Solids and Fluids
Solids
Solid is a state of matter with a fixed volume and fixed shape.
The effect of force on a solid is either to change motion or shape. The change
of motion aspect of it has been described in previous chapters. The change of
shape aspect of it will be discussed in this chapter.
Physics of Deformation
There are two physical quantities used to
describe deformation of an object. These are stress and strain. Stress
is a measure of a force's ability to deform an object. It is proportional to the
magnitude of the force and inversely proportional to the area of the surface
upon which the force is applied. It is defined to be the ratio between the force
( F ) and the area of the surface ( A ) over which the force is
applied.
Stress = F ⁄ A
The unit of stress is N ⁄ m 2 which is
defined to be the Pascal abbreviated as Pa. Strain is a physical quantity
used as a measure of deformation. It is defined to be the ratio between the
change and the original value. For example if the change is change in length,
strain is defined to be ratio between the change in length and the original
length. Since strain is ratio between the same physical quantitities, it is unit-less.
Relationship between Stress and Strain
Stress and strain are directly
proportional as stress is increased from zero to a certain value. At a certain
value the proportionality between stress and strain ceases to apply. This point
where the proportionality between stress and strain breaks down is called the
elastic limit. As the stress is increased further beyond the elastic
limit, at a certain value the material breaks down. This point where the
material breaks down is called the breaking point of the material. The constant
of proportionality between stress and strain is called modulus.
modulus = stress ⁄ strain
The unit of measurement for modulus is Pascal.
There are three kinds of stresses. These are tensile stress, shear stress and
bulk stress. Tensile stress is a stress where the force is applied
parallel to the length of the material and perpendicular to the cross-sectional area
of the material. The deformation is change in length and the strain is defined
to be the ratio between the change in length ( ΔL ) and the original
length ( L ). The modulus associated with this kind of stress is called
Young's modulus ( Y ).
Y = ( F ⁄ A ) ⁄ ( ΔL ⁄ L)
Shear stress is a stress where the force is applied parallel to the
surface. The effect of this stress is to produce deformation parallel to the
surface. Its strain is defined to be the ratio between the deformation parallel
to the surface ( x ) and the height ( h ) of the material in
a direction perpendicular to this surface. The modulus associated with this kind
of stress is called shear modulus ( S ).
S = (F ⁄ A ) ⁄ ( x ⁄ h)
Bulk stress is a stress where the stress is applied perpendicularly
over the entire surface area of an object. The effect of this stress is to bring
change in volume and its strain is defined to be the ratio between the change in
volume ( ΔV ) and the original volume ( V ). The modulus
associated with this kind of stress is called bulk modulus ( B ).
B = (-F ⁄ A ) ⁄ ( ΔV ⁄ V)
The negative is introduced to make the bulk modulus positive, since the
change in volume is negative; that is a decrease.
Fluids
Fluid is a state of matter with a fixed volume but not
fixed shape. It takes the shape of its container. The density ( ρ ) of
fluid is defined to be the ratio between its mass ( m ) and its volume (
V ).
ρ = m ⁄ V
The unit of measurement for density is kg ⁄ m
3.
Fluid Statics
Fluid statics is the study of fluids at rest. There
can not be shear stress on a fluid at rest because if there were, the molecules
of the fluid would be moving. At any volume element (part) of the fluid the
forces act over the entire surface area of the volume element perpendicularly.
The force ( F ) per unit area ( A ) is called pressure ( P
).
P = F ⁄ A
Unit of measurement for pressure is Pascal. The pressure due to air molecules
is called atmospheric pressure. The value of atmospheric pressure at sea
level is equal to 1.013e5 Pa. Atmospheric pressure decreases with the
increase of altitude. A unit of pressure called atm is defined to be equal to
atmospheric pressure at sea level.
Dependence of Fluid Pressure on Depth
Let's consider a part of a fluid
in rest in cylinderical shape with the base of the cylinder parallel to the
surface of the fluid. Let the base area of the cylinder be A and its
height h. Since the fluid is at rest, this cylinder is in equilibrium and
hence the net force acting on it must be zero. The forces acting on the cylinder
are its weight and the force due to pressure difference between that at the
bottom surface ( P ) and that at the top surface (
P0 ). The direction of the force due to
pressure difference must be upwards since the direction of weight is downwards.
The force due to pressure difference is equal to (P -
Po )A. The weight of the cylinder is m|g|.
Its mass is equal to the product of the density ( ρ ) of the fluid and its
volume ( V ) and its volume is equal to the product of its base area (
A ) and its height ( h ). Therefore the weight of the cylinder is
equal to Ahρ|g| . Equating its weight to the force due to pressure
difference, the following equation for the dependence of pressure on depth can
be obtained.
P = Po + ρ|g|h
Po is pressure at the upper level and
P is pressure at the lower level. h is the separation between both
levels. Pressure increases with depth linearly.
Measuring Pressure
Pressure is measured by a device called
manometer. A manometer is essentially a U shaped tube filled with a fluid
( most of the time mercury). Fom the knowledge of the pressure on one side of
the U tube and the difference of the fluid levels on both sides, an unknown
pressure on the other side of the tube can be calculated. There are two types of
manometers. These are the closed manometer and the open manometer.
An open manometer is a manometer with one of the sides of the U tube
open (exposed to air molecules). Thus the pressure at the surface of the fluid
on this side of the tube is equal to atmospheric pressure. The gas of unknown
pressure is connected to the other side of the tube and the difference between
the fluid levels on both sides measured. Let atmospheric pressure be represented
by Pat and the unknown gas pressure be denoted
by Pg. If the level of the fluid on the gas
side is lower than the other side, then Pg = P
(pressure at lower level) and Pat =
Po ( pressure at higher level ). Therefore,
Pg =
Pat + ρ|g|h
where ρ is the density of the fluid and h is the separation
between the fluid levels on both sides. Similarly if the fluid level at the air
side is lower than the fluid level at the gas side,
Pg = Po and
Pat = P; and thus
Pg =
Pat - ρ|g|h
A closed manometer is a manometer with one side closed. The pressure
on the surface of the fluid on the closed side is approximately zero because it
is made to be approximately vacuum. Thus the upper level pressure is zero and the
lower level pressure is the gas pressure.
Pg = ρ|g|h
Measuring Atmospheric Pressure
Atmospheric pressure is measured by a
device called barometer. A barometer is essentially a closed manometer. A
closed tube is made to be approximately vacuum and inserted in a dish of mercury. The
pressure on the closed tube is approximately zero. The mercury in the dish is
exposed to atmospheric pressure. Because of the pressure difference, the mercury
rises to a height h. The upper level pressure is zero and the lower level
pressure is atmospheric pressure ( Pat ). Therefore,
Pat = ρ|g|h