Wtotal

Wtotal, often written as W_tot or W_total, denotes the total work performed on a body by all external forces during a specified displacement. It is defined as the line integral of the net force along the actual path of motion: W_total = ∫_C F · dr, where F is the net force acting on the object and dr is an infinitesimal displacement along the trajectory C. If several forces Fi act on the body, the total work is the sum of the works done by each force: W_total = ∑ ∫_C Fi · dr.

The work-energy relationship states that W_total equals the change in the object's kinetic energy: W_total = ΔK = K2 − K1. If the forces are conservative, the work done by those forces is equal to the negative change in potential energy: W_conservative = −ΔU. In general, W_total can be separated into conservative and non-conservative contributions: W_total = W_conservative + W_nonconservative.

Path dependence: The total work depends on the actual path taken if non-conservative forces are present. For purely conservative forces, the work around a closed path is zero; with non-conservative forces, a nonzero value is possible.

Examples: A block pushed along a horizontal track with friction: the applied work minus the frictional work changes the kinetic energy accordingly. If a block starts at speed v0 and comes to rest against friction, W_total = −½ m v0^2.

Applications: W_total is a fundamental quantity in mechanics, energy analysis, and simulations, used to connect forces, displacements, and energy changes.

See also: work, work-energy theorem, potential energy, conservative force.