Chapter 12: Equilibrium and Elasticity

Conditions Under Which a Rigid Object is in Equilibrium

Problem-Solving StrategyElasticity

Equilibrium:

An object at equilibrium is either ...

• at rest and staying at rest (i.e., static equilibrium) , or

• in motion and continuing in motion with the constant

velocity and constant angular momentum.

For the object in equilibrium,

• the linear momentum ( ) of its center of mass is constant.

• the angular momentum ( ) about its center of mass, or any other point, is constant.

vmP

)( vrmPrL

Conditions of Equilibrium:

amrdt

Ldam

dt

PdF netnet

Net force: Net torque:

Conditions of equilibrium:

torques)of (balance 0 and forces) of (balance 0 net

netF

0

0

0

,

,

,

znet

ynet

xnet

F

F

F

0

0

0

,

,

,

znet

ynet

xnet

Another requirements for static equilibrium: 0P

The center or gravity:

The gravitational force on a body effectively acts at a single point, called the center of gravity (cog) of the body.•the center of mass of an object depends on its shape and its density

•the center of gravity of an object depends on its shape, density, and the external gravitational field.

Does the center of gravity of the body always coincide with the center of mass (com)?

Yes, if the body is in a uniform gravitational field.

How is the center of gravity of an object determined?

The center of gravity (cog) of a regularly shaped body of uniform composition lies at its geometric center.

The (cog) of the body can be located by suspending it from several different points. The cog is always on the line-of-action of the force supporting the object.

cog

Problem-Solving Strategy:

• Define the system to be analyzed

• Identify the forces acting on the system

• Draw a free-body diagram of the system and show all the forces acting on the system, labeling them and making sure that their points of application and lines of action are correctly shown.

• Write down two equilibrium requirements in components and solve these for the unknowns

Sample Problem 12-1:• Define the system to be analyzed: beam & block

• Identify the forces acting on the system:

the gravitational forces: mg & Mg,

the forces from the left and the right scales: Fl & Fr

• Draw a force diagram

• Write down the equilibrium requirements in components and solve these for the unknowns

00)()(: torquesof balance

0:forces of balance

41

21

lr

rl

FMgLmgLLF

mgMgFF

O

ElasticitySome concepts:

• Rigid Body:

• Deformable Body:

elastic body: rubber, steel, rock…

plastic body: lead, moist clay, putty…

• Stress: Deforming force per unit area (N/m2)

• Strain: unit deformation

Strain

Stressmodulus Elastic

Young’s Modulus: Elasticity in Length

The Young’s modulus, E, can be calculated by dividing the stress by the strain, i.e.

   

where (in SI units)E is measured in newtons per square metre (N/m²). F is the force, measured in newtons (N)A is the cross-sectional area through which the force is applied, measured in square metres (m2)L is the extension, measured in metres (m)L is the natural length, measured in metres (m)

L

L

A

F

LL

AF

strain

stressE

/

/

Material Density (kg/m3)

Young’s Modulus E

(109N/m2)

Ultimate Strength Su

(106N/m2)

Yield Strength Sy

(106N/m2)

Steel 7860 200 400 250

Aluminum 2710 70 110 90

Glass 2190 65 50

Concrete 2320 30 40

Wood 525 13 50

Bone 1900 9 170

Polystyrene 1050 3 48

Table 12-1:

Some elastic properties of selected material of engineering interest

Shear Modulus: Elasticity in Shape

The shear modulus, G, can be calculated by dividing the shear stress by the strain, i.e.

   

where (in SI units)G is measured in newtons per square metre (N/m²) F is the force, measured in newtons (N)A is the cross-sectional area through which the force is applied, measured in square metres (m2)x is the horizontal distance the sheared face moves, measured in metres (m)L is the height of the object, measured in metres (m)

x

L

A

F

Lx

AFG

/

/

strainshear

stressshear

Bulk Modulus: Elasticity in Volume

The bulk modulus, B, can be calculated by dividing the hydraulic stress by the strain, i.e.

   

where (in SI units)B is measured in newtons per square metre (N/m²) P is measured in in newtons per square metre (N/m²) V is the change in volume, measured in metres (m3)V is the original volume, measured in metres (m3)

V

Vp

VV

pB

/strain hydraulic

pressure hydraulic

Young’s modulus Shear modulus Bulk modulus

Under tension and

compressionUnder shearing Under hydraulic

stress

Strain is Strain is Strain is

L

L

A

F

Strain

StressE

x

L

A

F

Strain

StressG

V

Vp

Strain

StressB

LL / Lx / VV /

Summary:• Requirements for Equilibrium:

• The cog of an object coincides with the com if the object is in a uniform gravitational field.

• Solutions of Problems:

• Elastic Moduli:tension and compressionshearinghydraulic stress

strain modulusstress

L

LE

A

F

L

xG

A

F

V

VBp

0 and 0 net

netF

•Define the system to be analyzed

• Identify the forces acting on the system

• Draw a force diagram

• Write down the equilibrium requirements in components and solve these for the unknowns

Sample Problem 12-2:• Define the object to be analyzed: firefighter & ladder

• Identify the forces acting on the system:

the gravitational forces: mg & Mg,

the force from the wall: Fw

the force from the pavement: Fpx & Fpy

• Draw a force diagram

• Write down the equilibrium requirements in components and solve these for the unknowns

22

31

21

where,

0)()(: torquesof balance

0

0:forces of balance

hLa

hFmgaMga

mgMgF

FF

w

py

pxw

Sample Problem 12-3:

• Define the object to be analyzed: Beam

• Identify the forces acting on the system:

the gravitational force (mg),

the force from the rope (Tr)

the force from the cable (Tc), and

the force from the hinge (Fv and Fh)

• Draw a force diagram

• Write down the equilibrium requirements in components and solve these for the unknowns

Na

mMgb

mgbbTaT rcznet

6093)(

T

0))((

21

c

21

,

NTF

TFF

ch

chxnet

6093

0,

NMgmgF

TmgFF

v

rvynet

5047

0,

NFFF vh 790022

Balance of forces:

Balance of torques:

Sample Problem 12-6:

• Define the system to be analyzed: table plus steel cylinder.

• Identify the forces acting on the object:

the gravitational force (Mg),

the forces on legs from the floor (F1= F2=

F3 and F4).

• Draw a force diagram43

4321

3

FF

FFFF

MgFg

• Write down the equilibrium requirements in components and solve these for the unknowns

Balance of forces: 03 43 MgFFFnet

dAE

LF

AE

LF

dLLL

LE

A

FL

LE

A

F

34

34

44

33

If table remains level:

NF

NF

1200

550

4

3