| 1.1.6 Work, Energy, Power and Efficiency |
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Mechanical energy analysis links three closely related quantities: work, energy and power.
Work done on an object equals the component of force along the displacement multiplied by that displacement, given by
ΔW = FΔs cos θ
where θ is the angle between the force and the direction of displacement; work done is measured in joules (J).
When a resultant force does work on an object, its kinetic energy changes by an equal amount (the work–energy principle). Kinetic energy is the energy of a moving object and is given by
Eₖ = ½mv² W = ½mv² − ½mu²
where m is the mass and v is the speed, so the work done by the resultant force equals the change in kinetic energy.
When an object is raised vertically near the Earth’s surface, work is done against gravity and its gravitational potential energy increases by
ΔEₚ = mgΔh
where g is the gravitational field strength and Δh is the vertical height gained.
Energy is conserved in every process: it is transferred between stores such as kinetic, gravitational potential, elastic and thermal, but is never created or destroyed.
Power measures how quickly energy is transferred or how quickly work is done,
P = E/t = W/t
measured in the SI unit watt (W), where 1 W = 1 J s⁻¹.
For an object moving at constant velocity under a constant force, this combines into
P = Fv
Real devices are never perfectly efficient, because some input energy is always transferred to non-useful stores, especially thermal stores from friction or resistance.
Key Definition Efficiency is the fraction of the input energy that is transferred usefully. It can be calculated using either the energy or power form of the same ratio, and is often expressed as a percentage.
efficiency = (useful output energy / total input energy) × 100% efficiency = (useful power output / total power input) × 100%
Mastery of these ideas requires careful identification of forces, vertical heights, energy stores, and time intervals, plus the discipline to track every joule from input to output.