along the axis of the pencil, as shown below, what is the minimum angle at which the pencil can stand without slipping? Ignore the weight of the pencil.

Figure shows a pencil that rests against a corner. The eraser end touches a rough horizontal floor. Angle between pencil and ground is Theta.

tan −1 ( 1 / μ s ) = 51.3 °

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A pencil rests against a corner, as shown below. The sharpened end of the pencil touches a smooth vertical surface and the eraser end touches a rough horizontal floor. The coefficient of static friction between the eraser and the floor is μ s = 0.80 . The center of mass of the pencil is located 9.0 cm from the tip of the eraser and 11.0 cm from the tip of the pencil lead. Find the minimum angle θ for which the pencil does not slip.

Figure shows a pencil that rests against a corner. The sharpened end of the pencil touches a smooth vertical surface and the eraser end touches a rough horizontal floor. Angle between pencil and ground is Theta. Center of mass is 9 cm from the eraser and 11 cm from the sharpened end.
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Questions & Answers

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The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
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Anatomy is the identification and description of the structures of living things
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Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
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Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
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Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)
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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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