Work-Energy Principle for Particles

Core Idea

The work-energy principle states that net work done on a particle equals its change in kinetic energy: U₁₋₂ = T₂ − T₁, where T = ½mv². Work by a force along a path is U = ∫F·dr. Conservative forces (gravity, springs) have associated potential energy: V_g = mgh, V_e = ½kx². For conservative systems, total mechanical energy is conserved: T₁ + V₁ = T₂ + V₂. When non-conservative forces (friction, applied forces) act, the work they do modifies the energy balance: T₁ + V₁ + U₁₋₂(nc) = T₂ + V₂.

How It's Best Learned

Classify every force as conservative or non-conservative. For conservative systems, apply energy conservation directly between two states without integrating equations of motion. For problems with friction or variable applied forces, compute work integrals explicitly.

Common Misconceptions

• Normal forces and forces perpendicular to displacement do zero work — forgetting this inflates the work calculation.
• Applying conservation of energy when friction is present without including the energy dissipation term.
• Using the spring energy formula ½kx² with x measured incorrectly (x must be the spring deformation from natural length).

Explainer

Newton's second law and the work-energy principle are two ways to analyze the same motion, but they are not equally convenient for every problem. Newton's law requires tracking acceleration at each instant; work-energy connects two states — a start and an end — using scalar energy quantities, avoiding vectors and integration of force over time. Whenever you care only about how fast a particle is moving at one point given its speed at another, work-energy is usually faster.

The central equation is U₁₋₂ = T₂ − T₁, where U₁₋₂ = ∫F·dr is the net work done on the particle between states 1 and 2, and T = ½mv² is kinetic energy. Work is a dot product: only the component of force along the displacement contributes. This is why a normal force — always perpendicular to motion along a surface — does zero work. Gravity does work equal to mg times the vertical drop, regardless of the path taken; this path-independence is the hallmark of a conservative force.

For conservative forces (gravity and ideal springs), we define potential energy so that work can be expressed as a change in stored energy: U_gravity = −ΔV_g = −mgΔh, U_spring = −ΔV_e = −Δ(½kx²). When only conservative forces act, T + V = constant, which is the conservation of mechanical energy. This is a powerful shortcut: pick any two points on the path, write the energy balance, and solve — no need to integrate equations of motion.

When non-conservative forces act — friction, air drag, applied motors — they add or remove mechanical energy. The modified equation is T₁ + V₁ + U₁₋₂(nc) = T₂ + V₂. Friction always contributes a negative U(nc) equal to the friction force times the distance slid. The key habit is first classifying every force: conservative (gets a potential energy term) or non-conservative (its work must be computed explicitly and added to the equation). Missing a friction term, or including the normal force as a contributor, are the two most common sources of wrong answers.

Spring problems carry an additional trap: the deformation x in ½kx² must be measured from the spring's natural (unstretched) length, not from an arbitrary reference position. If the spring is pre-compressed by x₀ and then further compressed to x₁, the change in spring potential energy is ½k(x₁² − x₀²), not ½k(x₁ − x₀)². Computing ½k(Δx)² instead of Δ(½kx²) is a systematic error that gives wrong answers whenever the spring has any initial deformation.