AP Physics 1 Unit 3 Notes: Building Understanding of Work, Energy, and Power

Work (physics)

Energy transferred into or out of a system by a force acting through a displacement; if there is no displacement, no work is done.

Work by a constant force

For straight-line motion with constant force, work is W = Fd cos(θ), where θ is the angle between force and displacement.

Joule (J)

SI unit of work/energy; 1 J = 1 N·m.

Angle θ (between force and displacement)

The angle used in W = Fd cos(θ); it determines how much of the force contributes to energy transfer along the displacement.

Parallel component of force (F∥)

The component of a force along the displacement; F∥ = F cos(θ), and W = F∥ d.

Positive work

Work done when a force has a component in the same direction as displacement (0° ≤ θ < 90°), transferring energy into the object.

Negative work

Work done when a force component is opposite displacement (90° < θ ≤ 180°), removing mechanical energy from the object.

Zero work

Occurs when force is perpendicular to displacement (θ = 90°) or when displacement is zero; no energy is transferred by that force.

Work sign from cosine

In W = Fd cos(θ), cos(θ) automatically gives the sign: positive for acute θ, zero at 90°, negative for obtuse θ.

Normal force (work)

Often does zero work because it is perpendicular to motion on a surface, but it can do work if displacement has a component along the normal (e.g., accelerating elevator platform).

Gravity (work vs. potential)

Gravity can be treated as doing work (Wg) in an object-only system or as changing gravitational potential energy (ΔUg) in an object+Earth system—do not do both in the same equation.

Tension (work)

A force from a rope/cable; it does work only if it has a component along the object’s displacement (often zero for circular motion with radial tension).

Kinetic friction (work)

A nonconservative force that typically does negative work when it opposes motion; often modeled as Wf = −fk d.

Spring force

A restoring force for an ideal spring described by Hooke’s law; its magnitude changes with displacement from equilibrium.

Hooke’s law

Relationship for an ideal spring: Fs = kx, where k is spring constant and x is displacement from equilibrium.

Work done on a system

Energy transferred to the system by external forces (work input increases the system’s energy).

Work done by a system

Energy transferred from the system to the surroundings (the system does work on something else).

Net work (Wnet)

Sum of the works done by all forces on an object: Wnet = ΣWi.

Force–position (F vs. x) graph work

Work done by a force over a displacement equals the signed area under the F vs. x curve.

Area under an F vs. x curve

Graphical method for work; for constant force it’s a rectangle (FΔx), and for piecewise linear forces it can be found using triangle/trapezoid geometry.

Kinetic energy (K)

Energy of motion for mass m at speed v: K = (1/2)mv² (a scalar quantity).

Work–energy theorem

The net work done on an object equals the change in its kinetic energy: Wnet = ΔK.

Change in kinetic energy (ΔK)

Difference between final and initial kinetic energies: ΔK = Kf − Ki.

Speed-squared dependence of kinetic energy

Because K ∝ v², doubling speed increases kinetic energy by a factor of 4.

Conservative force

A force whose work depends only on initial and final positions (not path); it allows a potential energy function to be defined.

Nonconservative force

A force whose work depends on path (e.g., friction, air resistance); it changes mechanical energy into other forms like thermal.

Closed-path work for conservative forces

For a conservative force, the net work done around a closed loop is zero.

Potential energy (U)

Energy stored due to system configuration (e.g., Earth–object or spring–object interaction), not an intrinsic property of a lone object.

Work–potential energy relation for conservative forces

For a conservative force, work equals the negative change in potential energy: Wc = −ΔU.

Gravitational potential energy change near Earth (ΔUg)

Near Earth’s surface, ΔUg = mgΔy, where Δy is vertical position change (final minus initial).

Zero level of gravitational potential energy

A reference choice; only changes in gravitational potential energy (ΔUg) affect the physics, not the absolute zero.

Work done by gravity (Wg)

Related to gravitational potential energy by Wg = −ΔUg; gravity does positive work when an object moves downward.

Elastic potential energy (Us)

Energy stored in an ideal spring: Us = (1/2)kx².

Change in spring potential energy (ΔUs)

ΔUs = (1/2)k(xf² − xi²); depends on displacement from equilibrium, not distance traveled.

Work done by a spring (Ws)

Work by the spring force equals the negative change in its potential energy: Ws = −ΔUs.

Mechanical energy (Emech)

Sum of kinetic and potential energies from conservative interactions: Emech = K + U.

Conservation of mechanical energy condition

Mechanical energy is conserved when only conservative forces do work (or nonconservative forces do zero work).

Mechanical energy conservation equation

When conserved: Ki + Ui = Kf + Uf.

Energy “snapshot” method

Solving by comparing initial and final states: choose system, pick states, write K and relevant U terms, set initial total equal to final total (if applicable).

Mass cancels in frictionless height-to-speed problems

In frictionless gravitational motion, speed from a given vertical drop depends on Δy (via g), not mass, because m cancels in energy equations.

Tension does no work in ideal pendulum motion

In a pendulum, tension is perpendicular to the arc displacement at each moment, so it does zero work (idealized case).

General energy relation with nonconservative work

Work by nonconservative forces equals change in mechanical energy: Wnc = ΔEmech = (Kf + Uf) − (Ki + Ui).

Thermal energy increase (ΔEth) from friction

Energy is not destroyed by friction; mechanical energy converted by friction appears as increased thermal energy of the interacting surfaces.

System boundary (energy problems)

The chosen set of objects included in analysis (e.g., object-only vs. object+Earth); it determines whether forces are treated as external work or internal potential energy changes.

Object-only system (energy approach)

A system choice where gravity, friction, and applied forces are external and do work; aligns naturally with Wnet = ΔK.

Object + Earth (and/or spring) system

A system choice where gravity and/or spring forces are internal and represented via potential energy (Ug, Us); external work may include friction or applied pushes from outside the system.

Double-counting gravity (common error)

Including both Wg and ΔUg in the same energy equation; you must choose either the work view (object-only) or the potential-energy view (object+Earth).

Power (P)

Rate of energy transfer or work done: Pavg = W/Δt; SI unit is the watt (W = J/s).

Instantaneous power for a force

P = Fv cos(θ), where θ is the angle between force and velocity; only the component along motion transfers energy.

Efficiency

Ratio of useful output energy (or power) to input energy (or power): efficiency = useful output/input; ≤ 1 (≤ 100%).

Energy bar chart

A qualitative representation that tracks energy storage forms (K, Ug, Us) and transfers (external work) and/or conversions (thermal energy) between initial and final states.