Conservation of energy with friction
Energy is never lost, but friction quietly converts the useful kind into heat, and the accounting must include it.
Kinetic and potential energy
Kinetic energy is energy of motion, (1/2)mv^2, and gravitational potential energy is energy of position, mgh relative to a chosen zero level. When only gravity does work, the total is conserved: whatever kinetic energy the object loses climbing, it regains descending. A ball dropped from 2 m arrives at the ground at sqrt(2(9.8)(2)) = 6.3 m/s no matter its mass.
KE + PE = \text{constant} \quad (\text{only conservative forces})Conservative vs nonconservative
A force is conservative when the work it does depends only on the endpoints: gravity and ideal springs qualify, friction does not. Drag a box across a table and back, and friction has done negative work the whole way, leaving thermal energy behind. Friction is always nonconservative, and its work equals friction force times the path length slid.
KE_i + U_i = KE_f + U_f + f_k dWorked example
A 2.0 kg block starts from rest and slides 3.0 m down a 30 degree incline with mu_k = 0.20. It drops h = 3.0 sin(30) = 1.5 m, releasing U = (2.0)(9.8)(1.5) = 29.4 J. The normal force is mg cos(30) = 16.97 N, so friction is (0.20)(16.97) = 3.39 N and absorbs 3.39 times 3.0 = 10.2 J. The remaining 19.2 J is kinetic, giving v = sqrt(2(19.2)/2.0) = 4.4 m/s, slower than the frictionless 5.4 m/s.
\tfrac{1}{2}mv^2 = mgh - \mu_k mgd\cos\theta = 29.4 - 10.2 = 19.2 \text{ J}Flashcard
Gravitational potential energy near Earth
Check your understanding
A pendulum bob is released from rest with the string taut and horizontal, so the bob sits 0.20 m above the lowest point of the swing. What is its speed at the bottom? Use g = 9.8 m/s^2.