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Chapter 12 Equilibrium and Elasticity

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Chapter 12 Equilibrium and Elasticity Equilibrium and Elasticity Equilibrium - Definition - Requirements - Static equilibrium II. Center of gravity III. – PowerPoint PPT presentation

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Title: Chapter 12 Equilibrium and Elasticity


1
Chapter 12 Equilibrium and Elasticity
2
Equilibrium and Elasticity
  • Equilibrium
  • - Definition
  • - Requirements
  • - Static equilibrium
  • II. Center of gravity
  • III. Elasticity
  • - Tension and
    compression
  • - Shearing
  • - Hydraulic stress

3
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.
4
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).
5
.
- Requirements of equilibrium
Balance of forces ? translational equilibrium
Balance of torques ? rotational equilibrium
6
- 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
7
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.
8
Each force Fgi produces a torque ti on the
element of mass about the origin O, with moment
arm xi.
9
  • 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.

10
  • 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?

11
  • 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.

12
  • 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.

13
  • 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.

14
  • 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.

15
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

16
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

17
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

18
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

19
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
20
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

21
What is happening in detail?
  • Bonds between atoms are compressed or put in
    tension

22
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

23
Limits of elastic behaviour
24
Shear Modulus
  • Shear modulus characterises a bodys deformation
    under a sideways (tangential) force

A
25
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

26
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)

27
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
28
Density
  • Density of a pure solid, liquid, or gas is its
    mass per unit volume
  • Density r ? m/V (units are kg/m3)

29
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.
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