work (idea)

In physics

The work done on a body as it travels a path from one point to another is the average of the force along the direction of motion (the component tangent to the curve) over that path times the length of the path. If the force F is constant in magnitude along the path of motion and parallel to it, then

W=|F|*x

where x is the length of the path (and |F| is the magnitude of F). If the force makes an angle θ with the path of the body, then the work can also be expressed as

W=|F|*x*Cos(θ)

In the most general case work done depends on the path taken and the endpoints, but in many cases of interest the work done only depends on the end points of the path. In that case F is called a conservative force.

From these definitions, we can see that the work done on the body is positive if the force acts in the direction of motion and negative if the force acts opposite the direction of motion. The work done on the body is the amount of energy that the body gains, or in the case of negative work the amount that it loses. This is probably the most fundamental definition of energy. Work is measured in units of Joules in SI. If we think only about mechanical energy, then the work done on a body is equal to the change in kinetic energy. The kinetic energy is KE=¹/₂*m*v², so if we consider work W₁₂ to be done on the body while crossing the path between points 1 and 2, with the speed at point 1 being v₁ and v₂ at point 2, then

W₁₂=KE₂-KE₁=¹/₂*m*v₂²-¹/₂*m*v₁²

In more complicated systems, work is converted into other forms of energy, which can include potential energy, thermal energy, elastic energy, and other forms. The important thing is that energy is conserved. Besides, the fact that you can do work to get other sorts of energy, like electrical or thermal, another consequence of the definition of work and energy conservation is simple machines.

Now two examples:

1. A man in a bobsled race pushed his bobsled horizontally with a force of 500 N over a distance of 4 m at the beginning of a race. At the end of that length, the man has done work of
   W = F*x = (500 N)*(4 m) = 2000 J
   If it started from rest it will have a kinetic energy of
   W = KE₂-KE₁ = KE₂-0 = KE₂=2000 J
   If the bobsled has a mass of 40 kg then at it would have a speed of
   KE = ¹/₂*m*v²
   v = sqrt(2*KE/m) = 10 m/s

2. A boy pushes a wheelbarrow with a force of 150 N. Because he has to hold up the wheelbarrow as he pushed it, the force is directed at an angle of 30 degrees up from the horizontal. If he pushes for a distance of 10 m, the work done on the wheelbarrow is
   
   W=F*x*Cos(θ)=(150 N)*(10 m)*(Cos(30 degrees))=750 J

Finally, for the more advanced, the exact definition of work is as follows: Given a force F(x) acting on a body with a path described by the oriented curve C, then the work W done by the force F(x) is defined to be the line integral of F along the curve C, or

W = ∫ F(x) dx
    C

This represents the energy imparted to the body by the force F(x). In the case of a conservative force, W depends only on the endpoints of C and, thus, W = 0 if C is a closed curve. By invoking Newton's second law F = m*a, one may prove that the work done is equal to the change in kinetic energy:

W = ∫ F dx = m*∫ dv/dt dx = m*∫ v*dv/dt dt

W = m*∫ ½*d/dt(v²) dt = ½*m*(v₂²-v₁²)

Where "*" with two vectors is intended to mean the dot product.