Loyola College Of Education: July 2017

DENSITY AND PRESSURE

The density of a material is defined as the mass per unit volume.

density = r = mass/Volume (the Greek letter rho)

The units are = kilograms/meter³ = kg/m³

It’s a measure of how tightly the atoms of a material are packed. It has nothing to do with the hardness of the material.

Examples:

Material	Density (kg/m³)
air	1.29
ice	917
water	1000
aluminum	2700
lead	11300
gold	19300

The specific gravity of a material is the ratio of its density to that of water. For example, the specific gravity of aluminum would be 2.7. This number is dimensionless.

The pressure, P, is defined as the ratio of force to area:

The units of pressure are: Newtons/meter² = N/m² = Pascals = Pa

Example:

A hammer supplies a force of 700 N. The hammer head has an area of 7.1 x 10^-4 m². What is the pressure?

P = F/A

P = (700 N/7.1 x 10^-4 m²)

P = 9.86 x 10⁵ N/m²

P = 9.86 x 10⁵ Pa

VARIATION OF PRESSURE WITH DEPTH

If a fluid is at rest, then all points at the same depth must be at the same pressure (otherwise it would be moving!). However the pressure WILL vary with depth: it will have the weight of the fluid on top of it.

P = pressure, P_o = pressure due to the air (atmospheric pressure)

Because the fluid is at rest, the net force will be _________ ?

Pressure at a point within a fluid

Consider a fluid at rest as shown. From around the point of interest, P in the fluid let us pull out a small wedge of dimensions dx x dz x ds . Let the depth normal to the plane of paper be b. In some of the derivations we chose z to be the vertical coordinate. This is consistent with the use of z as the elevation or height in many applications involving atmosphere or an ocean. Let us now mark the surface and body forces acting upon the wedge.

Pressure at a point

The surface forces acting on the three faces of the wedge are due to the pressures,

and

as shown. These forces are normal to the surface upon which they act. We follow the usual convention that compression pressure is positive in sign. We again remind ourselves that since the fluid is at rest there is no shear force acting. In addition we have a body force, the weight, W of the fluid within the wedge acting vertically downwards.

Summing the horizontal and the vertical forces we have,

$\displaystyle \Sigma F_x$	$\displaystyle =0;~ \texttt{ie.}, p_x ~ dz~b ~-~ p_n ~ b ~ ds ~ sin \theta = 0$
$\displaystyle \Sigma F_z$	$\displaystyle =0;~ \texttt{ie.}, p_z ~ dx~b ~-~ p_n ~ b ~ ds ~ cos \theta~-~W ~~= ~~0$
$\displaystyle \texttt{i.e.,}$	$\displaystyle = p_z ~ dx~b ~-~ p_n ~ b ~ ds ~ cos \theta~-~{1 \over 2} \rho~g ~b ~dx~ dz ~~= ~~0$

Noting that

$\displaystyle ds~sin\theta~=~dz,~~\texttt{and }~~~ ds~cos\theta~=~dx$

we have after simplification,

$\displaystyle p_x~=~p_n,~~~\texttt{and}~~~p_z~=~p_n~+~{1 \over 2}~ \rho~g~dz$

We note that the pressure in the horizontal direction does not change, which is a consequence of the fact that there is shear in a fluid at rest. In the vertical direction there is a change in pressure proportional to density of the fluid, acceleration due to gravity and difference in elevation.

Now if we take the limit as the wedge volume decreases to zero, i.e., the wedge collapses to the point P, we have,

$\displaystyle p_x~=~p_z~=p_n~=~p$

This equation is known as Pascal's Law. It is important to note that it is valid only for a fluid at rest. In the case of a moving fluid, pressures in different directions could be different depending upon fluid accelerations in different directions. Hence, for a moving fluid pressure is defined as an average of the three normal stresses acting upon the fluid element.

Archimedes principle

Archimedes’ principle

Archimedes’ principle, physical law of buoyancy, discovered by the ancient Greek mathematician and inventor Archimedes, stating that any body completely or partially submerged in a fluid (gas or liquid) at rest is acted upon by an upward, or buoyant, force the magnitude of which is equal to the weight of the fluid displaced by the body. The volume of displaced fluid is equivalent to the volume of an object fully immersed in a fluid or to that fraction of the volume below the surface for an object partially submerged in a liquid. The weight of the displaced portion of the fluid is equivalent to the magnitude of the buoyant force. The buoyant force on a body floating in a liquid or gas is also equivalent in magnitude to the weight of the floating object and is opposite in direction; the object neither rises nor sinks. For example, a ship that is launched sinks into the ocean until the weight of the water it displaces is just equal to its own weight. As the ship is loaded, it sinks deeper, displacing more water, and so the magnitude of the buoyant force continuously matches the weight of the ship and its cargo.

Illustration of Archimedes’ principle of buoyancy. Here a 5-kg object immersed in water is shown …

Encyclopædia Britannica, Inc.

If the weight of an object is less than that of the displaced fluid, the object rises, as in the case of a block of wood that is released beneath the surface of water or a helium-filled balloon that is let loose in air. An object heavier than the amount of the fluid it displaces, though it sinks when released, has an apparent weight loss equal to the weight of the fluid displaced. In fact, in some accurate weighings, a correction must be made in order to compensate for the buoyancy effect of the surrounding air.

The buoyant force, which always opposes gravity, is nevertheless caused by gravity. Fluid pressure increases with depth because of the (gravitational) weight of the fluid above. This increasing pressure applies a force on a submerged object that increases with depth. The result is buoyancy.

Tuesday, 18 July 2017

DENSITY AND PRESSURE

Pressure at a point within a fluid

Pressure at a point within a fluid

Archimedes principle