In physics, a constant force F does work when it undergoes a displacement in the direction of the force. If an object moves a distance s under this force, the work done by the force is given by the dot product of force and displacement vectors.

      W = F • s

If the angle between the force and the moving direction is θ, then the equation can be written in scalar forma

      W = F s cosθ

Work of Available Force

If the force F is not a constant, the work done by this force can be derived using integration over the path.

Suppose an object moves along the x-axis in the positive direction from x = a to x = b. At each point between a and b a force f(x) acts on the object, where f is a continuous function. First divide [a, b] into n small subintervals, so the typical interval is [x_i, x_i + dx]. Let x_i^* be a point in this interval. The force at this point will be f(x_i^*). When this object moves from x_i to x_i+ dx, the work done by force f(x) can be approximated as

      dW = f(x_i^*) dx

Using Riemann sum, the total work done by f(x) from x = a to x = b can be approximated as

This approximation becomes better as n becomes larger. The work done in moving the object from a to b is defined as the limiting value of this quantity as n approaches infinite, and it is a definite integral: 
     
      

If the force and the moving direction has an angle θ, the work done by the force is