Work, energy and the conservation law

Energy is the quantitative measure of different forms of motion and interaction; its unit is 1 J. The scalar product of a force and the displacement is the mechanical work: A = F·s·cosα, where α is the angle between the force and the displacement. For α < 90° the work is positive, for α > 90° negative, for α = 90° zero. The work of gravity depends not on the shape of the path but only on the difference of heights: A = mg(h₁ – h₂); such forces are called potential (conservative), and on a closed path their work is zero. Kinetic energy is Eₖ = mυ²/2, potential energy Eₚ = mgh, total mechanical energy E = Eₖ + Eₚ. In a closed system where only conservative forces do work, E = const – the law of conservation of mechanical energy; in general energy is neither lost nor created, it only passes from one form to another. Efficiency η = Aᵤ/Aₜ·100%, where Aᵤ is the useful work and Aₜ the total work; friction makes η < 100%. An inclined plane gives a gain in force but none in work (the “golden rule of mechanics”). Raising a load m uniformly up an incline of length l and height h, the useful work is mgh, the total work is mgh(1 + μ·cotα) and η = 1/(1 + μ·cotα).