Sunday, September 2, 2007
2ND QUARTER OF PHYSICS-IV
Static is the branch of physics concerned with the analysis of loads (force, torque/moment) on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at rest under the action of external forces of equilibrium. When in static equilibrium, the system is either at rest, or moving at constant velocity through its center of mass.
By Newton's second law, this situation implies that the net force and net torque (also known as moment) on every body in the system is zero, meaning that for every force bearing upon a member, there must be an equal and opposite force. From this constraint, such quantities as stress or pressure can be derived. The net forces equalling zero is known as the first condition for equilibrium, and the net torque equalling zero is known as the second condition for equilibrium. See statically determinate.
Statics is thoroughly used in the analysis of structures, for instance in architectural and structural engineering. Strength of materials is a related field of mechanics that relies heavily on the application of static equilibrium.
Hydrostatics, also known as fluid statics, is the study of fluids at rest. This analyzes systems in static equilibrium which involve forces due to mechanical fluids. The characteristic of any fluid at rest is that the force exerted on any particle of the fluid is the same in every direction. If the force is unequal the fluid will move in the direction of the resulting force. This concept was first formulated in a slightly extended form by the French mathematician and philosopher Blaise Pascal in 1647 and would be later known as Pascal's Law. This law has many important applications in hydraulics. Galileo also was a major figure in the development of hydrostatics.
In economics, "static" analysis has substantially the same meaning as in physics. Since the time of Paul Samuelson's Foundations of Economic Analysis (1947), the focus has been on "comparative statics", i.e., the comparison of one static equilibrium to another, with little or no discussion of the process of going between them – except to note the exogenous changes that caused the movement.
In exploration geophysics, "statics" is used as a short form for "static correction", referring to bulk time shifts of a reflection seismogram to correct for the variations in elevation and velocity of the seismic pulse through the weathered and unconsolidated upper layers.
Equilibrium is the condition of a system in which competing influences are balanced and it may refer to:
Equilibrioception, the sense of balance present in humans and animals
Homeostasis, the ability of an open system, especially living organisms, to regulate its internal environment
Genetic equilibrium, theoretical state in which a population is not evolving
Punctuated equilibrium, theory in evolutionary biology
Sedimentation equilibrium, analytical ultracentrifugation method for measuring protein molecular masses in solution
Ecological homeostasis, a property of some ecosystems
Chemical equilibrium, the state in which the concentrations of the reactants and products have no net change over time
Diffusion equilibrium, when the concentrations of the diffusing substance in the two compartments are equal
Donnan equilibrium, the distribution of ion species between two ionic solutions separated by a semipermeable membrane or boundary
Dynamic equilibrium, the state in which two reversible processes occur at the same rate
Equilibrium constant, a quantity characterizing a chemical equilibrium in a chemical reaction
Equilibrium unfolding, the process of unfolding a protein or RNA molecule by gradually changing its environment
Partition equilibrium, a type of chromatography that is typically used in GC
Quasistatic equilibrium, the quasi-balanced state of a thermodynamic system near to equilibrium in some sense or degree
Schlenk equilibrium, a chemical equilibrium named after its discoverer Wilhelm Schlenk taking place in solutions of Grignard reagents
Solubility equilibrium, any chemical equilibrium between solid and dissolved states of a compound at saturation
Thermodynamic equilibrium, the state of a thermodynamic system which is in thermal, mechanical, and chemical equilibrium
Competitive equilibrium, economic equilibrium when all buyers and sellers are small relative to the market
Economic equilibrium, a condition in economics
Equilibrium price, the price at which supply equals demand
General equilibrium, a branch of theoretical microeconomics
Intertemporal equilibrium, an equilibrium concept over time
Lindahl equilibrium, a method proposed by Erik Lindahl for financing public goods
Partial equilibrium, one part of the general economic equilibrium
Radner equilibrium, an economic concept defined by economist Roy Radner in the context of general equilibrium
Static equilibrium (economics), the intersection of supply and demand in any market
Sunspot equilibrium, an economic equilibrium in which nonfundamental factors affect prices or quantities
Underemployment equilibrium, a situation in Keynesian economics with a persistent shortfall relative to full employment and potential output
Correlated equilibrium, a solution concept in game theory that is more general than the well known Nash equilibrium
Nash equilibrium, a solution concept in game theory involving two or more players
Quasi-perfect equilibrium, a refinement of Nash Equilibrium for extensive form games due to Eric van Damme
Sequential equilibrium, a refinement of Nash Equilibrium for extensive form games due to David M. Kreps and Robert Wilson
Symmetric equilibrium, in game theory, an equilibrium where all players use the same strategy
Trembling hand perfect equilibrium assumes that the players, through a "slip of the hand" or tremble, may choose unintended strategies
Equilibrium (band), a German Viking metal band
"IX Equilibrium", the third LP by the Black metal band Emperor
Equilibrium (record label), a Portuguese record label
Equilibrium mode distribution, involves light traveling in an optical waveguide or fiber
Hydrostatic equilibrium, the state of a system in which compression due to gravity is balanced by a pressure gradient force
Hyperbolic equilibrium point, a mathematical concept in physics
Mechanical equilibrium, the state in which the sum of the forces, and torque, on each particle of the system is zero
Secular equilibrium, a state of radioactive elements in which the production rate of a daughter nucleus is balanced by its own decay rate
Equilibrium (2002 film), a science-fiction film
"Equilibrium" (seaQuest 2032 episode), episode of seaQuest 2032
"Equilibrium" (DS9 episode), an episode of Star Trek: Deep Space Nine
Equilibrium moisture content, the moisture content at which the wood is neither gaining nor losing moisture
Equilibrium point, node in mathematics
Patterson equilibrium, a theory used in highway traffic modelling, air-traffic modelling, business modelling, and electrical grid operation
Reflective equilibrium, the state of balance or coherence among a set of beliefs arrived at by a process of deliberative mutual adjustment
Social equilibrium, a system in which there is a dynamic working balance among its interdependent parts
Equilibrium (puzzle), a 3D interlocking type puzzle made of 6 half circle pieces
Static equilibrium refers to objects that are not moving. There are many objects that we would like to have in static equilibrium; your kitchen cabinents or bedroom wall, for example. If an object is in static equilibrium, there are two very specific things the object is not doing that are important to understand. It is neither translating nor is it rotating. The first condition, no translation, requires that the net force (meaning the sum of all forces) acting on the object must be zero. The second condition, no rotation, requires that the net torque or moment also be zero. We will carefully define and examine both of these ideas.
Weight and mass
In modern usage in the field of mechanics, weight and mass are fundamentally different quantities: mass is an intrinsic property of matter, whereas weight is a force that results from the action of gravity on matter.
However, the recognition of this difference is, historically, a relatively recent development – and in many everyday situations the word "weight" continues to be used when strictly speaking "mass" is meant. For example, we say that an object "weighs one kilogram", even though the kilogram is actually a unit of mass. This common parlance usage is due to the early proliferation of force-based measurement systems widely being displaced with the more scientific and mass-based SI system. This transition has led to the common and even legal intertwining of "weight" and "mass" as equivalents.
The distinction between mass and weight is unimportant for many practical purposes because, to a reasonable approximation, the strength of gravity is the same everywhere on the surface of the Earth. In such a constant gravitational field, the gravitational force exerted on an object (its weight) is directly proportional to its mass. So, if object A weighs, say, 10 times as much as object B, then object A's mass is 10 times that of object B. This means that an object's mass can be measured indirectly by its weight (for conversion formulas see below). For example, when we buy a bag of sugar we can measure its weight (how hard it presses down on the scales) and be sure that this will give a good indication of the quantity that we are actually interested in, which is the mass of sugar in the bag. Nevertheless, slight variations in the Earth's gravitational field do exist (see Earth's gravity), and these must be taken into account in high precision weight measurements.
The use of "weight" for "mass" also persists in some scientific terminology – for example, in the chemical terms "atomic weight", "molecular weight", and "formula weight", rather than the preferred "atomic mass" etc.
The difference between mass and force becomes obvious when
objects are compared in different gravitational fields, such as away from the Earth's surface. For example, on the surface of the Moon, gravity is only about one-sixth as strong as on the surface of the Earth. A one-kilogram mass is still a one-kilogram mass (as mass is an intrinsic property of the object) but the downwards force due to gravity is only one-sixth of what the object would experience on Earth. Weight is relative to the local gravitional field, mass is not.
masses are considered in the context of a lever, such as a cantilever structure.
locating the center of gravity of an object.
Units of weight (force)
Systems of units of weight (force) and mass have a tangled history, partly because the distinction was not properly understood when many of the units first came into use.
In most modern scientific work, physical quantities are measured in SI units. The SI unit of mass is the kilogram. The SI unit of force (and hence weight) is the newton (N) – which can also be expressed in SI base units as kg·m/s² (kilograms times meters per second squared).
The kilogram-force is a non-SI unit of force, defined as the force exerted by a one-kilogram mass in standard Earth gravity (equal to about 9.8 newtons).
The gravitational force exerted on an object is proportional to the mass of the object, so it is reasonable to think of the strength of gravity as measured in terms of force per unit mass, that is, newtons per kilogram (N/kg). However, the unit N/kg resolves to m/s²; (metres per second per second), which is the SI unit of acceleration, and in practice gravitational strength is usually quoted as an acceleration.
The pound and related units
In United States customary units, the pound can be either a unit of force or a unit of mass. Related units used in some distinct, separate subsystems of units include the poundal and the slug. The poundal is defined as the force necessary to accelerate a one-pound object at 1 ft/s², and is equivalent to about 1/32 of a pound (force). The slug is defined as the amount of mass that accelerates at 1 ft/s² when a pound of force is exerted on it, and is equivalent to about 32 pounds (mass).
Conversion between weight (force) and mass
To convert between weight (force) and mass we use Newton's second law, F = ma (force = mass × acceleration). Here, F is the force due to gravity (i.e. the weight force), m is the mass of the object in question, and a is the acceleration due to gravity, on Earth approximately 9.8 m/s² or 32 ft/s²). In this context the same equation is often written as W = mg, with W standing for weight, and g for the acceleration due to gravity.
When applying the equation it is essential to use consistent units otherwise a meaningless answer will result. In SI units we see that a one-kilogram mass experiences a gravitational force of 1 kg × 9.8 m/s² = 9.8 newtons; that is, its weight is 9.8 newtons. In general, to convert mass in kilograms to weight (force) in newtons (at the earth's surface), multiply by 9.8. Conversely, to convert newtons to kilograms divide by 9.8.
Sensation of weight
See also: apparent weight
The weight force that we actually sense is not the downward force of gravity, but the normal force (an upward contact force) exerted by the surface we stand on, which opposes gravity and prevents us falling to the center of the Earth. This normal force, called the apparent weight, is the one that is measured by a spring scale.
For a body supported in a stationary position, the normal force balances the earth's gravitational force, and so apparent weight has the same magnitude as actual weight. (Technically, things are slightly more complicated. For example, an object immersed in water weighs less, according to a spring scale, than the same object in air; this is due to buoyancy, which opposes the weight force and therefore generates a smaller normal. These and other factors are explained further under apparent weight.)
If there is no contact with any surface to provide such an opposing force then there is no sensation of weight (no apparent weight). This happens in free-fall, as experienced by sky-divers (until they approach terminal velocity) and astronauts in orbit, who feel "weightless" even though their bodies are still subject to the force of gravity: they're just no longer resisting it. The experience of having no apparent weight is also known as microgravity.
A degree of reduction of apparent weight occurs, for example, in elevators. In an elevator, a spring scale will register a decrease in a person's (apparent) weight as the elevator starts to accelerate downwards. This is because the opposing force of the elevator's floor decreases as it accelerates away underneath one's feet.
Main article: Weighing scale
Weight is commonly measured using one of two methods. A spring scale or hydraulic or pneumatic scale measures weight force (strictly apparent weight force) directly. If the intention is to measure mass rather than weight, then this force must be converted to mass. As explained above, this calculation depends on the strength of gravity. Household and other low precision scales that are calibrated in units of mass (such as kilograms) assume roughly that standard gravity will apply. However, although nearly constant, the apparent or actual strength of gravity does in fact vary very slightly in different places on the earth (see standard gravity, physical geodesy, gravity anomaly and gravity). This means that same object (the same mass) will exert a slightly different weight force in different places. High precision spring scales intended to measure mass must therefore be calibrated specifically according their location on earth.
Mass may also be measured with a balance, which compares the item in question to others of known mass. This comparison remains valid whatever the local strength of gravity. If weight force, rather than mass, is required, then this can be calculated by multiplying mass by the acceleration due to gravity – either standard gravity (for everyday work) or the precise local gravity (for precision work).
Gross weight is a term that generally is found in commerce or trade applications, and refers to the gross or total weight of a product and its packaging. Conversely, net weight refers to the intrinsic weight of the product itself, discounting the weight of packaging or other materials.
Relative weights on the Earth, other planets and the Moon
The following is a list of the weights of a mass on the surface of some of the bodies in the solar system, relative to its weight on Earth:
Center of gravity
The center of gravity is a geometric property of any object. The center of gravity is the average location of the weight of an object. We can completely describe the motion of any object through space in terms of the translation of the center of gravity of the object from one place to another, and the rotation of the object about its center of gravity if it is free to rotate. If the object is confined to rotate about some other point, like a hinge, we can still describe its motion. In flight, both airplanes and rockets rotate about their centers of gravity. A kite, on the other hand, rotates about the bridle point. But the trim of a kite still depends on the location of the center of gravity relative to the bridle point, because for every object the weight always acts through the center of gravity.
Determining the center of gravity is very important for any flying object. How do engineers determine the location of the center of gravity for an aircraft which they are designing?
In general, determining the center of gravity (cg) is a complicated procedure because the mass (and weight) may not be uniformly distributed throughout the object. The general case requires the use of calculus which we will discuss at the bottom of this page. If the mass is uniformly distributed, the problem is greatly simplified. If the object has a line (or plane) of symmetry, the cg lies on the line of symmetry. For a solid block of uniform material, the center of gravity is simply at the average location of the physical dimensions. (For a rectangular block, 50 X 20 X 10, the center of gravity is at the point (25,10, 5) ). For a triangle of height h, the cg is at h/3, and for a semi-circle of radius r, the cg is at (4*r/(3*pi)) where pi is ratio of the circumference of the circle to the diameter. There are tables of the location of the center of gravity for many simple shapes in math and science books. The tables were generated by using the equation from calculus shown on the slide.
For a general shaped object, there is a simple mechanical way to determine the center of gravity:
1. If we just balance the object using a string or an edge, the point at which the object is balanced is the center of gravity. (Just like balancing a pencil on your finger!)
2. Another, more complicated way, is a two step method shown on the slide. In Step 1, you hang the object from any point and you drop a weighted string from the same point. Draw a line on the object along the string. For Step 2, repeat the procedure from another point on the object You now have two lines drawn on the object which intersect. The center of gravity is the point where the lines intersect. This procedure works well for irregularly shaped objects that are hard to balance.
If the mass of the object is not uniformly distributed, we must use calculus to determine center of gravity. We will use the symbol S dw to denote the integration of a continuous function with respect to weight. Then the center of gravity can be determined from:
cg * W = S x dw
where x is the distance from a reference line, dw is an increment of weight, and W is the total weight of the object. To evaluate the right side, we have to determine how the weight varies geometrically. From the weight equation, we know that:
w = m * g
where m is the mass of the object, and g is the gravitational constant. In turn, the mass m of any object is equal to the density, rho, of the object times the volume, V:
m = rho * V
We can combine the last two equations:
w = g * rho * V
dw = g * rho * dV
dw = g * rho(x,y,z) * dx dy dz
If we have a functional form for the mass distribution, we can solve the equation for the center of gravity:
cg * W = g * SSS x * rho(x,y,z) dx dy dz
where SSS indicates a triple integral over dx. dy. and dz. If we don't know the functional form of the mass distribution, we can numerically integrate the equation using a spreadsheet. Divide the distance into a number of small volume segments and determining the average value of the weight/volume (density times gravity) over that small segment. Taking the sum of the average value of the weight/volume times the distance times the volume segment divided by the weight will produce the center of gravity.