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Mechanical work

2007 Schools Wikipedia Selection. Related subjects: General Physics

   In physics, mechanical work is the amount of energy transferred by a
   force. Like energy, it is a scalar quantity, with SI units of joules.
   Heat conduction is not considered to be a form of work, since there is
   no macroscopically measurable force, only microscopic forces occurring
   in atomic collisions. In the 1830s, the French mathematician
   Gaspard-Gustave Coriolis, coined the term work for the product of force
   and distance.

   Positive and negative signs of work indicate whether the object
   exerting the force is transferring energy to some other object, or
   receiving it. A baseball pitcher, for example, does positive work on
   the ball, but the catcher does negative work on it. Work can be zero
   even when there is a force. The centripetal force in uniform circular
   motion, for example, does zero work because the kinetic energy of the
   moving object doesn't change. Likewise, when a book sits on a table,
   the table does no work on the book, because no energy is transferred
   into or out of the book.

   When the force is constant and along the same line as the motion, the
   work can be calculated by multiplying the force by the distance, W = Fd
   (letting both F and d have positive or negative signs, according to the
   coordinate system chosen). When the force does not lie along the same
   line as the motion, this can be generalized to the scalar product of
   force and displacement vectors.
   The baseball pitcher does work on the ball by transferring energy into
   it.
   Enlarge
   The baseball pitcher does work on the ball by transferring energy into
   it.

Calculation

   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 D \,\!

   where

          F is the force and
          D 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:

          W = \vec F \cdot \vec \mathbf{D} = |\mathbf{F}| |\mathbf{D}|
          \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 differentials to express the
   infinitesimal work done by the force over an infinitesimal
   displacement, thus:

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

   The integration of both sides of this equation yields the following
   line integral:

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

   where:

          C is the path or curve traversed by the object;
          \vec F is the force vector;
          \vec s 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

   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.

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, the force 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:

          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 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

   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).

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:

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

   Also, if we substitute the equation for kinetic energy that states E[k]
   = 1 / 2mv^2, we then get:

          W = \Delta (1/2 mv^2) = 1/2 mv_2 ^2 - 1/2 mv_1 ^2

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.

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

   where
     * E[k] is the kinetic energy, and
     * E[p] is the potential energy.

   The external work will usually be done by the friction force between
   the system on the motion or the internal-non conservative force in the
   system or loss of energy due to heat transfer.
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