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Chapter 9 Static Equilibrium; Elasticity and Fracture

Chapter 9 Static Equilibrium; Elasticity and Fracture

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Page 1: Chapter 9 Static Equilibrium; Elasticity and Fracture

Chapter 9Static Equilibrium; Elasticity and Fracture

Page 2: Chapter 9 Static Equilibrium; Elasticity and Fracture

Torque and Two Conditions For Equilibrium

An object in mechanical equilibrium must satisfy the following conditions:

1. The net external force must be zero:

ΣF = 0

2. The net external torque must be zero:

Στ = 0

Page 3: Chapter 9 Static Equilibrium; Elasticity and Fracture

Two Conditions of Equilibrium

First Condition of Equilibrium○ The net external force must be zero

Necessary, but not sufficientTranslational equilibrium

0

0 0x y

or

and

F

F F

Page 4: Chapter 9 Static Equilibrium; Elasticity and Fracture

Two Conditions of Equilibrium

Second Condition of Equilibrium○ The net external torque must be zero

Στ = 0 or

Στx = 0 and Στy = 0

Rotational equilibrium

Both conditions satisfy mechanical equilibrium

Page 5: Chapter 9 Static Equilibrium; Elasticity and Fracture

Two Conditions of Equilibrium

Objects in mechanical equilibrium

Rock on ridge

See-saw

Page 6: Chapter 9 Static Equilibrium; Elasticity and Fracture

Examples of Objects in Equilibrium

1. Draw a diagram of the system Include coordinates and choose a rotation

axis

2. Isolate the object being analyzed and draw a free body diagram showing all the external forces acting on the object

For systems containing more than one object, draw a separate free body diagram for each object

Page 7: Chapter 9 Static Equilibrium; Elasticity and Fracture

Examples of Objects in Equilibrium

3. Apply the Second Condition of Equilibrium

Στ = 0 This will yield a single equation, often with

one unknown which can be solved immediately

4. Apply the First Condition of Equilibrium

ΣF = 0 This will give you two more equations

4. Solve the resulting simultaneous equations for all of the unknowns

Solving by substitution is generally easiest

Page 8: Chapter 9 Static Equilibrium; Elasticity and Fracture

Examples of Objects in Equilibrium

Examples of Free Body Diagrams (forearm)

• Isolate the object to be analyzed• Draw the free body diagram for that object

• Include all the external forces acting on the object

Page 9: Chapter 9 Static Equilibrium; Elasticity and Fracture

Examples of Objects in Equilibrium

FBD - Beam

Fig 8.12, p.228

Slide 17

• The free body diagram includes the directions of the forces

• The weights act through the centers of gravity of their objects

Page 10: Chapter 9 Static Equilibrium; Elasticity and Fracture

Examples of Objects in Equilibrium FBD - Ladder

• The free body diagram shows the normal force and the force of static friction acting on the ladder at the ground

• The last diagram shows the lever arms for the forces

Page 11: Chapter 9 Static Equilibrium; Elasticity and Fracture

Examples of Objects in Equilibrium

Example:

Page 12: Chapter 9 Static Equilibrium; Elasticity and Fracture

Stress and Strain

So far, studying rigid bodiesthe rigid body does not ever stretch,

squeeze or twist

However, we know that in reality this does occur, and we need to find a way to describe it.

This is done by the concepts of stress, strain and elastic modulus.

Page 13: Chapter 9 Static Equilibrium; Elasticity and Fracture

Stress and Strain All objects are deformable

All objects are spring-like!

It is possible to change the shape or size (or both) of an object through the application of external forces

When the forces are removed, the object tends to its original shapeThis is a deformation that exhibits elastic

behavior (spring-like)

Page 14: Chapter 9 Static Equilibrium; Elasticity and Fracture

Elastic Properties Stress is the force per unit area causing the

deformation

Strain is a measure of the amount of deformation

Page 15: Chapter 9 Static Equilibrium; Elasticity and Fracture

Elastic Modulus The elastic modulus is the constant of

proportionality between stress and strainFor sufficiently small stresses, the stress is

directly proportional to the strainThe constant of proportionality depends on the

material being deformed and the nature of the deformation

Can be thought of as the stiffness of the materialA material with a large elastic modulus is very stiff

and difficult to deform○ Analogous to the spring constant

Page 16: Chapter 9 Static Equilibrium; Elasticity and Fracture

Young’s Modulus: Elasticity in Length

Tensile stress is the ratio of the external force to the cross-sectional areaTensile is because

the bar is under tension

The elastic modulus is called Young’s modulus

Page 17: Chapter 9 Static Equilibrium; Elasticity and Fracture

Young’s Modulus, cont.

SI units of stress are Pascals, Pa1 Pa = 1 N/m2

The tensile strain is the ratio of the change in length to the original lengthStrain is dimensionless

o

F LY

A L

s t r e s s = E l a s t i c m o d u l u s × s t r a i n

Page 18: Chapter 9 Static Equilibrium; Elasticity and Fracture

Stress and Strain, Illustrated A bar of material, with a force F

applied, will change its size by:ΔL/L = = /Y = F/AY

Strain is a very useful number, being dimensionless

Example: Standing on an aluminum rod: Y = 70109 N·m2 (Pa) say area is 1 cm2 = 0.0001 m2

say length is 1 m weight is 700 N = 7106 N/m2

= 104 ΔL = 100 m compression is width of human

hair

F F

A

L

L

= F/A

= ΔL/L

= Y·

Page 19: Chapter 9 Static Equilibrium; Elasticity and Fracture

Young’s Modulus, final

Young’s modulus applies to a stress of either tension or compression

It is possible to exceed the elastic limit of the material No longer directly

proportional Ordinarily does not return

to its original length

Page 20: Chapter 9 Static Equilibrium; Elasticity and Fracture

Breaking If stress continues, it surpasses its ultimate

strengthThe ultimate strength is the greatest stress the

object can withstand without breaking The breaking point

For a brittle material, the breaking point is just beyond its ultimate strength

For a ductile material, after passing the ultimate strength the material thins and stretches at a lower stress level before breaking

Page 21: Chapter 9 Static Equilibrium; Elasticity and Fracture

Shear Modulus:Elasticity of Shape

Forces may be parallel to one of the object’s faces

The stress is called a shear stress

The shear strain is the ratio of the horizontal displacement and the height of the object

The shear modulus is S

Page 22: Chapter 9 Static Equilibrium; Elasticity and Fracture

Shear Modulus, final

S is the shear modulusA material having a large shear modulus is difficult to bend

Fshear stress

Ax

shear strainh

F xS

A h

Page 23: Chapter 9 Static Equilibrium; Elasticity and Fracture

Bulk Modulus:Volume Elasticity Bulk modulus characterizes the

response of an object to uniform squeezingSuppose the forces are perpendicular to,

and act on, all the surfaces○ Example: when an object is immersed in a

fluid The object undergoes a change in

volume without a change in shape

Page 24: Chapter 9 Static Equilibrium; Elasticity and Fracture

Bulk Modulus, cont.

Volume stress, ΔP, is the ratio of the force to the surface areaThis is also the

Pressure

The volume strain is equal to the ratio of the change in volume to the original volume

Page 25: Chapter 9 Static Equilibrium; Elasticity and Fracture

Bulk Modulus, final

A material with a large bulk modulus is difficult to compress

The negative sign is included since an increase in pressure will produce a decrease in volumeB is always positive

The compressibility is the reciprocal of the bulk modulus

VP B

V

Page 26: Chapter 9 Static Equilibrium; Elasticity and Fracture

Notes on Moduli

Solids have Young’s, Bulk, and Shear moduli

Liquids have only bulk moduli, they will not undergo a shearing or tensile stressThe liquid would flow instead

Page 27: Chapter 9 Static Equilibrium; Elasticity and Fracture

Ultimate Strength of Materials The ultimate strength of a material is the

maximum force per unit area the material can withstand before it breaks or factures

Some materials are stronger in compression than in tension

Page 28: Chapter 9 Static Equilibrium; Elasticity and Fracture

Stress and Strain

Example: