Chapter 11 Equilibrium and Elasticity

Equilibrium

Two Conditions for Equilibrium To motivate these, recall:

Defining Equilibrium Equilibrium = no net external force or torque = no change in translation or rotation) your text says L=0; others allow nonzero L:

Defining Static Equilibrium ‘Static’ Equilibrium = the special case of no translation or rotation at all

Two Conditions for Equilibrium When applying these, we must consider all external forces But the gravitational force is rather subtle

Center of Gravity (cg) Gravity acts at every point of a body Let  = the torque on a body due to gravity Can find  by treating the body as a single particle (the ‘cg’)

Center of Mass (cm) it can be shown: if g = constant everywhere, then: center of gravity = center of mass

Using the Center of Gravity Pressent some more explanatory notes

Solving Equilibrium Problems

Two Conditions for Equilibrium From now on, in this chapter/lecture: center of mass = center of gravity ‘equilibrium’ means ‘static equilibrium’ write:  F and  for  F ext and  ext

First Condition for Equilibrium

Second Condition for Equilibrium

Exercise Work through Exercise 11-11

Exercise Work through Exercise 11-14

A different version of Example 11-3 The ‘Leaning Ladder’ Problem Work through the variation the the text’s leaning ladder problem

Problem ‘Wheel on the Curb’ Problem Work through Problem 11-62

Elasticity

Real bodies are not perfectly rigid They deform when forces are applied Elastic deformation: body returns to its original shape after the applied forces are removed

Stress and Strain stress: describes the applied forces strain: describes the resulting deformation Hooke’s Law: stress = modulus × strain modulus: property of material under stress (large modulus means small deformation)

Hooke’s Law and Beyond O to a : small stress, strain Hooke’s Law: stress=modulus×strain a < b : stress and strain are no longer proportional

Units stress = modulus × strain stress (‘applied force’): pascal= Pa=N/m 2 strain (‘deformation’): dimensionless modulus: same unit as stress

Types of Stress and Strain Applied forces are perpendicular to surface: tensile stress bulk (volume) stress Applied forces are parallel to surface: shear stress

Tensile Stress and Strain tensile stress = F/A tensile strain =  l/l 0 Young’s modulus = Y

Tensile Stress and Strain Work through Exercise 11-22

Compression vs. Tension tension (shown): pull on object compression: push on object (reverse direction of F shown at left) Y compressive = Y tensile Work through Exercise 11-26

Tension and Compression at once

Bulk Stress and Strain pressure: p=F/A bulk stress =  p bulk strain =  V/V 0 bulk modulus = B

Bulk Stress and Strain B > 0 negative sign above:  p and  V have opposite signs Work through Exercise 11-30

Shear Stress and Strain

shear stress = F 7 /A shear strain = x/h = tan  shear modulus = S

Shear Stress and Strain Do Exercise 11-32

Regimes of Deformation O to a : (small stress, strain) stress=modulus×strain elastic, reversible a < b : elastic, reversible but stress and strain not proportional

Regimes of Deformation From point O to b : elastic, reversible from point b to d: plastic, irreversible ductile materials have long c–d curves brittle materials have short c–d curves

Demonstation Tensile Strength and Fracture