Chapter 12: Equilibrium and Elasticity Conditions Under Which a Rigid Object is in Equilibrium ...
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Transcript of Chapter 12: Equilibrium and Elasticity Conditions Under Which a Rigid Object is in Equilibrium ...
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