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## PowerPedia:Mechanical work

Lasted edited by Andrew Munsey, updated on June 15, 2016 at 1:21 am.

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Mechanical work is a force applied through a distance, defined mathmatically as the There was an error working with the wiki: Code[1] and displacement vectors. Work is a There was an error working with the wiki: Code[2] 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 There was an error working with the wiki: Code[14] 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 There was an error working with the wiki: Code[3], please see "Simpler formulae" below.

definition 1: Work is defined as the following There was an error working with the wiki: Code[15]:

: W = \int_{C} \vec F \cdot d\vec{s} \,\!

where:

:C is the path or There was an error working with the wiki: Code[16] traversed by the object

: \vec F is the Force vector

:\vec s is the There was an error working with the wiki: Code[17].

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 There was an error working with the wiki: Code[18] 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: There was an error working with the wiki: Code[19] (the integral of force over time). Impulse measures change in a body's There was an error working with the wiki: Code[20], 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 There was an error working with the wiki: Code[4]'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 There was an error working with the wiki: Code[21] (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 There was an error working with the wiki: Code[22], the There was an error working with the wiki: Code[23], the There was an error working with the wiki: Code[24], and the There was an error working with the wiki: Code[25].

#### 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

: W = F s \,\!

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 There was an error working with the wiki: Code[5] quantities, and combined using the There was an error working with the wiki: Code[26]:

: W = \vec F \cdot \vec {s} = |{F}| |{s}| \cos\phi \,\!

where \phi 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 There was an error working with the wiki: Code[6] to express the infinitesimal work done by the force over an infinitesimal displacement, thus:

: dW = \vec F \cdot d\vec{s} \,\!

The There was an error working with the wiki: Code[7] 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 There was an error working with the wiki: Code[8]d particles as they move through a medium.

One mechanism of There was an error working with the wiki: Code[27] 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

There was an error working with the wiki: Code[28] studies PV work, which occurs when the volume of a fluid changes. PV work is represented by the following There was an error working with the wiki: Code[29]:

:dW = -P dV \,

where:

W = work done on the system

P = external pressure

V = volume

Therefore, we have:

:W=-\int_{V_i}^{V_f} P\,dV

Like all work functions, PV work is There was an error working with the wiki: Code[9]. (The path in question is a curve in the There was an error working with the wiki: Code[30] specified by the fluid's There was an error working with the wiki: Code[31] and There was an error working with the wiki: Code[32], and infinitely many such curves are possible.) From a thermodynamic perspective, this fact implies that PV work is not a There was an error working with the wiki: Code[33]. This means that the differential dW is an There was an error working with the wiki: Code[34] to be more rigorous, it should be written ?W (with a line through the d).

From a mathematical point of view, that is to say, dW is not an There was an error working with the wiki: Code[10] There was an error working with the wiki: Code[35]. This line through is merely a flag to warn us there is actually no function (There was an error working with the wiki: Code[36]) W which is the There was an error working with the wiki: Code[37] of dW. If there were, indeed, this function W, we should be able to just use There was an error working with the wiki: Code[38], and evaluate this putative function, the potential of dW, at the There was an error working with the wiki: Code[39] 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 There was an error working with the wiki: Code[40]al work, but not easily decreased) and There was an error working with the wiki: Code[41] (which is constant so long as the There was an error working with the wiki: Code[42] remains the same).

##### What is it?

The study of There was an error working with the wiki: Code[11] of There was an error working with the wiki: Code[12] and the Forces that act upon them. Most people are familiar with systems described by There was an error working with the wiki: Code[43] - objects that sit around, move, collide, and are influenced by Gravity. Mechanical energy includes things like the kinetic energy of a moving There was an error working with the wiki: Code[44], or the potential energy a There was an error working with the wiki: Code[45] at the top of its track.

The physics of There was an error working with the wiki: Code[13], 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 There was an error working with the wiki: Code[46], 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.

There was an error working with the wiki: Code[47] studies how rigid bodies behave in response to external forces. There was an error working with the wiki: Code[48] studies the internal motion of There was an error working with the wiki: Code[49]s, 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 There was an error working with the wiki: Code[50]), whereas "mechanical work" describes the amount of mechanical energy a component has gained or lost.

The There was an error working with the wiki: Code[51] 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 There was an error working with the wiki: Code[52] 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 There was an error working with the wiki: Code[53] (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 There was an error working with the wiki: Code[54]s and considering all four There was an error working with the wiki: Code[55]s. This is usually only done for very small systems, such as those studied in There was an error working with the wiki: Code[56].

##### 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 There was an error working with the wiki: Code[57] studied in Chemistry

There was an error working with the wiki: Code[58], energy stored in interactions between the particles in the There was an error working with the wiki: Code[59] studied in There was an error working with the wiki: Code[60]

Electromagnetic energy, in the form of electric charges, magnetic fields, and There was an error working with the wiki: Code[61] from the study of Electromagnetism

Various forms of energy in Quantum mechanics for example, the There was an error working with the wiki: Code[62]s 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 There was an error working with the wiki: Code[63], 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 Ek1 to Ek2, then:

:W = \Delta E_k = E_{k2} - E_{k1}\,

##### Conservation of mechanical energy

The principle of conservation of mechanical energy states that, if a system is subject only to There was an error working with the wiki: Code[64]s (e.g. only to a There was an error working with the wiki: Code[65]), 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.

: (E_k + E_p)_1 = (E_k + E_p)_2 \,\!

where

E_k is the Kinetic energy, and

E_p is the Potential energy.

Energy

#### References and external articles

Mark T. Holtzapple and W. Dan Reece, "Foundations of Engineering", 2/e, Glossary, 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

There was an error working with the wiki: Code[1], Wikipedia: The Free Encyclopedia. Wikimedia Foundation.