Unit 3: Work, Energy, and Power

Work (W)

Energy transferred into or out of a system by a force acting through a displacement; in general, W = ∫ F⃗ · d r⃗ .

Mechanical work (vs. effort)

Mechanical work depends on force and displacement; if displacement is zero (object held still), mechanical work on the object is zero even if you feel tired.

Displacement (Δ r⃗ )

The vector change in position; work depends on the displacement through which a force acts.

Infinitesimal displacement (d r⃗ )

A tiny vector step along a path used in calculus-based work calculations.

Infinitesimal work (dW)

Differential work done over an infinitesimal displacement: dW = F⃗ · d r⃗ .

Work line integral

General definition of work over a path: W = ∫_1^2 F⃗ · d r⃗ ; needed when force or direction changes or the path is curved.

Dot product (F⃗ ·Δ r⃗ )

Operation that picks out the component of force along the displacement: F⃗ ·Δ r⃗ = FΔr cosθ.

Constant-force work

For constant force and straight-line displacement: W = F⃗ ·Δ r⃗ = FΔr cosθ.

Cosine factor (cosθ)

Accounts for the angle between force and displacement; only the parallel component contributes to work.

Parallel component of force

The component along displacement: F∥ = F cosθ; it is the only part that does work.

Positive work

Work is positive when the force component is in the same direction as displacement (θ = 0° gives maximum positive work).

Negative work

Work is negative when the force component opposes the displacement (θ = 180°).

Zero work

Work is zero when force is perpendicular to displacement (θ = 90°), even if the object moves a distance.

Joule (J)

SI unit of work/energy.

Newton-meter (N·m)

Equivalent unit to the joule: 1 J = 1 N·m.

1D work integral

If motion is along x and the force component is Fx(x), then W = ∫{x1}^{x2} F_x(x) dx.

Variable force

A force whose magnitude and/or direction changes; work must be found with an integral, not just Fd.

Force–position graph (F_x vs x)

A graph used to compute work via W = ∫ F_x dx.

Signed area under an F–x curve

Geometric meaning of ∫ F_x dx; area above the x-axis gives positive work and area below gives negative work.

Net work (W_net)

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

Work–energy theorem

Net work equals the change in kinetic energy: W_net = ΔK.

Kinetic energy (K)

Energy of motion: K = (1/2)mv^2 (scalar and nonnegative).

Change in kinetic energy (ΔK)

ΔK = Kf − Ki = (1/2)mvf^2 − (1/2)mvi^2.

Kinetic energy not always conserved

K stays constant only in special cases (e.g., zero net work); often it transforms into potential or thermal energy.

Conservative force

A force whose work between two points depends only on endpoints (not on the path taken).

Path independence

Defining property of conservative forces: work from A to B is the same for any path connecting A and B.

Closed-loop work criterion

For a conservative force, the work around any closed path is zero: ∮ F⃗ · d r⃗ = 0.

Potential energy (U)

Energy stored in configuration, defined for conservative forces so that changes in U track conservative work.

Conservative work–potential relation (W_cons = −ΔU)

For a conservative force: Wcons = −(Uf − U_i); positive work by the conservative force means U decreases.

Gravitational potential energy (near Earth)

Often written as U_g = mgy (choice of zero is arbitrary).

Change in gravitational potential energy (ΔU_g)

Near Earth with constant g: ΔU_g = mgΔy.

Hooke’s law

Ideal spring force: F_s = −kx (restoring force opposite displacement).

Spring potential energy

Energy stored in a spring: U_s = (1/2)kx^2, where x is displacement from equilibrium.

Nonconservative force

A force (like friction/drag) whose work depends on path length and typically converts mechanical energy into thermal/internal energy.

Kinetic friction does negative work

Friction usually opposes motion, so its work is negative (e.g., Wf = −fk d along the path).

Mechanical energy (E_mech)

Sum of kinetic and potential energies: E_mech = K + U.

Conservation of mechanical energy

If only conservative forces do work (or Wnc = 0), then Ki + Ui = Kf + U_f.

Nonconservative work equation

When nonconservative forces do work: Wnc = ΔK + ΔU (equivalently Ki + Ui + Wnc = Kf + Uf).

Choosing the system

Defining the system determines whether a force is treated via work (external) or via potential energy (internal); you must be consistent.

Double-counting gravity

Common mistake: including both gravitational potential energy U_g and work by gravity in the same energy equation for the same system.

Power (P)

Rate of doing work or transferring energy.

Average power (P_avg)

P_avg = W/Δt.

Instantaneous power

P = dW/dt.

Power from force and velocity (P = F⃗ · v⃗ )

Using dW = F⃗ · d r⃗ and v⃗ = d r⃗ /dt gives P = F⃗ · v⃗ (depends on the component of force along velocity).

Watt (W)

SI unit of power: 1 W = 1 J/s.

Power depends on force component along velocity

If force is along motion, P = Fv; if opposite, power is negative; if perpendicular, power is zero at that instant.

Force from a potential (1D)

Relationship between force and potential energy in one dimension: F_x = −dU/dx.

Allowed region on a U(x) graph

For conservative motion with total energy E, motion is possible only where U(x) ≤ E (since K = E − U ≥ 0).

Turning point

A position where speed is zero and K = 0, so U(x) = E on a potential-energy graph.

Unstable equilibrium

An equilibrium point where dU/dx = 0 and U(x) is a local maximum; small displacements tend to move the object away.