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In
physics,
mechanical work is the amount of
energy transferred by a
force. Like energy, it is a scalar (physics) 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 (physics) and displacement vectors.
Calculation:In the simplest case of a body initially at rest acted on by a constant force parallel to that direction, the work is given by these formulas
W = F d \,\! (1)
W = \frac{1}{2}m v^2 \,\! (derived from the above equation)
where
F is the portion of the force acting in the same direction as the motion, and
d is the distance traveled by the object (note that distance is a scalar quantity and so, too, is work).
The work is taken to be negative when the force opposes the motion. More generally, the force and distance are taken to be vector (spatial) quantities, and combined using the
dot product:
W = \bold{F} \cdot \bold{d} = F d \cos\phi (2)
where \phi is the angle between the force and the displacement vector. In order for this formula to be valid, the force and angle must remain constant. The object's path must always remain on a single, straight line, though it may change directions while moving along the line.
In situations where the force changes over time, and/or the path deviates from a straight line, equation (1) is not directly applicable. However, under mild restrictions, it is possible to divide the motion into small steps, such that the force and motion are well approximated as being constant for each step, and then to express the overall work as the sum over these steps. This is formalized by the following line integral, which can be taken as a rather general definition of work:
W_C := \int_{C} \bold{F} \cdot \mathrm{d}\bold{s} (3)
where:
C is the path or curve traversed by the object;
F is the
force vector;
s is the
position vector.
It must be emphasized that W_C is explicitly a function of the path C. If work were a
potential it would depend only on the endpoints of the path, but this is not the case; in general the work W_C depends on every detail of the path C. As a related matter, it is not proper to write d
W =
F·d
s nor d
W = anything (except perhaps in trivial cases, which we exclude from further consideration). This is because the notation d
W implies that d
W is an exact differential, whereas the correct expression
F·d
s is an
inexact differential. It is fairly common to see \mathrm{d}W used as shorthand
F·d
s, but this must be considered highly informal and mathematically unjustifiable. Certainly there is no function W that can be differentiated to give
F·d
s.
Equation (3) 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 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 Nicolas Léonard 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, through 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 electric 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 (Pressure-Volume) work
Chemical thermodynamics studies
PV work, which occurs when the volume of a fluid changes. PV work is represented by the following equation:
W_C =-\int_C P\,\mathrm{d}V (4)
where:
- W is the work done on the system
- P is the external pressure
- V is the volume
Like all work functions, PV work is
Path dependent (mathematics) on the path C. (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 P \mathrm{d}V is an
inexact differential. Some prefer to write the 'd' with a line through or use \delta W instead to signal this condition.
From a mathematical point of view, that is to say, \mathrm{d}W is not an
exact form one-form. The use of a different symbol for the differential warns there is actually no function (0-form) W which is the
potential of \mathrm{d}W. If there were, indeed, this function W, we should be able to use the
Stokes Theorem, and calculate the above integral by just evaluating this putative function, the potential of \mathrm{d}W, at the
Boundary (topology) 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 liter-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 as 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
Ek1 to
Ek2, then:
W = \Delta E_k = E_{k2} - E_{k1}\,\!
Also, if we substitute the equation for kinetic energy that states E_k = \frac{1}{2} mv^2 , we then get:
W = \Delta E_k = \frac{1}{2} mv_2 ^2 - \frac{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 total 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.
References
External links
- Work - a chapter from an online textbook
- Work, Power, Kinetic Energy on Project PHYSNET
In physics,
mechanical work is the amount of
energy transferred by a
force. Like energy, it is a scalar (physics) 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 (physics) and displacement vectors.
Calculation:In the simplest case of a body initially at rest acted on by a constant force parallel to that direction, the work is given by these formulas
W = F d \,\! (1)
W = \frac{1}{2}m v^2 \,\! (derived from the above equation)
where
F is the portion of the force acting in the same direction as the motion, and
d is the distance traveled by the object (note that distance is a scalar quantity and so, too, is work).
The work is taken to be negative when the force opposes the motion. More generally, the force and distance are taken to be vector (spatial) quantities, and combined using the dot product:
W = \bold{F} \cdot \bold{d} = F d \cos\phi (2)
where \phi is the angle between the force and the displacement vector. In order for this formula to be valid, the force and angle must remain constant. The object's path must always remain on a single, straight line, though it may change directions while moving along the line.
In situations where the force changes over time, and/or the path deviates from a straight line, equation (1) is not directly applicable. However, under mild restrictions, it is possible to divide the motion into small steps, such that the force and motion are well approximated as being constant for each step, and then to express the overall work as the sum over these steps. This is formalized by the following line integral, which can be taken as a rather general definition of work:
W_C := \int_{C} \bold{F} \cdot \mathrm{d}\bold{s} (3)
where:
C is the path or curve traversed by the object;
F is the
force vector;
s is the
position vector.
It must be emphasized that W_C is explicitly a function of the path C. If work were a
potential it would depend only on the endpoints of the path, but this is not the case; in general the work W_C depends on every detail of the path C. As a related matter, it is not proper to write d
W =
F·d
s nor d
W = anything (except perhaps in trivial cases, which we exclude from further consideration). This is because the notation d
W implies that d
W is an exact differential, whereas the correct expression
F·d
s is an
inexact differential. It is fairly common to see \mathrm{d}W used as shorthand
F·d
s, but this must be considered highly informal and mathematically unjustifiable. Certainly there is no function W that can be differentiated to give
F·d
s.
Equation (3) 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 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
Nicolas Léonard 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, through 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 electric 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 (Pressure-Volume) work
Chemical thermodynamics studies
PV work, which occurs when the volume of a fluid changes. PV work is represented by the following equation:
W_C =-\int_C P\,\mathrm{d}V (4)
where:
- W is the work done on the system
- P is the external pressure
- V is the volume
Like all work functions, PV work is
Path dependent (mathematics) on the path C. (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 P \mathrm{d}V is an inexact differential. Some prefer to write the 'd' with a line through or use \delta W instead to signal this condition.
From a mathematical point of view, that is to say, \mathrm{d}W is not an exact form
one-form. The use of a different symbol for the differential warns there is actually no function (0-form) W which is the potential of \mathrm{d}W. If there were, indeed, this function W, we should be able to use the Stokes Theorem, and calculate the above integral by just evaluating this putative function, the potential of \mathrm{d}W, at the Boundary (topology) 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 liter-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 as 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
Ek1 to
Ek2, then:
W = \Delta E_k = E_{k2} - E_{k1}\,\!
Also, if we substitute the equation for kinetic energy that states E_k = \frac{1}{2} mv^2 , we then get:
W = \Delta E_k = \frac{1}{2} mv_2 ^2 - \frac{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 total 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
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.
References
External links
- Work - a chapter from an online textbook
- Work, Power, Kinetic Energy on Project PHYSNET
Mechanical work - Wikipedia, the free encyclopedia
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See the calculation of work done moving an object. Work is equal to the force exerted on an object in the direction of motion times the displacement.
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