Chapter 12 Equilibrium and Elasticity

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Chapter 12 Equilibrium and Elasticity
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Equilibrium and Elasticity

• Equilibrium
• - Definition
• - Requirements
• - Static equilibrium
• II. Center of gravity
• III. Elasticity
• - Tension and
  compression
• - Shearing
• - Hydraulic stress

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I. Equilibrium
An object is in equilibrium if
- Definition
- The linear momentum of its center of mass is
constant.
- Its angular momentum about its center of mass
is constant
Example block resting on a table, hockey puck
sliding across a frictionless surface with
constant velocity, the rotating blades of a
ceiling fan, the wheel of a bike traveling across
a straight path at constant speed.
Objects that are not moving either in TRANSLATION
or ROTATION
- Static equilibrium
Example block resting on a table.
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Stable static equilibrium
If a body returns to a state of static
equilibrium after having been displaced from it
by a force ? marble at the bottom of a spherical
bowl.
Unstable static equilibrium
A small force can displace the body and end the
equilibrium
.
(1) Torque about supporting edge by Fg0 because
line of action of Fg through rotation axis ?
domino in equilibrium.
(1) Slight force ends equilibrium ? line of
action of Fg moves to one side of supporting edge
? torque due to Fg increases domino rotation.
(3) Not as unstable as (1) ? to topple it one
needs to rotate it through beyond balance
position in (1).
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.
- Requirements of equilibrium
Balance of forces ? translational equilibrium
Balance of torques ? rotational equilibrium
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- Equilibrium

• Vector sum of all external torques that act on
  the body, measured about any possible point must
  be zero.
• Vector sum of all external forces that act on
  body must be zero.

Balance of forces ? Fnet,x Fnet,y Fnet,z 0
Balance of torques ? tnet,x tnet,y tnet,z 0
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II. Center of gravity
Gravitational force on extended body ? vector sum
of the gravitational forces acting on the
individual bodys elements (atoms) .
cog Bodys point where the gravitational force
effectively acts.
The center of gravity is at the center of mass.
- This course initial assumption
Assumption valid for every day objects ? g
varies only slightly along Earths surface and
decreases in magnitude slightly with altitude.
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Each force Fgi produces a torque ti on the
element of mass about the origin O, with moment
arm xi.
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• A baseball player holds a 36-oz bat (weight
  10.0 N) with one hand at the point O . The bat is
  in equilibrium. The weight of the bat acts along
  a line 60.0 cm to the right of O. Determine the
  force and the torque exerted by the player on the
  bat around an axis through O.

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• A uniform beam of mass mb and length l supports
  blocks with masses m1 and m2 at two positions.
  The beam rests on two knife edges. For what value
  of x will the beam be balanced at P such that the
  normal force at O is zero?

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• A circular pizza of radius R has a circular
  piece of radius R/2 removed from one side as
  shown in Figure. The center of gravity has moved
  from C to C along the x axis. Show that the
  distance from C to C is R/6. Assume the
  thickness and density of the pizza are uniform
  throughout.

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• Pat builds a track for his model car out of
  wood, as in Figure. The track is 5.00 cm wide,
  1.00 m high and 3.00 m long and is solid. The
  runway is cut such that it forms a parabola with
  the equation
• Locate the horizontal coordinate of the center
  of gravity of this track.

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• Find the mass m of the counterweight needed to
  balance the 1 500-kg truck on the incline shown
  in Figure. Assume all pulleys are frictionless
  and massless.

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• A 20.0-kg floodlight in a park is supported at
  the end of a horizontal beam of negligible mass
  that is hinged to a pole, as shown in Figure. A
  cable at an angle of 30.0 with the beam helps to
  support the light. Find (a) the tension in the
  cable and (b) the horizontal and vertical forces
  exerted on the beam by the pole.

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States of Matter

• Matter is characterised as being solid, liquid or
  gas
• Solids can be thought of as crystalline or
  amorphous
• For a single substance it is normally the case
  that
• solid state occurs at lower temperatures than
  liquid state and
• liquid state occurs at lower temperatures than
  gaseous state

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Solids and Fluids

• Solids are what we have assumed all objects are
  up to this point (rigid, compact, unchanging,
  simple shapes)
• We will now look at bulk properties of particular
  solids and also at fluid
• Fluids include both liquids and gases
• fluids assume the shape of their container

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Deformation of Solids

• All states of matter (S,L,G) can be deformed
• it is possible to change the shape and volume of
  solids and
• the volumes of liquids and gases
• Any external force will deform matter
• for solids the deformation is usually small in
  relation to its overall size when everyday
  forces are applied

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Stress and Strain

• When force is removed, the object will usually
  return to its original shape and size
• Matter is elastic
• We characterise the elastic properties of solids
  in terms of stress (amount of force applied) and
  strain (extent of deformation) that occur.
• The amount of stress required to produce a
  particular amount of strain is a constant for a
  particular material
• this constant is called the elastic modulus

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Youngs Modulus (length elasticity)

• F creates a tensile stress of F/A.
• Units of tensile stress are Pascal
• 1 Pa 1 N/m2
• Change in length created
• by stress is
• DL/Lo (no unit)
• This quantity is the
• tensile strain

F
A
Lo
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Youngs modulus

• Youngs modulus is the ratio of tensile stress to
  tensile strain.
• Y has units of Pa
• Youngs modulus applies to rod or wire under
  tension (stretching) or compression

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What is happening in detail?

• Bonds between atoms are compressed or put in
  tension

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Elastic behaviour

• Notice that the Stress is proportional to the
  Strain
• This is similar to the relation we had between
  spring force and its extension
• F kx
• We can identify k with YA/Lo
• F kx so F/A kx/AYDL/Lo

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Limits of elastic behaviour
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Shear Modulus

• Shear modulus characterises a bodys deformation
  under a sideways (tangential) force

A
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Bulk Modulus

• Response of body to uniform squeezing
• Bulk modulus is ratio of the change in the normal
  force per unit area to the relative volume change

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Values for Y, S and B
Y (N/m2) S (N/m2) B (N/m2)
Al 7.0 x 1010 2.5 x 1010 7.0 x 1010
Water - - 0.21 x 1010
Tungsten 35 x 1010 14 x 1010 20 x 1010
Glass 7 x 1010 3 x 1010 5.2 x 1010

• Note that liquids do not have Y defined. S for
  liquids is called viscosity.
• In liquids and gases S and B are strongly
  dependent on temperature (more later)

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Example - Youngs modulus

• How much force must be applied to a 1 m long
  steel rod that has end area 1 cm2 to fit it
  inside a 0.999 m long case? (Ysteel20x1010N/m2)

1 cm2
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Density

• Density of a pure solid, liquid, or gas is its
  mass per unit volume
• Density r ? m/V (units are kg/m3)

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Pressure

• Pressure is the force per unit area
• P ? F/A
• Pressure at any point can be measured by
  placing a plate of known area in (e.g.) a liquid
  and measuring the force (compression) on a spring
  attached to it.