3.3 Conservation of Energy

Total Mechanical Energy

Mechanical energy is the sum of kinetic energy (KE) and potential energy (PE): E = KE + PE. In a system with only conservative forces (gravity, springs), total mechanical energy stays constant. This lets you equate initial and final energy states without tracking forces over time, a huge shortcut for solving motion problems.

Conservative vs. Nonconservative Forces

Conservative forces (gravity, spring force) do work that depends only on position, not path, and can be stored as potential energy. Nonconservative forces (friction, air resistance) dissipate energy as heat, so mechanical energy is *not* conserved. The work-energy theorem generalizes: W_nc = ΔKE + ΔPE, accounting for energy lost to nonconservative forces.

Applying Conservation of Energy

For an isolated system with no nonconservative forces, set E_initial = E_final: KE_i + PE_i = KE_f + PE_f. This works for gravitational PE (mgh), elastic PE (½kx²), and combinations. When friction acts, include W_friction = -f·d as an energy loss term: KE_i + PE_i + W_nc = KE_f + PE_f.

Key terms

Conservation of Mechanical Energy

In a system with only conservative forces, KE + PE remains constant: KE_i + PE_i = KE_f + PE_f.

Conservative force

A force whose work depends only on initial/final position, not path (e.g., gravity, spring force). Can be stored as PE.

Nonconservative force

A force (e.g., friction) that dissipates mechanical energy as heat; its work depends on path taken.

Generalized work-energy theorem

W_nc = ΔKE + ΔPE — accounts for energy lost or gained due to nonconservative forces.

Elastic potential energy

PE stored in a spring, given by ½kx², where k is spring constant and x is displacement from equilibrium.

Try an AP-style question

U(x) = 2x³ (J). What is the force at x = 1 m?

1. −2 N
2. −6 N
3. 6 N
4. 2 N