M13_Equilibrium, Elasticity and Fluid Mechanics.pptx

SFT 3033
MECHANICS
WEEK 13 : EQUILIBRIUM, ELASTICITY & FLUID
MECHANICS
Dr Anis Diyana Bt Halim

Copyright © 2012 Pearson Education Inc.
Introduction
• Many bodies, such as
bridges, aqueducts, and
ladders, are designed so they
do not accelerate.
• Real materials are not truly
rigid. They are elastic and do
deform to some extent.
• We shall introduce concepts
such as stress and strain to
understand the deformation
of real bodies.

Conditions for equilibrium (For extended body)
• First condition: The sum of all
the forces is equal to zero:
Fx = 0 Fy = 0 Fz = 0
• Second condition: The sum of
all torques about any given point
is equal to zero.

Center of Mass
• For a collection of particles with masses m1, m2, …..and coordinates (x1, y1, z1),
(x2,y2,z2) the coordinates xcm, ycm and zcm is given by :

Center of gravity
• We can treat a body’s
weight as though it all
acts at a single point
—
the center of gravity.
• If we can ignore the
variation of gravity
with altitude, the
center of gravity is the
same as the center of
mass.

Calculation : Walking the plank

Strain, stress, and elastic moduli
• Stretching, squeezing, and twisting a real body causes it to deform,
as shown in Figure below. We shall study the relationship between
forces and the deformations they cause.
• Stress is the force per unit area and strain is the fractional
deformation due to the stress. Elastic modulus is stress divided by
strain.
• The proportionality of stress and strain is called Hooke’s law.

Strain, stress, and elastic moduli
• The proportionality of stress and strain is called Hooke’s law.

Tensile and compressive stress and strain
• Tensile stress = F /A and tensile strain = l/l0. Compressive
stress and compressive strain are defined in a similar way. (See
Figures 11.13 and 11.14 below.)
• Young’s modulus is tensile stress divided by tensile strain, and is
given by Y = (F/A)(l0/l).

Some values of elastic moduli

Tensile stress and strain
• In many cases, a body can experience both tensile and
compressive stress at the same time, as shown in figure
below

Tensile stress and strain

Bulk stress and strain
• Pressure in a fluid is force
per unit area: p = F/A.
• Bulk stress is pressure
change p and bulk strain
is fractional volume change
V/V0. (See Figure
11.16.)
• Bulk modulus is bulk stress
divided by bulk strain and is
given by B =
–p/(V/V0).

Sheer stress and strain
• Sheer stress is F||/A and
sheer strain is x/h, as
shown in Figure 11.17.
• Sheer modulus is sheer
stress divided by sheer
strain, and is given by
S = (F||/A)(h/x).

Sheer stress and strain

INTRODUCTION
• Why must the shark keep moving to
stay afloat while the small fish can
remain at the same level with little
effort?
• We begin with fluids at rest and then
move on to the more complex field of
fluid dynamics.

DENSITY
• The density of a material is its mass per
unit volume:  = m/V.
• The specific gravity of a material is its
density compared to that of water at 4°C.

PRESSURE IN A FLUID
• The pressure in a fluid is the normal force per
unit area: p = dF/dA.

PRESSURE AT DEPTH IN A FLUID
• The pressure at a depth h in a fluid of uniform
density is given by P = P0 + gh. As Figure
12.6 at the right illustrates, the shape of the
container does not matter.
• The gauge pressure is the pressure above
atmospheric pressure. The absolute pressure is
the total pressure.

PASCAL’S LAW
• Pascal’s law: Pressure applied to an enclosed
fluid is transmitted undiminished to every
portion of the fluid and the walls of the
containing vessel.
• The hydraulic life shown an application of
Pascal’s law.

TWO TYPES OF PRESSURE GAUGE
• Figure 12.8 below shows two types of gauges for measuring
pressure.

ARCHIMEDES PRINCIPLE
• Archimedes’ Principle: When a body is completely or partially
immersed in a fluid, the fluid exerts an upward force (the “buoyant
force”) on the body equal to the weight of the fluid displaced by the
body. (See Figure 12.11 below.)

SURFACE TENSION
• The surface of a liquid
behaves like a membrane
under tension, so it resists
being stretched. This force is
the surface tension. (See
Figure 12.15 at the right.)
• The surface tension allows the
insect shown at the right to
walk on water.

FLUID FLOW
• The flow lines in the bottom figure
are laminar because adjacent layers
slide smoothly past each other.
• In the figure at the right, the upward
flow is laminar at first but then
becomes turbulent flow.

The continuity equation
• The figure at the right shows a flow
tube with changing cross-sectional area.
• The continuity equation for an
incompressible fluid is A1v1 = A2v2.
• The volume flow rate is dV/dt = Av.

Bernoulli’s equation
• Bernoulli’s equation is
p1 + gy1 + 1/2 v1
2
=
p2 + gy2 + 1/2 v2
2
• Refer to Figure 12.22 at the right.

Water pressure in the home

The Venturi meter

Viscosity and turbulence
• Viscosity is internal friction
in a fluid. (See Figures 12.27
and 12.28 at the right.)
• Turbulence is irregular
chaotic flow that is no longer
laminar. (See Figure 12.29
below.)

A curve ball (Bernoulli’s equation applied to sports)
• Does a curve ball really curve? Follow Conceptual Example
12.11 and Figure 12.30 below to find out.

THANK YOU

M13_Equilibrium, Elasticity and Fluid Mechanics.pptx

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