jw Physics 2111 Understanding Physics
Chapter 12 1
Fundamentals of Physics
Chapter 12 Equilibrium & Elasticity
1. Equilibrium
2. The Requirements of Equilibrium
3. The Center of Gravity
4. Some Examples of Static Equilibrium
Problem Solving Tactics
5. Indeterminate Structures
6. Elasticity
Tension & Compression
Shearing
Hydraulic Stress
Review & Summary
Questions
Exercises & Problems
jw Physics 2111 Understanding Physics
Chapter 12 2
Equilibrium
Equilibrium: P and L are constant.
Static Equilibrium:
P
L
0LP
Stable Static Equilibrium: body returns to static equilibrium after being displaced by a force.
Unstable Equilibrium: a “small” force can displace the body and end the equilibrium,
jw Physics 2111 Understanding Physics
Chapter 12 3
Requirements of Equilibrium
Requirements for Equilibrium:• The vector sum of all the external forces that act on the body must be zero.• The vector sum of all the external torques that act on the body, measured about
any possible point, must be zero.
Simplification in Chapter 12: forces are in the xy-plane (torques in z-direction)
Newton’s 2nd Law:
vector equations
0dt
PdFnet
0dt
Ldnet
jw Physics 2111 Understanding Physics
Chapter 12 4
Center of Gravity
The gravitational force Fg on a body effectively acts at a stable
point, called the Center of Gravity of the body.
If g is the same for all parts of a body, then the body’s Center of
Gravity is coincident with the body’s Center of Mass.
gF
jw Physics 2111 Understanding Physics
Chapter 12 5
Equilibrium
Stable Equilibrium Neutral Equilibrium
jw Physics 2111 Understanding Physics
Chapter 12 6
Equilibrium & Center of Gravity
cog over the base of support
cog beyond the base of support
jw Physics 2111 Understanding Physics
Chapter 12 7
Problem Solving:
• Draw a picture relevant to the problem.
• Choose the system: e.g. “the ladder & man”
• Draw the corresponding free-body diagram for the system.
– Forces
– Distances
• Choose coordinate axes.
– Component Force Equations along axes.
– Torque Equations (Choose the axis).
• Algebraically solve the equations.
– Solve for the unknowns.
– Check that units are correct.
• Plug in the numbers.
• Set significant figures for the answer(s).
jw Physics 2111 Understanding Physics
Chapter 12 8
What are the forces on the ladder from the wall and pavement?
half-way up ladder
frictionless wall
L = 12 m
ML = 45 kg
h = 9.3 m
com = L/3 from bottom
MF = 72 kg climbs halfway up
Forces on ladder from wall, pavement
jw Physics 2111 Understanding Physics
Chapter 12 9
Fireman on the Ladder
Choose rotation axis here
hFww
2agMMg
3agmmg
0pypx FF
Fr
net 0
h
jw Physics 2111 Understanding Physics
Chapter 12 10
Elasticity
A rigid body is a 3D matrix of atoms (or molecules) held together by inter-atomic forces.
All rigid bodies are elastic - deform under stress.
Stress = “Modulus of Elasticity” X Strain
Force Stress Deformation Strain
jw Physics 2111 Understanding Physics
Chapter 12 11
Elasticity
Apply a stretching force to a bar:
This force gets distributed over the surface area of the bar:
Force per unit area is defined as the stress.
jw Physics 2111 Understanding Physics
Chapter 12 12
Elasticity
If this sample bar is pulled on, what happens?
jw Physics 2111 Understanding Physics
Chapter 12 13
Elasticity
Tension & Compression
Shearing Hydraulic
Stress
Young’s Modulus(aka Y)
Shear Modulus Bulk Modulus
L
LE
A
F L
xG
A
F
V
VB
A
Fp
jw Physics 2111 Understanding Physics
Chapter 12 14
tension in rope & normal force from wall?
N
T
mg
T N
T mg
sin
cos
0
0
Tmg L r
L
Nmgr
L
2 2
cos L
L r2 2
Equilibrium: Fnet 0
vector equation
jw Physics 2111 Understanding Physics
Chapter 12 15
forces on the 4 tires of a car?
mg
Ffront
Frear
L
Lcog
Vertical forces: 2 2F F mgfront rear
F x Nfront 2 77 103. F x Nrear 389 103.
car
m
mL
kgm
78.1L
wheelbase""05.3
1360
cog
Torque about rear axle: 0)(2)L-(Lmg- cog LFfront
jw Physics 2111 Understanding Physics
Chapter 12 16
Example: biceps lifts a weight; force on elbow?
mg N60
?mF
?elbowF
elbowF
jw Physics 2111 Understanding Physics
Chapter 12 17
Example: biceps lifts a weight; force on elbow?
0
34 30 0
530
about O gives F
F cm mg cm
F N
m
m
m
.
.
elbowF
mg N60
NF
mgFFF
elbow
elbowmy
470
0
jw Physics 2111 Understanding Physics
Chapter 12 18
R h
FN
Fe
mg
F when the wheel is about to go over the edgeN 0, .
Take torques about the edge F R h Mg x
trigonometry x R h R
:
:
02 2 2
F
Mg h R h
R h
2
Example: force to raise the wheel over the edge?
Forces on the wheel
jw Physics 2111 Understanding Physics
Chapter 12 19
T = ? Force on Hinge = ?
mg
Mg
T
FX
FY
T L M g L mgL
T M m g
T N
O O O
O
O
sin sin sin
sin
sin.
15 452
45 0
45
156 63 10
12
3
F T
F T Mg mgX
O
Y
O
cos
sin
30 0
30 0
F N
F NX
Y
574 10
596 10
3
3
.
.
Mm
mstrut=25kg
jw Physics 2111 Understanding Physics
Chapter 12 20
nut cracks @ 40 N on both sides; F ?
Torques about hinge:
F F
F Nnut 2 6 12 0
8 7
.
.
“Mechanical Advantage”: F
Fnut
4 6.