EngArc - L - Work

Work

Quick
Work is the scalar product of force and displacement, where the work is relative to the force with respect to the displacement of an object. Work [J (joules)] is force [N (newtons)] times distance [m (meters)]. Work is energy.

Equations

(Eq1)

W = Fx

Work defining a force acting through a distance

(Eq2)

W =

∫

F dx

Work expressed as an integral

(Eq3)

W = Fx cosθ =

⋅

= Fx (for θ = 90°)

There are many different equations for work, as work is energy and there are many different types of energy with which work can be equated to, such as heat.

Nomenclature

symbol	description
W	work
F	force (a vector quantity)
x	displacement (a vector quantity)

Details

Work is a broad topic and expands far and wide. Work is covered in, but not limited to, the following subjects: thermodynamics, chemistry, physics, mechanics of materials, statics, and dynamics. It is hard work to pull, push, or lift a heavy object. In this sense, the word 'work' relates to the everyday meaning of work. In science and engineering, work has a much more precise definition. It is found that the total work done on a particle by all forces acting on it equals the change in its kinetic energy, which is related to the particle's speed. This relationship holds even when the forces acting on the particle are not constant. In each case of pulling, pushing, and lifting, work is being performed through the exertion of force on a body while that body moves from one position to another, undergoing a displacement. If the force or displacement increases, so does the work. A rising piston, a rotating shaft, and an electric wire crossing the system boundary are all associated with work interactions. Compressors, pumps, fans, and mixers consume work. In fluid mechanics, work-consuming devices transfer energy to a fluid, such as a fan.

Consider a body, assuming it is treated as a particle, which experiences a displacement due to a force which acts in the same direction as the displacement. The body starts at point a and moves to point b. During the displacement from point a to point b the force remains constant. At point b the force is removed. From point a to point b the body has experienced a displacement of magnitude x. The displacement occurs along a straight line.

The work, W, done by a constant force, F, acting on the body under these conditions is defined as:

(Eq1) W = Fx

The work done on the body is greater if either the force, F, or the displacement, x, is greater, as stated above. Work is energy.

The SI unit of work is the joule, and is abbreviated by J.

Looking at the pictures below, if a force pushes on a block in the direction of motion of the block, then the work equals the force applied multiplied by the displacement of the block. If the same force pushes on the block, but at an angle to the motion of the block, then the work equals the component of the force parallel to the motion of the block multiplied by the displacement of the block. Therefore, for the same force applied to the block in both situations, the situation where the force is applied at an angle will have smaller work because only part of the force is contributing to the motion of the block.

W = Fx
W = Fx cosθ = ⋅

It is assumed that F and θ are constant during the displacement, and that no other forces are acting on the block, such as friction or wind resistance. If θ = 0, as in the case of the first situation, then cosθ = 1, whereby the equation for work then becomes Eq1.

The second equation above has the form of the scalar product of two vectors, because both the force, and displacement are vectors.

It is important to understand that work is a scalar quantity, as defined by the scalar product.

Energy transfer as work is 100% available. By definition work is 100% exergy or availability.

Work and Angle of Application (Sign Convention)

Work is respective to the force used to calculate it. Look at the following pictures. If an object moved in a direction, and the force contributed to the direction of movement, then the value of work is positive. If an object moved in a direction, and the force worked against the motion of that object, then the work is negative. If an object moved in a direction, and the force did not contribute whatsoever to the motion of the object, then the work is zero, and zero contribution also indicates that the force is acting perpendicular to the object, as shown. Therefore, the work can give an indication of the angle and direction of the force applied on an object with respect to the motion of that object.

work is positive (+)	force contributes to direction of movement
work is negative (−)	force acts against direction of movement
work is zero, W = 0	force has no contribution to movement

Keep in mind that the force under consideration need not be the force causing the displacement with regards to calculating work. For the previous pictures, visualize, for example, a vehicle whose wheels are locked forward and is rolling in a straight direction, the force is what you are acting on the vehicle. Obviously the first and third scenarios wouldn't make much sense if you want the car to stop. The second case may be for example if you pushing back on the car at an angle.

There are many situations in which forces act but do little to zero work. Structures, such as buildings or bridges, may bear great loads but experience very minimal displacement, so the work would be very small relative to the supported loads.

From newton's third law of motion, when one body does negative work on a second body, the second body does an equal amount of positive work on the first body, and vice versa.

Work may also be calculated for the sum of a number of forces acting on a body, by taking the sum of the forces contributing to the motion of the body multiplied by the displacement.

Work with Respect to a Fixed Coordinate System

Consider a particle which moves from point A to point A' as shown below. If x denotes the position vector corresponding to point A, the small vector joining A and A' can be denoted by the differential dx; the vector dx is the displacement of the particle.

Now, assume that a force F is acting on the particle. The work of the force F corresponding to the displacement dx is defined as the quantity:

dW = F dx

This is the work contributed by the force F to the displacement dx. The formula is obtained by forming the scalar product of the force F and the displacement dx. Denoting by F and dx respectively, the magnitudes of the force and of the displacement, and by θ the angle formed by F and dx, and recalling the definition of the scalar product of two vectors, it can be written:

dW = F dx cos θ

Using this formula from the scalar product lesson, the work can be expressed in terms of the rectangular componenets of the force and displacement:

dW = F_xdx + F_ydy + F_zdz

Being a scalar quantity, work has a magnitude and a sign but no direction. The work should be expressed in units obtained by multiplying units of length by units of force. The work is positive if the angle θ is acute and negative if θ is obtuse.

Work Explained Through Thermodynamics and Mechanics of Materials

In thermodynamics, typically the displacement is in the same direction as the force, mainly dealing with vertical motion. In contrast to Eq1 from above, work can also be expressed as an integral:

(Eq2)

W =

∫

F dx

Eq2 enables the work to be found in order to raise a weight, to stretch a wire, or to move a charged particle through a magnetic field. This equation can also apply in mechanics of materials, where for a rod, W may be the strain energy and F the load applied as shown in the following picture:

The definition of work can be tied in with the concepts of systems, properties, and processes. Viewing work in this perspective, it can be defined in the following way: Work is done by the system if the sole effect on the surroundings (everything external to the system) could be the raising of a weight. Notice that the raising of a weight is in effect a force acting through a distance. Notice also that the definition does not state that a weight was actually raised or that a force actually acted through at a given distance, but that the sole effect external to the system could be the raising of a weight. Work done by a system is considered positive and work done on a system is considered negative. The symbol W designates the work done by a system. Work is the form of energy that fulfills this definition. In general, work is a form of energy in transit, that is, energy being transferred across a system boundary.

This definition can be illustrated:

For part B, as the motor turns, the weight is raised, and the sole effect external to the system is the raising of a weight. So, for part A, work is crossing the boundary of the system, because the sole effect external to the system could be the raising of a weight. Looking at part C, the only limiting factor in having the sole external effect be the raising of a weight is the inefficiency of the motor. However, as a more efficient motor is designed, with lower bearing and electrical losses, a certain limit can be approached that meets the requirement of having the only external effect be the raising of a weight. Therefore, it can be concluded that when there is a flow of electricity across the boundary of a system, it is work.