Physics 201: Lecture 20

1
Physics 201 Lecture 20

• Statics
• Stable and Unstable Equilibrium
• Elastic Properties of Solids
• Moduli
• Tensile (Youngs)
• Shear
• Bulk

2
Statics

• Make a diagram and the indicate all the forces
• as vectors

• In general, we can use the two equations
• to solve any statics problem.
• When choosing axes about which to calculate
  torque,
• the choice can make the problem easy....

3
Example Ladder against smooth wall

• Bill (mass M) is climbing a ladder (length L,
  mass m) that leans against a smooth wall (no
  friction between wall and ladder). A frictional
  force F between the ladder and the floor keeps it
  from slipping. The angle between the ladder and
  the wall is ?.
• What is the magnitude of F as a function of
  Bills distance up the ladder?

?
L
m
Bill
F
4
Ladder against smooth wall...

• Consider all of the forces acting. In addition to
  gravity and friction, there will be normal forces
  Nf and Nw by the floor and wall respectively on
  the ladder.

Nw
L/2
?

• Again use the fact that FNET 0
  in both x and y directions
• x Nw F
• y Nf Mg mg

m
mg
d
Mg
F
Nf
5
Ladder against smooth wall...

• Since we are not interested in Nw, calculate
  torques about an axis through the top end of the
  ladder, in the z direction.

axis
Nw
L/2
?
m

• Substituting Nf Mg mg andsolving for F

mg
a
d
Mg
F
Nf
a
6
Example Ladder against smooth wall...

• For a given coefficient of static friction
  ?s,the maximum force of friction F that can
  beprovided is ?sNf ?s g(M m).
• The ladder will slip if F exceedsthis value.

?
m

• Moral
• Brace the bottom of ladders!
• Dont make ? too big!

d
F
7
Potential Energy Diagrams

• Consider a block sliding on a frictionless
  surface, attached to an ideal spring.
• F -dU/dx -slope

F
x
U
F
x
x
0
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Equilibrium

• F -dU/dx -slope
• So F 0 if slope 0.
• This is the case at the minimum or maximum of
  U(x).
• This is called an equilibrium position.
• If we place the block at rest at x 0, it wont
  move.

m
x
U
x
0
9
Equilibrium

• If small displacements from the equilibrium
  position result in a force that tends to move the
  system back to its equilibrium position, the
  equilibrium is said to be stable.
• This is the case if U is a minimum at the
  equilibrium position.
• In calculus language, the equilibrium is stable
  if the curvature (second derivative) is positive.

F
m
x
U
F
x
0
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Equilibrium
U

• Suppose U(x) looked like this
• This has two equilibrium positions, one is
  stable ( curvature) and one is unstable (-
  curvature).
• Think of a small object sliding on the U(x)
  surface
• If it wants to keep sliding when you give it a
  little push, the equilibrium is unstable.
• If it returns to the equilibrium position when
  you give it a little push, the equilibrium is
  stable.
• If the curvature is zero (flat line) the
  equilibrium is neutral.

unstable
neutral
stable
x
0
11
Solids

• Have definite volume
• Have definite shape
• Molecules are held in specific locations
• by electrical forces
• vibrate about equilibrium positions
• Can be modeled as springs connecting molecules

12
Liquid

• Has a definite volume
• No definite shape
• Exist at a higher temperature than solids
• The molecules wander through the liquid in a
  random fashion
• The intermolecular forces are not strong enough
  to keep the molecules in a fixed position

13
Gas

• Has no definite volume
• Has no definite shape
• Molecules are in constant random motion
• The molecules exert only weak forces on each
  other
• Average distance between molecules is large
  compared to the size of the molecules

14
Question

• Are atoms in a solid always arranged in an
  ordered structure?
• Yes
• No

15
Deformation of Solids

• All objects are deformable, i.e It is possible to
  change the shape or size (or both) of an object
  through the application of external forces
• Sometimes when the forces are removed, the object
  tends to its original shape, called elastic
  behavior
• Large enough forces will break the bonds between
  molecules and also the object

16
Elastic Properties

• Stress is related to the force causing the
  deformation
• Strain is a measure of the degree of deformation
• The elastic modulus is the constant of
  proportionality between stress and strain
• For sufficiently small stresses, the stress is
  directly proportional to the strain
• The constant of proportionality depends on the
  material being deformed and the nature of the
  deformation
• The elastic modulus can be thought of as the
  stiffness of the material

17
Youngs Modulus Elasticity in Length

• Tensile stress is the ratio of the external force
  to the cross-sectional area
• For both tension and compression
• The elastic modulus is called Youngs modulus
• SI units of stress are Pascals, Pa
• 1 Pa 1 N/m2
• The tensile strain is the ratio of the change in
  length to the original length
• Strain is dimensionless

18
Shear Modulus Elasticity of Shape

• Forces may be parallel to one of the objects
  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
• A material having a large shear modulus is
  difficult to bend

19
Beams
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Bulk Modulus Volume Elasticity

• Bulk modulus characterizes the response of an
  object to uniform squeezing
• Suppose the forces are perpendicular to, and acts
  on, all the surfaces -- as when an object is
  immersed in a fluid
• The object undergoes a change in volume without a
  change in shape

• Volume stress, DP, is the ratio of the force to
  the surface area
• This is also the Pressure
• The volume strain is equal to the ratio of the
  change in volume to the original volume

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Notes on Moduli

• Solids have Youngs, Bulk, and Shear moduli
• Liquids have only bulk moduli, they will not
  undergo a shearing or tensile stress
• The liquid would flow instead
• The negative sign is included since an increase
  in pressure will produce a decrease in volume --
  B is always positive
• The compressibility is the reciprocal of the bulk
  modulus

Ultimate Strength of Materials

• The ultimate strength of a material is the
  maximum stress the material can withstand before
  it breaks or factures
• Some materials are stronger in compression than
  in tension
• Linear to the Elastic Limit

22
Arches

• Which of the following two archways can you
  build bigger, assuming that the same type of
  stone is available in whatever length you desire?
• Post-and-beam (Greek) arch
• Semicircular (Roman) arch
• You can build big in either type

• Low ultimate tensile strength of sagging stone
  beams

Stability depends upon the compression of the
wedge-shaped stones
23
Gothic Arch

• First used in Europe in the 12th century
• Extremely high
• The flying buttresses are needed to prevent the
  spreading of the arch supported by the tall,
  narrow columns

24
Problem
A plastic box is sliding across a horizontal
floor. The frictional force between the box and
the floor causes the box to deform. To describe
the relationship between stress and strain for
the box, you would use

• Youngs modulus
• Shear modulus
• Bulk modulus
• None of the above

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Solution
FORCE
MOTION
FRICTION
SO THE SHEAR MODULUS IS THE CHOICE!
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Moduli Values
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Prestressed Concrete

• If the stress on a solid object exceeds a certain
  value, the object fractures
• The slab can be strengthened by the use of steel
  rods to reinforce the concrete
• The concrete is stronger under compression than
  under tension
• A significant increase in shear strength is
  achieved if the reinforced concrete is
  prestressed
• As the concrete is being poured, the steel rods
  are held under tension by external forces
• These external forces are released after the
  concrete cures
• This results in a permanent tension in the steel
  and hence a compressive stress on the concrete
• This permits the concrete to support a much
  heavier load