Post on 26-Dec-2015
Chapter 9Static 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
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
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
Two Conditions of Equilibrium
Objects in mechanical equilibrium
Rock on ridge
See-saw
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
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
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
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
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
Examples of Objects in Equilibrium
Example:
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.
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)
Elastic Properties Stress is the force per unit area causing the
deformation
Strain is a measure of the amount of deformation
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
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
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
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·
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
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
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
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
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
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
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
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
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
Stress and Strain
Example: