How to calculate the work done by a force field?

To calculate the work done by a force field, we need to integrate the dot product of the force and displacement vectors.

Work is defined as the product of force and displacement in the direction of the force. In the case of a force field, the force varies with position, so we need to integrate the dot product of the force and displacement vectors over the path of the object.

Let F(x,y,z) be the force field and r(t) = <x(t), y(t), z(t)> be the path of the object. Then the work done by the force field along the path is given by:

W = ∫ F(r(t)) · r'(t) dt

where · denotes the dot product and r'(t) is the derivative of r(t) with respect to t.

If the force field is conservative, then we can express it as the gradient of a scalar potential function, i.e. F = -∇φ. In this case, the work done by the force field along any closed path is zero, and the work done along any path depends only on the endpoints of the path.

In summary, to calculate the work done by a force field, we need to integrate the dot product of the force and displacement vectors over the path of the object. If the force field is conservative, we can express it as the gradient of a scalar potential function, and the work done along any closed path is zero.