Work, Energy, and Power
Work, Energy, and Power
Work is energy transferred by a force: W = F d cos θ for a constant force, or the integral of F·dr in general. The work-energy theorem, W_net = ΔK, bridges dynamics and energy. Conservative forces (gravity, ideal springs) permit a potential energy - U = mgh and U = ½kx² - while friction and applied pushes do not. When only conservative forces act, K + U is constant; otherwise write K_i + U_i + W_nc = K_f + U_f. Power is the rate of energy transfer, P = dW/dt = F v cos θ.
Traps: assigning work to forces perpendicular to the displacement (the normal force on a level floor and any centripetal force do zero work); forgetting that friction's work is negative and depends on path length; dropping the W_nc term when friction or an external push acts; and confusing the work done by one force with the net work done on the object.
Worked example: a 1 kg block slides 4 m down a 30° incline (vertical drop 2 m) with 4 J of energy lost to friction. Then ½mv² = mgh − 4 = 19.6 − 4 = 15.6 J, so v = sqrt(2(15.6)/1) ≈ 5.6 m/s.
Rule of thumb: before integrating a force, ask whether an energy ledger (initial K + U, plus nonconservative work, equals final K + U) solves the problem faster.