Mechanical work

# Mechanical work

Overview
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, work is a scalar
Scalar (physics)
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...

quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work. The term work was first coined in 1826 by the French mathematician Gaspard-Gustave Coriolis
Gaspard-Gustave Coriolis
Gaspard-Gustave de Coriolis or Gustave Coriolis was a French mathematician, mechanical engineer and scientist. He is best known for his work on the supplementary forces that are detected in a rotating frame of reference. See the Coriolis Effect...

.
Discussion

Encyclopedia
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, work is a scalar
Scalar (physics)
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...

quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work. The term work was first coined in 1826 by the French mathematician Gaspard-Gustave Coriolis
Gaspard-Gustave Coriolis
Gaspard-Gustave de Coriolis or Gustave Coriolis was a French mathematician, mechanical engineer and scientist. He is best known for his work on the supplementary forces that are detected in a rotating frame of reference. See the Coriolis Effect...

.

If a constant force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

of magnitude F acts on a point that moves s in the direction of the force, then the work W done by this force is calculated W=Fs. For example, if a force of 10 newtons (F=10 N) acts along a path of 2 metres (s =2 m), it will do work W equal to W =(10 N)(2 m) = 20 N*m =20 J, where joule
Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

(J) is the SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

unit for work (defined as the product N*m, so that a joule is a newton-meter).

For moving objects, the quantity of work/time enters calculations as distance/time, or velocity. Thus, at any instant, the rate of the work done by a force (measured in joules/second, or watts) is the scalar product of the force (a vector) with the velocity vector of the point of application. This scalar product of force and velocity is called instantaneous power
Power (physics)
In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...

. Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus
Fundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.

The first law of thermodynamics
First law of thermodynamics
The first law of thermodynamics is an expression of the principle of conservation of work.The law states that energy can be transformed, i.e. changed from one form to another, but cannot be created nor destroyed...

states that when work
Work (thermodynamics)
In thermodynamics, work performed by a system is the energy transferred to another system that is measured by the external generalized mechanical constraints on the system. As such, thermodynamic work is a generalization of the concept of mechanical work in mechanics. Thermodynamic work encompasses...

is done to a system (and no other energy is subtracted in other ways), the system's energy state changes by the same amount of the work input. This equates work
Work (thermodynamics)
In thermodynamics, work performed by a system is the energy transferred to another system that is measured by the external generalized mechanical constraints on the system. As such, thermodynamic work is a generalization of the concept of mechanical work in mechanics. Thermodynamic work encompasses...

and energy. In the case of rigid bodies, Newton's laws
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...

can be used to derive a similar relationship called the work-energy theorem.

## Units

The SI unit of work is the joule
Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

(J), which is defined as the work done by a force of one newton acting over a distance of one metre
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

. This definition is based on Sadi Carnot
Nicolas Léonard Sadi Carnot was a French military engineer who, in his 1824 Reflections on the Motive Power of Fire, gave the first successful theoretical account of heat engines, now known as the Carnot cycle, thereby laying the foundations of the second law of thermodynamics...

'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-metre (N·m) is sometimes used for work, but this can be confused with the units newton-metre of torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

.

Non-SI units of work include the erg
Erg
An erg is the unit of energy and mechanical work in the centimetre-gram-second system of units, symbol "erg". Its name is derived from the Greek ergon, meaning "work"....

, the foot-pound, the foot-poundal
Foot-poundal
The Foot-poundal is a unit of energy that is part of the foot-pound-second system of units, in Imperial units introduced in 1879, and is from the specialized subsystem of English Absolut ....

, and the litre-atmosphere. Other non-SI units for work are the horsepower-hour
Horsepower
Horsepower is the name of several units of measurement of power. The most common definitions equal between 735.5 and 750 watts.Horsepower was originally defined to compare the output of steam engines with the power of draft horses in continuous operation. The unit was widely adopted to measure the...

, the therm
Therm
The therm is a non-SI unit of heat energy equal to 100,000 British thermal units . It is approximately the energy equivalent of burning 100 cubic feet of natural gas....

, the BTU and Calorie
Calorie
The calorie is a pre-SI metric unit of energy. It was first defined by Nicolas Clément in 1824 as a unit of heat, entering French and English dictionaries between 1841 and 1867. In most fields its use is archaic, having been replaced by the SI unit of energy, the joule...

. It is important to note that heat and work are measured using the same units.

Heat conduction is not considered to be a form of work, since the energy is transferred into atomic vibration rather than a macroscopic displacement. However, heat conduction can perform work by expanding a gas in a cylinder such as in the engine of an automobile.

## Mathematical calculation

Calculating the work as "force times straight path segment" can only be done in the simple circumstances described above. If the force is changing, if the body is moving along a curved path, possibly rotating and not necessarily rigid, then only the path of the application point of the force is relevant for the work done, and only the component of the force parallel to the application point velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

is doing work (positive work when in the same direction, and negative when in the opposite direction of the velocity). This component of the force can be described by the scalar quantity called scalar tangential component (, where is the angle between the force and the velocity). And then the most general definition of work can be formulated as follows:
Work of a force is the line integral of its scalar tangential component along the path of its application point.

Simpler (intermediate) formulas for work and the transition to the general definition are described in the text below.

### Force and displacement

If a force F that is constant with respect to time acts on an object while the object is translationally displaced for a displacement vector d, the work done by the force on the object is the dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

of the vector
Vector (mathematics and physics)
In mathematics and physics, a vector is an element of a vector space. If n is a non negative integer and K is either the field of the real numbers or the field of the complex number, then K^n is naturally endowed with a structure of vector space, where K^n is the set of the ordered sequences of n...

s F and d:
(1)

where is the angle between the force and the displacement vector.
Whereas the magnitude and direction of the force must remain constant, the object's path may have any shape: the work done is independent of the path and is determined only by the total displacement vector . A most common example is the work done by gravity – see diagram. The object descends along a curved path, but the work is calculated from , which gives the familiar result .

More generally, if the force causes (or affects) rotation of the body, or if the body is not rigid, displacement of the point to which the force is applied (the application point) must be used to calculate the work. This is also true for the case of variable force (below) where, however, magnitude of can equally be interpreted as differential displacement magnitude or differential length of the path of the application point. (Although use of displacement vector most frequently can simplify calculation of work, in some cases simplification is achieved by use of the path length, as in the work of torque calculation below.)

In situations where the force changes over time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

, equation (1) is not generally applicable. But it is possible to divide the motion into small steps, such that the force is well approximated as being constant for each step, and then to express the overall work as the sum over these steps. This will give an approximate result, which can be improved by further subdivisions into smaller steps (numerical integration). The exact result is obtained as the mathematical limit of this process, leading to the general definition below.

The general definition of mechanical work is given by the following line integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...

:
(2)

where: is the path or curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

traversed by the application point of the force; is the force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

vector; is the position vector; and is its velocity.

The expression is an inexact differential
Inexact differential
An inexact differential or imperfect differential is a specific type of differential used in thermodynamics to express the path dependence of a particular differential. It is contrasted with the concept of the exact differential in calculus, which can be expressed as the gradient of another...

which means that the calculation of is path-dependent and cannot be differentiated to give .

Equation (2) explains how a non-zero 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. This is what happens during 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 illustrates the difference between work and a related quantity, impulse, which is the integral of force over time. Impulse measures change in a body's momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

, 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.

### Torque and rotation

Work done by a torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

can be calculated in a similar manner, as is easily seen when a force of constant magnitude is applied perpendicularly to a lever arm. After extraction of this constant value, the integral in the equation (2) gives the path length of the application point, i.e. the circular arc , and the work done is .

However, the arc length can be calculated from the angle of rotation (expressed in radians) as , and the ensuing product is equal to the torque applied to the lever arm. Therefore, a constant torque does work as follows:

## Work and kinetic energy

According to the work-energy theorem, if one or more external forces act upon a rigid object, causing its kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

to change from Ek1 to Ek2, then the work (W) done by the net force
Net force
In physics, net force is the total force acting on an object. It is calculated by vector addition of all forces that are actually acting on that object. Net force has the same effect on the translational motion of the object as all actual forces taken together...

is equal to the change in kinetic energy. For translational motion, the theorem can be specified as:

where m is the mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

of the object and v is the object's velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

.

The theorem is particularly simple to prove for a constant force acting in the direction of motion along a straight line. For more complex cases, however, it can be claimed that very concept of work is defined in such a way that the work-energy theorem remains valid.

### Derivation for a particle

In rigid body dynamics, a formula equating work and the change in kinetic energy of the system is obtained as a first integral of Newton's second law of motion
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...

.

To see this, consider a particle P that follows the trajectory X(t) with a force F acting on it. Newton's second law provides a relationship between the force and the acceleration of the particle as
where m is the mass of the particle.

The scalar product of each side of Newton's law with the velocity vector yields
which is integrated from the point X(t1) to the point X(t2) to obtain

The left side of this equation is the work of the force as it acts on the particle along the trajectory from time t1 to time t2. This can also be written as
This integral is computed along the trajectory X(t) of the particle and is therefore path dependent.

The right side of the first integral of Newton's equations can be simplified using the identity
which can be integrated explicitly to obtain the change in kinetic energy,
where the kinetic energy of the particle is defined by the scalar quantity,

The result is the work-energy principle for rigid body dynamics,
This derivation can be generalized to arbitrary rigid body systems.

## Frame of reference

The work done by a force acting on an object depends on the choice of reference frame
Reference frame
Reference frame may refer to:*Frame of reference, in physics*Reference frame , frames of a compressed video that are used to define future frames...

because displacements
Displacement (vector)
A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...

and velocities are dependent on the reference frame in which the observations are being made.

The change in kinetic energy also depends on the choice of reference frame because kinetic energy is a function of velocity. However, regardless of the choice of reference frame, the work energy theorem remains valid and the work done on the object is equal to the change in kinetic energy.

## Zero work

An important class of forces in mechanical systems perform zero work. These are constraint forces that restrict the relative movement of bodies. For example, the centripetal force
Centripetal force
Centripetal force is a force that makes a body follow a curved path: it is always directed orthogonal to the velocity of the body, toward the instantaneous center of curvature of the path. The mathematical description was derived in 1659 by Dutch physicist Christiaan Huygens...

exerted by a string on a ball a uniform circular motion
Circular motion
In physics, circular motion is rotation along a circular path or a circular orbit. It can be uniform, that is, with constant angular rate of rotation , or non-uniform, that is, with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of...

does zero work because this force is perpendicular to the velocity of the ball. As a result the kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

of the moving ball doesn't change.

Another example is a book at rest on a table, the table does no work on the book despite exerting a force equivalent to mg upwards, because no energy is transferred into or out of the book. On the other hand, if the table moves upward, then it performs work on the book, since the force of the table on the book will be acting through a distance.

A current that generates a magnetic field can also produce a magnetic force where a charged particle exerts a force on a magnetic field, but the magnetic force can do no work because the charge velocity is perpendicular to the magnetic field and in order for a force or an object to perform work, the force has to be in the same direction as the distance that it moves.