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Mechanical work is a force applied through a distance, defined mathmatically as the line integral of a scalar product of force and displacement vectors. Work is a scalar quantity which can be positive or negative. More simply, it is the energy related to the applied force over a distance.

The force can do positive, negative, or zero work. For instance, a centripetal force in uniform circular motion does zero work (because the scalar product of force and displacement vector is zero as they are orthogonal to each other). Another example is Lorentz magnetic force on moving electric charge which always does zero work because it is always orthogonal to the direction of motion of the charge.

Definition

Note: Readers not familiar with multivariate calculus or vectors, please see "Simpler formulae" below.

definition 1: Work is defined as the following line integral:

where:

C is the path or curve traversed by the object;

is the force vector;

is the position vector.

This formula readily explains how a nonzero force can do zero work. The simplest case is where the force is always perpendicular to the direction of motion, making the integrand always zero (viz. circular motion). However, even if the integrand sometimes takes nonzero values, it can still integrate to zero if it is sometimes negative and sometimes positive.

The possibility of a nonzero force doing zero work exemplifies the difference between work and a related quantity: impulse (the integral of force over time). Impulse measures change in a body's momentum, a vector quantity sensitive to direction, whereas work considers only the magnitude of the velocity. For instance, as an object in uniform circular motion traverses half of a revolution, its centripetal force does no work, but it transfers a nonzero impulse.

Units

In thermodynamics, thermodynamic work is the quantity of energy transferred from one system to another. It is a generalization of the concept of mechanical work in mechanics. The SI derived unit of work is the joule (J), which is defined as the work done by a force of one newton acting over a distance of one meter. This definition is based on Sadi Carnot's 1824 definition of work as "weight lifted through a height", which is based on the fact that early steam engines were principally used to lift buckets of water, though a gravitational height, out of flooded ore mines. The dimensionally equivalent newton-meter (N·m) is sometimes used instead; however, it is also sometimes reserved for torque to distinguish its units from work or energy.

Non-SI units of work include the erg, the foot-pound, the foot-poundal, and the liter-atmosphere.

Simpler formulae

In the simplest case, that of a body moving in a steady direction, and acted on by a constant force parallel to that direction, the work is given by the formula

where

F is the force and

s is the distance travelled by the object.

The work is taken to be negative when the force opposes the motion. More generally, the force and distance are taken to be vector quantities, and combined using the dot product:

where φ is the angle between the force and the displacement vector. This formula holds true even when the object changes its direction of travel throughout the motion.

To further generalize the formula to situations in which the force changes over time, it is necessary to use differentials to express the infinitesimal work done by the force over an infinitesimal displacement, thus:

The integration of both sides of this equation yields the most general formula, as given above.

Types of work

Forms of work that are not evidently mechanical in fact represent special cases of this principle. For instance, in the case of "electrical work", an electric field does work on charged particles as they move through a medium.

One mechanism of heat conduction is collisions between fast-moving atoms in a warm body with slow-moving atoms in a cold body. Although colliding atoms do work on each other, it averages to nearly zero in bulk, so conduction is not considered to be mechanical work.

PV work

Chemical thermodynamics studies PV work, which occurs when the volume of a fluid changes. PV work is represented by the following differential equation:

where:

• W = work done on the system
• P = external pressure
• V = volume

Therefore, we have:

Like all work functions, PV work is path-dependent. (The path in question is a curve in the Euclidean space specified by the fluid's pressure and volume, and infinitely many such curves are possible.) From a thermodynamic perspective, this fact implies that PV work is not a state function. This means that the differential dW is an inexact differential; to be more rigorous, it should be written đW (with a line through the d).

From a mathematical point of view, that is to say, dW is not an exact one-form. This line through is merely a flag to warn us there is actually no function (0-form) W which is the potential of dW. If there were, indeed, this function W, we should be able to just use Stokes Theorem, and evaluate this putative function, the potential of dW, at the boundary of the path, that is, the initial and final points, and therefore the work would be a state function. This impossibility is consistent with the fact that it does not make sense to refer to the work on a point; work presupposes a path.

PV work is often measured in the (non-SI) units of litre-atmospheres, where 1 L·atm = 101.3 J.

Mechanical energy

In physics, mechanical energy describes the potential energy and kinetic energy present in the components of a mechanical system. The mechanical energy of a body is that part of its total energy which is subject to change by mechanical work. It includes kinetic energy and potential energy. Some notable forms of energy that it does not include are thermal energy (which can be increased by frictional work, but not easily decreased) and rest energy (which is constant so long as the rest mass remains the same).

What is it?

The study of mechanics concerns the motion of physical bodies and the forces that act upon them. Most people are familiar with systems described by Newtonian mechanics - objects that sit around, move, collide, and are influenced by gravity. Mechanical energy includes things like the kinetic energy of a moving billiard ball, or the potential energy a roller coaster at the top of its track.

The physics of electromagnetism were not understood at the time of Newton, but in some situations, the mechanics (i.e. the mathematics of motion) of bodies influenced by electromagnetic forces is the same as that of those influenced by gravity. For example, two particles of opposite electrical charge experience an attractive force which is (allowing for certain idealizations) mathematically identical to the gravitational forces two passing planets experience. An electromechanical system might also involve the conversion of mechanical energy into electrical charges or magnetic fields, or vice versa.

Everyday objects are composed of atoms and molecules, which to some degree, are like billiard balls that are constantly bouncing off one another. "Mechanical energy" might include the kinetic energy of these particles, or potential energy stored in the physical arrangement. For example, a compressed solid exerts pressure because electromagnetic forces between particles tend to push them apart. Compressing a solid (moving the particles "uphill" against repulsive electromagnetic forces) stores potential energy in a similar way that pushing a boulder up a hill does (moving the object uphill against the attractive gravitational force of the Earth). On the other hand, a compressed gas exerts pressure because independently moving particles collide with the walls of the container and change direction. The particle is accelerated (its velocity vector changed), and the acceleration times the mass of the particle gives the force applied. Compressing a gas changes the average kinetic energy of the particles, which is reflected in the corresponding increase in the temperature of the gas. The pressure also increases, but this is because the same number of particles have been forced into a smaller volume, so they collide more often with the walls. The force of any given collision is the same, but the number of collisions has increased.

Potential energy does play a role in the pressure of a gas. During an individual collision, a gas molecule comes closer to the molecules of the container wall. The electric fields exert a force on the molecule, slowing it down and reducing its kinetic energy. This energy is temporarily stored as potential energy. Soon, the particle is nearly stationary (if it happened to approach head on), or at least, it is not approaching the wall any more. The electric fields continue to exert a force on the gas molecule. The force continues to change the velocity, and soon the molecule is moving away from the wall and gaining kinetic energy. Generally, the collision is elastic, and all of the kinetic energy is recovered and the particle continues moving with the same speed it had originally.

Solid mechanics studies how rigid bodies behave in response to external forces. Fluid mechanics studies the internal motion of liquids, gases, and other forms of matter. Mechanical energy can be expended in crushing a soda can, affecting the motion and positional arrangement of its component molecules. Mechanical energy can be transferred from the molecules of a solid to the molecules of a liquid when, for example, a glass of water is stirred.

Associated concepts

When a given quantity of mechanical energy is transferred (such as when throwing a ball, lifting a box, crushing a can, or stirring a beverage) it is said that this amount mechanical work has been done. Both mechanical energy and mechanical work are measured in the same units as energy in general. It is usually said that a component of a system has a certain amount of "mechanical energy" (i.e. it is a state function), whereas "mechanical work" describes the amount of mechanical energy a component has gained or lost.

The conservation of mechanical energy is a principle which states that, under certain conditions, the total mechanical energy of a system is constant. This rule does not hold when mechanical energy is converted to other forms, such as chemical, nuclear, or electromagnetic. However, the principle of general conservation of energy is so far an unbroken rule of physics - as far as we know, energy cannot be created or destroyed, only changed in form.

Simplifying assumptions

Scientists often make simplifying assumptions to make calculations about how mechanical systems behave. For example, instead of calculating the mechanical energy separately for each of the billions of molecules in a soccer ball, it is easier to treat the entire ball as one object. This means that only two numbers (one for kinetic mechanical energy, and one for potential mechanical energy) are needed for each dimension (for example, up/down, north/south, east/west) under consideration.

To calculate the energy of a system without any simplifying assumptions would require examining the state of all elementary particles and considering all four fundamental interactions. This is usually only done for very small systems, such as those studied in particle physics.

Distinguished from other types of energy

The classification of energy into different "types" often follows the boundaries of the fields of study in the natural sciences.

• Chemical energy, the kind of potential energy stored in chemical bonds; studied in chemistry
• Nuclear energy, energy stored in interactions between the particles in the atomic nucleus; studied in nuclear physics
• Electromagnetic energy, in the form of electric charges, magnetic fields, and photons; from the study of electromagnetism
• Various forms of energy in quantum mechanics; for example, the energy levels of electrons in an atom

In certain cases, it can be unclear what counts as "mechancial" energy. For example, is the energy stored in the structure of a crystal "mechanical" or "chemical"? Scientists generally use these "types" as convenient labels which clearly distinguish between different phenomena. It is not scientifically important to decide what is "mechanical" energy and what is "chemical". In these cases, usually there is a more specific name for the phenomenon in question. For example, in considering two bonded atoms, there are energy components from vibrational motion, from angular motions, from the electrical charge on the nuclei, secondary electromagnetic considerations like the Van der Waals force, and quantum mechanical contributions concerning the energy state of the electron shells.

The relation between work and kinetic energy

If an external work W acts upon a body, causing its kinetic energy to change from E_k1 to E_k2, then:

Conservation of mechanical energy

The principle of conservation of mechanical energy states that, if a system is subject only to conservative forces (e.g. only to a gravitational force), its mechanical energy remains constant.

For instance, if an object with constant mass is in free fall, the total energy of position 1 will equal that of position 2.

where

• E_k is the kinetic energy, and
• E_p is the potential energy.

Related concepts

• Energy

References and external articles

• Mark T. Holtzapple and W. Dan Reece, "Foundations of Engineering", 2/e, Glossary (http://highered.mcgraw-hill.com/sites/0072480823/student_view0/glossary.html), highered.mcgraw-hill.com.
• Serway, Raymond A.; Jewett, John W., "Physics for Scientists and Engineers" (6th ed.) Brooks/Cole, 2004 ISBN 0-534-40842-7
• Tipler, Paul, "Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics" (5th ed.) W. H. Freeman, 2004 | id=ISBN 0-7167-0809-4
• Wikipedia contributors (http://en.wikipedia.org/wiki/Special:Recentchanges), Wikipedia: The Free Encyclopedia. Wikimedia Foundation. <http://en.wikipedia.org>.

See also

- PowerPedia main index
- PESWiki home page
- PES Network, Inc. (http:pureenergysystems.com)