Work (physics)

In science, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. A force is said to do positive work if it has a component in the direction of the displacement of the point of application. A force does negative work if it has a component opposite to the direction of the displacement at the point of application of the force.
For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball multiplied by the distance to the ground. If the ball is thrown upwards, the work done by the gravitational force is negative, and is equal to the weight multiplied by the displacement in the upwards direction.
Both force and displacement are vectors. The work done is given by the dot product of the two vectors, where the result is a scalar. When the force is constant and the angle between the force and the displacement is also constant, then the work done is given by:
If the force and/or displacement is variable, then work is given by the line integral:
where is the infinitesimal change in displacement vector, is the infinitesimal increment of time, and represents the velocity vector. The first equation represents force as a function of the position and the second and third equations represent force as a function of time.
Work is a scalar quantity, so it has only magnitude and no direction. Work transfers energy from one place to another, or one form to another. The SI unit of work is the joule, the same unit as for energy.

History

The ancient Greek understanding of physics was limited to the statics of simple machines, and did not include dynamics or the concept of work. During the Renaissance the dynamics of the Mechanical Powers, as the simple machines were called, began to be studied from the standpoint of how far they could lift a load, in addition to the force they could apply, leading eventually to the new concept of mechanical work. The complete dynamic theory of simple machines was worked out by Italian scientist Galileo Galilei in 1600 in Le Meccaniche, in which he showed the underlying mathematical similarity of the machines as force amplifiers. He was the first to explain that simple machines do not create energy, only transform it.

Early concepts of work

Although work was not formally used until 1826, similar concepts existed before then. Early names for the same concept included moment of activity, quantity of action, latent live force, dynamic effect, efficiency, and even force. In 1637, the French philosopher René Descartes wrote:
In 1686, the German philosopher Gottfried Leibniz wrote:
In 1759, John Smeaton described a quantity that he called "power" "to signify the exertion of strength, gravitation, impulse, or pressure, as to produce motion." Smeaton continues that this quantity can be calculated if "the weight raised is multiplied by the height to which it can be raised in a given time," making this definition remarkably similar to Coriolis's.

Etymology and modern usage

The term work, and the use of the work-energy principle in mechanics, was introduced in the late 1820s independently by French mathematician Gaspard-Gustave Coriolis and French Professor of Applied Mechanics Jean-Victor Poncelet. Both scientists were pursuing a view of mechanics suitable for studying the dynamics and power of machines, for example steam engines lifting buckets of water out of flooded ore mines. According to Rene Dugas, French engineer and historian, it is to Solomon of Caux "that we owe the term work in the sense that it is used in mechanics now". The concept of virtual work, and the use of variational methods in mechanics, preceded the introduction of "mechanical work" but was originally called "virtual moment". It was re-named once the terminology of Poncelet and Coriolis was adopted.

Units

The SI unit of work is the joule, named after English physicist James Prescott Joule. According to the International Bureau of Weights and Measures it is defined as "the work done when the point of application of 1 MKS unit of force moves a distance of 1 metre in the direction of the force."
The dimensionally equivalent newton-metre is sometimes used as the measuring unit for work, but this can be confused with the measurement unit of torque. Usage of N⋅m is discouraged by the SI authority, since it can lead to confusion as to whether the quantity expressed in newton-metres is a torque measurement, or a measurement of work.
Another unit for work is the foot-pound, which comes from the English system of measurement. As the unit name suggests, it is the product of pounds for the unit of force and feet for the unit of displacement. One joule is approximately equal to 0.7376 ft-lbs.
Non-SI units of work include the newton-metre, erg, the foot-pound, the foot-poundal, the kilowatt hour, the litre-atmosphere, and the horsepower-hour. Due to work having the same physical dimension as heat, occasionally measurement units typically reserved for heat or energy content, such as therm, BTU and calorie, are used as a measuring unit.

Work and energy

The work done by a constant force of magnitude on a point that moves a displacement in a straight line in the direction of the force is the product
For example, if a force of 10 newtons acts along a point that travels 2 metres, then. This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity.
The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance.
Work is closely related to energy. Energy shares the same unit of measurement with work because the energy from the object doing work is transferred to the other objects it interacts with when work is being done. The work–energy principle states that an increase in the kinetic energy of a rigid body is caused by an equal amount of positive work done on the body by the resultant force acting on that body. Conversely, a decrease in kinetic energy is caused by an equal amount of negative work done by the resultant force. Thus, if the net work is positive, then the particle's kinetic energy increases by the amount of the work. If the net work done is negative, then the particle's kinetic energy decreases by the amount of work.
From Newton's second law, it can be shown that work on a free, rigid body, is equal to the change in kinetic energy corresponding to the linear velocity and angular velocity of that body,
The work of forces generated by a potential function is known as potential energy and the forces are said to be conservative. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object,
These formulas show that work is the energy associated with the action of a force, so work subsequently possesses the physical dimensions, and units, of energy.
The work/energy principles discussed here are identical to electric work/energy principles.

Constraint forces

Constraint forces determine the object's displacement in the system, limiting it within a range. For example, in the case of a slope plus gravity, the object is stuck to the slope and, when attached to a taut string, it cannot move in an outwards direction to make the string any 'tauter'. It eliminates all displacements in that direction, that is, the velocity in the direction of the constraint is limited to 0, so that the constraint forces do not perform work on the system.
For a mechanical system, constraint forces eliminate movement in directions that characterize the constraint. Thus the virtual work done by the forces of constraint is zero, a result which is only true if friction forces are excluded.
Fixed, frictionless constraint forces do not perform work on the system, as the angle between the motion and the constraint forces is always 90°. Examples of workless constraints are: rigid interconnections between particles, sliding motion on a frictionless surface, and rolling contact without slipping.
For example, in a pulley system like the Atwood machine, the internal forces on the rope and at the supporting pulley do no work on the system. Therefore, work need only be computed for the gravitational forces acting on the bodies. Another example is the centripetal force exerted inwards by a string on a ball in uniform circular motion sideways constrains the ball to circular motion restricting its movement away from the centre of the circle. This force does zero work because it is perpendicular to the velocity of the ball.
The magnetic force on a charged particle is, where is the charge, is the velocity of the particle, and is the magnetic field. The result of a cross product is always perpendicular to both of the original vectors, so. The dot product of two perpendicular vectors is always zero, so the work, and the magnetic force does not do work. It can change the direction of motion but never change the speed.

Mathematical calculation

For moving objects, the quantity of work/time is integrated along the trajectory of the point of application of the force. Thus, at any instant, the rate of the work done by a force is the scalar product of the force, and the velocity vector of the point of application. This scalar product of force and velocity is known as instantaneous power. Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.
Work is the result of a force on a point that follows a curve, with a velocity, at each instant. The small amount of work that occurs over an instant of time is calculated as
where the is the power over the instant. The sum of these small amounts of work over the trajectory of the point yields the work,
where C is the trajectory from x to x. This integral is computed along the trajectory of the particle, and is therefore said to be path dependent.
If the force is always directed along this line, and the magnitude of the force is, then this integral simplifies to
where is displacement along the line. If is constant, in addition to being directed along the line, then the integral simplifies further to
where s is the displacement of the point along the line.
This calculation can be generalized for a constant force that is not directed along the line, followed by the particle. In this case the dot product, where is the angle between the force vector and the direction of movement, that is
When a force component is perpendicular to the displacement of the object, no work is done, since the cosine of 90° is zero. Thus, no work can be performed by gravity on a planet with a circular orbit. Also, no work is done on a body moving circularly at a constant speed while constrained by mechanical force, such as moving at constant speed in a frictionless ideal centrifuge.