Elasticity

1
Elasticity

• Hookes Law
• Shear
• Elasticity of Volume
• Springs

2
Elasticity

• The property of a body by which it experiences a
  change in size or shape whenever a deforming
  force acts on the body.
• When the force is removed, the body returns to
  its original size and shape.
• Applies to all solids.

3
Atomic Nature of Solids

• Solids have lattice structure (3-D grid of
  atoms).
• Held together due to the interactions of their
  electrons.

4
Atomic Nature of Solids

• In equilibrium (no outside forces being applied),
  the net force of interactions is zero.
• The attractive and repulsive forces are balanced.
• The lattice is stable.

5
Atomic Nature of Solids

• If we stretch or compress the material, we move
  the atoms away from their equilibrium positions.
• The displacement is small but billions of atoms
  are displaced.
• This results in macroscopic lengthening or
  shortening.
• Once the force is removed, atoms return to
  equilibrium position and the material returns to
  its original length.

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Elasticity Factors Involved

• Force.
• Directly Proportional.

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Elasticity Factors Involved

• Cross-sectional area.
• The larger the area, the more atoms the force has
  to act over.
• Inversely proportional.

8
Elasticity Factors Involved

• Original length.
• If there are twice as many atoms to shift, the
  change in length will be twice as much.
• Directly proportional.

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Elasticity Factors Involved

• Putting it all together

10
Elasticity Factors Involved

• Commonly written as
• Ratio of applied force to cross-sectional area is
  call the STRESS on the wire.
• Ratio of the change in length to the original
  length of the wire is called the STRAIN.
• Stress is what is applied. Strain is the result.

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Types of Stresses

• Tensile Stress the bar, wire, pole, etc. is
  being stretched. Forces pull the ends of the
  object in opposite directions.
• Compressive Stress the bar, wire, pole, etc. is
  being compressed. Forces act inwardly on the
  body.

12
Elasticity Factors Involved

• To make the ratio into an equation, we must add a
  constant.
• Youngs modulus of elasticity (Y) is a measure of
  the stiffness of the material.
• The resulting equation

Thomas Young (1773-1829) Young was an English
physicist, physician, Egyptologist and
gynecologist. He is most famous for Youngs
double-slit experiment which displayed the
interference of light. This suggested that light
was composed of waves. He is sometimes
considered to be the last person to know
everything.
13
Hookes Law

• This equation of elasticity is called Hookes law
  of elasticity.
• Equation states that stress is proportional to
  strain.
• Applying twice the force results in twice the
  stretch.
• (We touched on this with the pendulum!)

Robert Hooke (1635-1703) Hooke was an English
Renaissance man whose studies included
mathematics, physics, biology, architecture, and
astronomy. He worked as an assistant of Robert
Boyle and possibly formally state Boyles Law.
(Boyle was not a mathematician.) In 1660, he
discovered Hookes law of elasticity. He also is
known for coining the biological term cell.
Hooke and Newton were heated enemies. In 1679,
Hooke wrote to Newton advocating the inverse
square law of gravitation but was unable to
formally prove it. Eight years later, Newton
published his Principia Mathematica and included
a proof of an inverse square law. He did not
credit Hooke.
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Elastic Limit

• We cannot stretch material indefinitely and stay
  in the elastic regime.
• Excess stress will permanently move atoms away
  from equilibrium.
• The material will experience some permanent
  stretching.

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Elastic Limit

• Elastic limit permanent stretching, object will
  not return to original length, stress and strain
  no longer proportional
• Ultimate stress highest point on stress-strain
  curve, greatest stress the material can bear,
  break shortly after
• Breaking point

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Example 1 Hookes Law

• A steel wire 1.00 m long with a diameter d 1.00
  mm has a 10.0-kg mass hung from it. (Youngs
  modulus for steel is 21 x 1010 N/m2.) (a)
  How much will the wire stretch? (b) What is the
  stress on the wire? (c) What is the strain?

(a)
(b)
(c)
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Types of Stresses

• Tensile Stress the bar, wire, pole, etc. is
  being stretched. Forces pull the ends of the
  object in opposite directions.
• Compressive Stress the bar, wire, pole, etc. is
  being compressed. Forces act inwardly on the
  body.
• Shear Stress the object has equal and opposite
  forces applied across its opposite faces
  (parallel to the surfaces).

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Elastic Deformation of Shape - Shear

• In addition to changing of length, elasticity
  applies to changing of shape.
• Shear stress is the application of equal and
  opposite forces parallel to the face of the
  object.
• Force results in torque but body is not free to
  rotate.
• Force applied to the body causes the layers of
  atoms to be displaced sideways.
• Body changes shape (shear).

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Elastic Deformation of Shape - Shear

• Angle of shear ( ) a measure of the
  deformation of the body (radians!).
• Shear modulus measures resistance to shear.
• Hookes law for elastic deformation of shape
• Tangential force divided by area of the top of
  the shape equals the shear modulus times the
  angle of shear (in radians).

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Types of Stresses

• Tensile Stress the bar, wire, pole, etc. is
  being stretched. Forces pull the ends of the
  object in opposite directions.
• Compressive Stress the bar, wire, pole, etc. is
  being compressed. Forces act inwardly on the
  body.
• Shear Stress the object has equal and opposite
  forces applied across its opposite faces
  (parallel to the surfaces).
• Pressure uniform force is exerted on all sides
  of an object.

21
Elasticity of Volume

• Another variety of elastic deformation is
  elasticity of volume.
• Uniform force is exerted on all sides of an
  object.
• Example object is submerged in water.
• Volume of object decreases.
• Hookes law for elasticity of volume includes the
  bulk modulus.

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Table of Constants
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Example 2 Elasticity of Volume

• A solid copper sphere of 0.500-m3 volume is
  placed 100 feet below the ocean surface where the
  pressure is 3.00 x 105 N/m2. What is the change
  in volume of the sphere? (The bulk modulus of
  copper is 14 x 1010 N/m2.)

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Types of Stresses
25
Springs

• Springs display the properties of Hookes law of
  elasticity, as well.
• Springs are considered massless and frictionless.
  
• Springs (like the pendulum) are examples of
  simple harmonic oscillators!

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Springs as SHOs

• Equilibrium position position of object if no
  force has been applied to the system.
• Displacement distance object is pulled from
  equilibrium.
• Amplitude maximum displacement.
• Restoring force force from spring on object
  which tries to bring object back to equilibrium
  position.

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Hookes Law for Springs

• F -kx.
• k is the spring constant in units of N/m. It
  is a measure of the stiffness of the spring.
• x is the displacement in units of m.
• F is the restoring force of the spring on the
  object in units of N.
• Proportional to the displacement.
• Opposite direction of the displacement.
• The force you must apply to stretch the spring is
  then F kx.

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Natural Frequency

• If you stretch a spring to any amplitude (within
  the elastic regime), it will oscillate with its
  natural frequency.
• Angular frequency
• Frequency/period
• Like all SHOs, we want to be able to describe the
  motion of the object using a sinusoidal wave

29
Springs and Work

• Your hand must do work to compress or stretch a
  spring.
• In the past, work force x distance.
• Problem our force is not constant! It varies
  with distance. We cannot use our old work
  equation.

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Springs and Work

• The answer to this problem is calculus! ?
• In reality, work is the integral of force with
  respect to displacement.

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Springs and Work

• First, lets look at the situation where the
  force is constant.
• The force in this situation can be represented by
  a constant value F.

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Springs and Work

• Now the situation of the spring where the force
  is not constant.
• In the case of the spring, F kx.

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Springs and Work

• For the spring, the work your hand does to
  displace the object is .
  
• Also, remember that the integral of a function
  gives the area under the curve.
• Using simple geometry, the area under the
  constant force curve is force x distance.
• Using simple geometry, the area under the F kx
  force curve is (½)(base)(height) (½)(x)(F)
  (½)(x)(kx) ½kx2.
• With this knowledge, we could find the work done
  by any force over any distance.

34
Springs and Potential Energy

• If we graph the work function, we end up with a
  parabola representing the energy of the spring as
  it oscillates.
• This function is typical of all simple harmonic
  oscillators.
• Amplitude give total energy of the system (all
  potential energy).
• System oscillates between kinetic and potential
  energy with the total energy remaining the same.

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Example 3 - Springs

• A ball of mass m 2.60 kg, starting from rest,
  falls a vertical distance h 55.0 cm before
  striking a vertical coiled spring. The ball
  compresses the spring y 15.0 cm. Determine the
  spring constant of the spring.