Physics Tutorials

Capillary Rise

2008-09-03T21:21:00.000-07:00

Capillary action, capillarity, capillary motion, or wicking is the ability of a substance to draw another substance into it. The standard reference is to a tube in plants but can be seen readily with porous paper. It occurs when the adhesive intermolecular forces between the liquid and a substance are stronger than the cohesive intermolecular forces inside the liquid. The effect causes a concave meniscus to form where the substance is touching a vertical surface. The same effect is what causes porous materials such as sponges to soak up liquids.

A common apparatus used to demonstrate capillary action is the capillary tube. When the lower end of a vertical glass tube is placed in a liquid such as water, a concave meniscus forms. Surface tension pulls the liquid column up until there is a sufficient mass of liquid for gravitational forces to overcome the intermolecular forces. The weight of the liquid column is proportional to the square of the tube's diameter, but the contact length (around the edge) between the liquid and the tube is proportional only to the diameter of the tube, so a narrow tube will draw a liquid column higher than a wide tube.

For example, a glass capillary tube 0.5 mm in diameter will lift approximately a 2.8 mm column of water.

Examples

In hydrology, capillary action describes the attraction of water molecules to soil particles. Capillary action is responsible for moving groundwater from wet areas of the soil to dry areas. Differences in soil matric potential (Ψm) drive capillary action in soil.

Stoke's Law

2008-08-20T19:52:00.000-07:00

In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force — also called drag force — exerted on spherical objects with very small Reynolds numbers (e.g., very small particles) in a continuous viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the generally unsolvable Navier–Stokes equations

where:
Fd is the frictional force (in N),
μ is the fluid's dynamic viscosity (in Pa s),
R is the radius of the spherical object (in m), and
V is the particle's velocity (in m/s).
If the particles are falling in the viscous fluid by their own weight due to gravity, then a terminal velocity, also known as the settling velocity, is reached when this frictional force combined with the buoyant force exactly balance the gravitational force. The resulting settling velocity (or terminal velocity) is given by:

where:
Vs is the particles' settling velocity (m/s) (vertically downwards if ρp > ρf, upwards if ρp < ρf ), g is the gravitational acceleration (m/s2), ρp is the mass density of the particles (kg/m3), and ρf is the mass density of the fluid (kg/m3). Note that for molecules Stokes' law is used to define their Stokes radius.

Surface Energy and Surface Tension:

2008-08-04T21:21:00.001-07:00

Surface energy quantifies the disruption of intermolecular bonds that occurs when a surface is created. In the physics of solids, surfaces must be intrinsically less energetically favourable than the bulk of a material; otherwise there would be a driving force for surfaces to be created, and surface is all there would .The surface energy may therefore be defined as the excess energy at the surface of a material compared to the bulk.
For a liquid, the surface tension (force per unit length) and the surface energy density are identical. Water, a special case, has a surface energy density of 0.08 J/m2 and a surface tension of 0.08 N/m. However, in general, the surface energy of a solid is not equal to its surface tension.

Cutting a solid body into pieces disrupts its bonds, and therefore consumes energy. If the cutting is done reversibly then conservation of energy means that the energy consumed by the cutting process will be equal to the energy inherent in the two new surfaces created. The unit surface energy of a material would therefore be half of its energy of cohesion, all other things being equal; in practice, this is true only for a surface freshly prepared in vacuum. Surfaces often change their form away from the simple "cleaved bond" model just implied above. They are found to be highly dynamic regions, which readily rearrange or react, so that energy is often reduced by such processes as passivation or adsorption.

Measuring the surface energy of a liquid:
As first described by Thomas Young in 1805 in the Philosophical Transactions of the Royal Society of London, it is the interaction between the forces of cohesion and the forces of adhesion which determines whether or not wetting, the spreading of a liquid over a surface, occurs. If complete wetting does not occur, then a bead of liquid will form, with a contact angle which is a function of the surface energies of the system.

Contact angle and surface energy measurements can be made using a contact angle goniometer.
Surface energy is most commonly quantified using a contact angle goniometer and a number of different methods.
Thomas Young described surface energy as the interaction between the forces of cohesion and the forces of adhesion which, in turn, dictate if wetting occurs. If wetting occurs, the drop will spread out flat. In most cases, however, the drop will bead to some extent and by measuring the contact angle formed where the drop makes contact with the solid the surface energies of the system can be measured.

Young's equation
Young established the well-regarded Young's Equation which defines the balances of forces caused by a wet drop on a dry surface. If the surface is hydrophobic then the contact angle of a drop of water will be larger. Hydrophilicity is indicated by smaller contact angles and higher surface energy. (Water has rather high surface energy by nature; it is polar and forms hydrogen bonds). The Young equation gives the following relation,

where γSL, γLV, and γSV are the interfacial tensions between the solid and the liquid, the liquid and the vapor, and the solid and the vapor, respectively.

The equilibrium contact angle that the drop makes with the surface is denoted by θc. To derive the Young equation, normally the interfacial tensions are described as forces per unit length and from the one-dimensional force balance along the x axis Young equation is obtained.
The Young equation assumes a perfectly flat surface, and in many cases surface roughness and impurities cause a deviation in the equilibrium contact angle from the contact angle predicted by Young's equation. Even in a perfectly smooth surface a drop will assume a wide spectrum of contact angles ranging from the so called advancing contact angle, θA, to the so called receding contact angle, θR. The equilibrium contact angle (θc) can be calculated from θA and θR as was shown by Tadmor as,

where

Bouyancy and Archimedes' Principle

2008-08-02T23:55:00.000-07:00

Bouyancy:
Buoyancy is the upward force on an object produced by the surrounding liquid or gas in which it is fully or partially immersed, due to the pressure difference of the fluid between the top and bottom of the object. The net upward buoyancy force is equal to the magnitude of the weight of fluid displaced by the body. This force enables the object to float or at least to seem lighter. Buoyancy is important for many vehicles such as boats, ships, balloons, and airships, and plays a role in diverse natural phenomena such as sedimentation.

Archimedes' principle
It is named after Archimedes of Syracuse, who first discovered this law. Vitruvius (De architectura IX.9–12) recounts the famous story of Archimedes making this discovery while in the bath (for which see eureka) but the actual record of Archimedes' discoveries appears in his two-volume work, On Floating Bodies.

This is true only as long as one can neglect the surface tension (capillarity) acting on the body.
The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (specifically if the surrounding fluid is of uniform density). Thus, among objects with equal masses, the one with greater volume has greater buoyancy.
Suppose a rock's weight is measured as 10 newtons when suspended by a string in a vacuum. Suppose that when the rock is lowered by the string into water, it displaces water of weight 3 newtons. The force it then exerts on the string from which it hangs will be 10 newtons minus the 3 newtons of buoyant force: 10 − 3 = 7 newtons. This same principle even reduces the apparent weight of objects that have sunk completely to the sea floor, such as the sunken battleship USS Arizona at Pearl Harbor, Hawaii. It is generally easier to lift an object up through the water than it is to finally pull it out of the water. And, it also works with boiled eggs, salt, and fresh water.
The density of the immersed object relative to the density of the fluid is easily calculated without measuring any volumes:

Pascal’s Law

2008-07-31T07:41:00.000-07:00

Pascal's law or Pascal's principle states that "a change in the pressure of an enclosed incompressible fluid is conveyed undiminished to every part of the fluid and to the surfaces of its container."
The difference of pressure due to a difference in elevation within a fluid column

is given by:

where, using SI units,
ΔP is the hydrostatic pressure (in pascals), or the difference in pressure at two points within a fluid column, due to the weight of the fluid;
ρ is the fluid density (in kilograms per cubic meter);
g is sea level acceleration due to Earth's gravity (in meters per second square);
Δh is the height of fluid above (in meters), or the difference in elevation between the two points within the fluid column.
The intuitive explanation of this formula is that the change in pressure between two elevations is due to the weight of the fluid between the elevations.
Note that the variation with height does not depend on any additional pressures. Therefore Pascal's law can be interpreted as saying that any change in pressure applied at any given point of the fluid is transmitted undiminished throughout the fluid.

Applications
Pascal's principle underlies the hydraulic press.
Used in artesian wells, water towers, and dams.
'Pascal's burst barrel demonstration': a long and narrow vertical pipe is connected to the contents of a large, sealed barrel. Adding water to the pipe increases the pressure throughout the system. Adding a small amount of water to the pipe is enough to burst the barrel. Scuba divers must understand this principle. At a depth of 10 meters under water, pressure is twice the atmospheric pressure at sea level, and increases by about 100 kPa for each increase of 10 m depth.
Atmospheric pressure diminishes with height, a fact first verified on the Puy-de-Dôme and the Saint-Jacques Tower in Paris, on the instigation of Blaise Pascal himself.

Young's Modulus

2008-07-30T07:20:00.000-07:00

Young's modulus (E) is a measure of the stiffness of a material. It is also known as the Young modulus, modulus of elasticity, elastic modulus (though the Young's modulus is actually one of several elastic moduli such as the bulk modulu and the shear modulus) or tensile modulus. It is defined as the ratio of stres over strain in the region in which Hooke's Law is obeyed for the material. This can be experimentally determined from the slope of a stress-strain curve created during tensile tests conducted on a sample of the material.
Young's modulus is named after Thomas Young, the 18th century British scientist.

Units
Young's modulus is the ratio of stress, which has units of pressure, to strain, which is dimensionless; therefore Young's modulus itself has units of pressure.
The SI unit of modulus of elasticity (E, or less commonly Y) is the pascal.

Usage
The Young's modulus allows the behavior of a material under load to be calculated. For instance, it can be used to predict the amount a wire will extend under tension or buckle under compression. Some calculations also require the use of other material properties, such as the shear modulus, density, or Poisson's ratio.
Linear vs non-linear
For many materials, Young's modulus is essentially constant over a range of strains. Such materials are called linear, and are said to obey Hooke's law. Examples of linear materials include steel, carbon fiber, and glass. Rubber and soils (except at very small strains) are non-linear materials.
Directional materials
Most metals and ceramics, along with many other materials, are isotropic: their mechanical properties are the same in all directions. However metals and ceramics can be treated with certain impurities to give them a “grain”. The grain of these, and other composites of two or more ingredients, is a mechanical structure of various orientations and sizes, which makes the material anisotropic. This means that Young's modulus will change depending on which direction the force is applied from. As a result, these anisotropic materials have different mechanical properties when load is applied in different directions. For example, carbon fiber is much stiffer (higher Young's modulus) when loaded parallel to the fibers (along the grain), and is an example of a material with transverse isotropy. Other such materials include wood and reinforced concrete. Engineers can use this directional phenomenon to their advantage in creating various structures in our environment.
Copper as an excellent electrical conductor is used to transmit electricity over long distance cables, however although copper has a relatively high value for Young's modulus at 130 GPa, it has a very low value for yield strength, and thus easily deforms in tension. When the copper cable is co-wound with hardened steel wire the stretching can largely be prevented, as the steel (with a higher value of Young's modulus in tension and much higher yield strength) takes up the tension that the copper would otherwise experience.
Calculation
Young's modulus, E, can be calculated by dividing the tensile stress by the tensile strain:
where

E is the Young's modulus (modulus of elasticity)
F is the force applied to the object;
A0 is the original cross-sectional area through which the force is applied;
ΔL is the amount by which the length of the object changes;
L0 is the original length of the object.

Example:

Pressure in a fluid:
Fluid pressure is the pressure at some point within a fluid, such as water or air.
Fluid pressure occurs in one of two situations:
1. an open condition, such as the ocean, a swimming pool, or the atmosphere; or
2. a closed condition, such as a water line or a gas line.
Pressure in open conditions usually can be approximated as the pressure in "static" or non-moving conditions (even in the ocean where there are waves and currents), because the motions create only negligible changes in the pressure. Such conditions conform with principles of fluid statics. The pressure at any given point of a non-moving (static) fluid is called the hydrostatic pressure.
Closed bodies of fluid are either "static," when the fluid is not moving, or "dynamic," when the fluid can move as in either a pipe or by compressing and air gap in a closed container. The pressure in closed conditions conforms with the principles of fluid dynamics.
The concepts of fluid pressure are predominantly attributed to the discoveries of Blaise Pascal and Daniel Bernoulli.

Hooke's law

2008-07-20T07:28:00.000-07:00

Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress).

Materials for which Hooke's law is a useful approximation are known as linear-elastic or "Hookean" materials.

Hooke's law is named after the 17th century British physicist Robert Hooke.
For systems that obey Hooke's law, the extension produced is directly proportional to the load

F=-kx
where:
x is the distance that the spring has been stretched or compressed away from the equilibrium position, which is the position where the spring would naturally come to rest (usually in meters),
F is the restoring force exerted by the material (usually in newtons), and
k is the force constant (or spring constant). The constant has units of force per unit length (usually in newtons per meter).

When this holds, we say that the behavior is linear. If shown on a graph, the line should show a direct variation. There is a negative sign on the right hand side of the equation because the restoring force always acts in the opposite direction of the x displacement (when a spring is stretched to the left, it pulls back to the right).