Energy, Work, and Power
Energy methods are the shortcut that makes force methods unnecessary for a huge class of problems: the work-energy theorem and conservation of mechanical energy let students skip finding acceleration entirely and jump straight from a start state to an end state. This unit also introduces the discipline of tracking where energy goes when it *isn't* conserved (friction, air resistance) and how to read work directly off a force-vs-displacement graph -- both skills that come back explicitly in Unit 8's rotational energy.
What you'll learn
- Calculate work by a constant force, including when force and displacement aren't parallel.
- Apply the work-energy theorem to relate net work to change in kinetic energy.
- Calculate kinetic, gravitational potential, and elastic potential energy.
- Apply conservation of mechanical energy with and without non-conservative forces.
- Calculate power as the rate of energy transfer.
- Interpret force-vs-displacement graphs to find work.
- Distinguish conservative from non-conservative forces and their effect on mechanical energy.
1. Work Done by a Constant Force
Work measures the energy transferred to or from an object by a force acting over a displacement: \(W = Fd\cos\theta\), where \(\theta\) is the angle between the force and the displacement direction. Work is a scalar (no direction), but it can be positive, negative, or zero.
Work is exactly zero whenever the force is perpendicular to the displacement (\(\theta = 90°\)) or when there is no displacement at all — holding a heavy box motionless takes effort, but does zero physics work on the box, since \(d = 0\).
2. The Work-Energy Theorem
The work-energy theorem states that the net work done on an object equals its change in kinetic energy: \(W_{net} = \Delta KE = \tfrac{1}{2}mv_f^2 - \tfrac{1}{2}mv_0^2\). This is often the fastest route to a final speed when forces are constant but the details of the motion (time, acceleration) are not directly asked for.
Note carefully: it's the *net* work — the work done by the combined effect of every force — that equals the change in KE, not the work done by any single force in isolation.
3. Gravitational and Elastic Potential Energy
Gravitational potential energy near Earth's surface is \(PE_g = mgh\), measured relative to any reference height you choose — only *changes* in \(PE_g\) have physical meaning, so the choice of reference level never affects an answer as long as it's used consistently.
Elastic potential energy stored in an ideal spring is \(PE_s = \tfrac{1}{2}kx^2\), measured from the spring's natural (unstretched) length. Because \(x\) is squared, \(PE_s\) is always positive whether the spring is stretched or compressed.
4. Conservation of Mechanical Energy
When only conservative forces do work (gravity, ideal springs — no friction or air resistance), total mechanical energy is conserved: \(KE_i + PE_i = KE_f + PE_f\). This lets you find a final speed from a height or spring compression without ever needing to know the time elapsed or the exact shape of the path.
5. Non-Conservative Forces and Energy Loss
When friction or air resistance act, mechanical energy is not conserved — some is converted to thermal energy (heat) and sometimes sound. The bookkeeping becomes: \(KE_i + PE_i = KE_f + PE_f + |W_{friction}|\), where \(|W_{friction}|\) is the magnitude of energy dissipated by friction. Total energy (including thermal) is still conserved — it's only *mechanical* energy that decreases.
6. Power
Power is the rate of energy transfer or work done: \(P = W/t\). When force and velocity are constant and aligned, this simplifies to \(P = Fv\) — useful for problems describing a motor or engine working at constant speed.
7. Reading Work from Force-vs-Displacement Graphs
Work equals the area between a force-vs-displacement graph and the horizontal axis, even when the force isn't constant — a spring's force-vs-displacement graph, for instance, is a straight line through the origin, and the "area" under it is a triangle. Break irregular graphs into triangles and rectangles and add the areas (with a negative sign for any area below the axis, since that represents negative work).
Key equations
- W = Fd cosθ — Work by a constant force
- KE = 1/2 mv^2 — Any motion-related energy problem
- PEg = mgh — Gravitational PE near Earth's surface
- PEs = 1/2 kx^2 — Elastic PE in springs
- Wnet = ΔKE — Work-energy theorem
- P = W/t = Fv — Power problems