Chapter 11 Equilibrium and Elasticity Equilibrium Two Conditions

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 t = the torque on a body due to gravity • Can find t 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: SF and St for SFext and Stext

First Condition for Equilibrium

Second Condition for Equilibrium

Exercise 11 -11 Work through Exercise 11 -11

Exercise 11 -14 Work through Exercise 11 -14

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

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

Elasticity

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 = Dl/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) • Ycompressive = Ytensile Work through Exercise 11 -26

Tension and Compression at once

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

Bulk Stress and Strain • B>0 • negative sign above: Dp and DV have opposite signs Work through Exercise 11 -30

Shear Stress and Strain

Shear Stress and Strain • shear stress = F 7/A • shear strain = x/h = tanf • 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