Revision summary

Name the system, force, displacement and interval. Conditions determine which equation can be used and which energy changes belong in the account.

Energy accounts

Total energy change = transfers in - transfers out for a fixed collection of matter. Kinetic, gravitational, electric, elastic, chemical and internal stores are distinct from transfer processes. Work is a transfer; power is a rate.

Count a transfer and its resulting store increase once. Dissipated energy remains in the total account, even when it is less useful for the intended task. Kinetic-plus-potential energy need not stay constant when internal energy increases.

Work and kinetic energy

For constant force, θ is between force and displacement. Work is positive along motion, negative against it and zero for a perpendicular force. For a varying force component, use signed force-component/displacement area.

Derive kinetic energy with W_net = F_resultants = mas and v² - u² = 2as for straight-line constant acceleration. The derivation assumes constant mass and no relevant rotational/internal change. Squared speed makes kinetic energy a scalar; reversing velocity need not change it.

Fields and force direction

Gravitational field

g = gravitational force / mass, in N/kg. Force on a positive mass is along g. Uniform lines are parallel; radial lines point inward towards a spherical mass.

Electric field

E = force / positive test charge, in N/C. F = qE: positive charge along E, negative charge opposite E. Uniform plate lines run positive to negative; radial lines point away from positive and towards negative point charges.

Field-line tangents give direction; spacing represents strength under a consistent convention. Lines do not cross where the field is defined. They are not compulsory trajectories. A smaller test body's smaller force need not mean a smaller field.

H2 equipotentials: potential is constant along an equipotential surface. A nonzero static field is perpendicular to it. Movement of a fixed mass or charge along it gives zero field work, even though the field force is nonzero. Uniform fields have equipotential planes; radial fields have concentric spherical surfaces.

Field work and potential energy

Apply this to the static gravitational/electric field's work. Positive field work decreases potential energy; negative field work increases it. Use the charge's actual force direction. Work by an external agent equals +ΔE_p only when kinetic energy is unchanged and there are no other energy transfers.

Gravitational and electric potential energy depend on interactions and relative positions; elastic potential energy depends on deformation. Changes are independent of a common shift in the energy reference.

Elastic graph area

For slow, reversible loading with negligible kinetic change and dissipation, elastic-energy increase equals stretching-force/extension area. A linear spring through the unloaded origin gives ½kx². Between nonzero extensions, subtract energies or find only the intervening area.

Trapezia are exact for supplied straight segments and estimates for sampled smooth curves. Convert extension to metres for an area in N m = J. Loading work alone does not prove that all energy can be recovered.

Power and efficiency

Mechanical power refers to the named force; θ is its angle to velocity. A zero resultant and constant speed do not imply zero input power. The useful/input power ratio gives efficiency when both rates refer to the same interval.

1 W = 1 J/s, while 1 kWh = 3.6 × 10⁶ J. Equal energies transferred in half the time mean twice the powers, not twice the energies.

For a lift, match load force, vertical rise, time and measured input energy to the same interval. Check whether kinetic energy changes. A label rating or force × speed does not measure the actual electrical input energy.

Read symbols in context. W can label work or energy in J, weight in N, or the unit watt when it follows a power value. E can label energy in J or electric field in N/C. Here s or x denotes displacement in m and v denotes velocity in m/s. Efficiency is a ratio with no unit.