5.2 Work, Power & Energy

Work, Power & Energy

Energy methods often solve problems more quickly than forces and accelerations alone. Module 2 requires you to define work, power, kinetic energy (KE), gravitational potential energy (PE), state conservation of mechanical energy where it applies, compute efficiency, and recognise heat as a form of energy — not as something that “destroys” energy.

Work

Work is done when a force causes (or contributes to) a displacement of its point of application. For a constant force:

W = F d cos θ

where F is force (N), d is displacement (m), and θ is the angle between the force and the displacement. When force and displacement are parallel, cos 0° = 1 and W = F d. The SI unit is the joule (J):

1 J = 1 N·m = 1 kg·m²/s²

Important special cases:

• If d = 0 (no movement), W = 0 even if a large force is applied (holding a heavy component stationary does no mechanical work).
• If force is perpendicular to displacement (θ = 90°, cos 90° = 0), W = 0 (ideal centripetal force does no work; it changes direction of velocity, not speed).
• Work can be positive (force component in the direction of motion — engine thrust doing work on the aircraft) or negative (force opposite to motion — drag or brakes removing mechanical energy).

Worked example — cargo push. A mechanic pushes a 300 N horizontal force over 12 m on a crate (frictionless idealisation for the work done by the push). Work by the push: W = 300 × 12 = 3 600 J = 3.6 kJ.

Worked example — angled force. A 200 N force at 30° above the horizontal moves a trolley 5 m horizontally. Work by that force: W = 200 × 5 × cos 30° = 1 000 × 0.866 ≈ 866 J. Only the horizontal component does work on the horizontal path.

Worked example — lifting. Raising a 50 kg mass vertically at constant speed through 2.0 m. Force upward ≈ weight = m g = 50 × 9.81 = 491 N. Work against gravity: W = 491 × 2.0 ≈ 982 J. This work is stored as gravitational potential energy (below).

Power

Power is the rate of doing work (or transferring energy):

P = W / t

For a constant force parallel to constant velocity v:

P = F v

SI unit: watt (W) = 1 J/s. In engineering, kilowatts (kW) are common: 1 kW = 1 000 W. (Older horsepower units appear in some manuals; Module 2 expects SI fluency.)

Worked example — winch. A winch lifts a 1 000 N load 6 m in 12 s at constant speed. Work = 1 000 × 6 = 6 000 J. Power = 6 000 / 12 = 500 W = 0.5 kW.

Worked example — thrust power. An aircraft engine produces 15 000 N thrust while true airspeed is 80 m/s (idealised propulsive power). P = F v = 15 000 × 80 = 1 200 000 W = 1.2 MW. Real propulsive efficiency is less than 100%; this is the mechanical power associated with thrust × speed.

High force at low speed or moderate force at high speed can give the same power. Brakes that stop a heavy aircraft quickly must absorb energy at a high rate — high power — which is why brake energy and cooling limits matter after a rejected take-off.

Kinetic Energy

Kinetic energy is energy of motion:

KE = ½ m v²

• m in kg, v in m/s, KE in joules.
• KE depends on speed squared: double the speed → four times the KE.
• Mass is linear: double mass at same speed → double KE.

Worked example. Aircraft mass 40 000 kg at 70 m/s (about 136 kt):

KE = 0.5 × 40 000 × 70² = 20 000 × 4 900 = 98 000 000 J = 98 MJ

That energy must be removed by drag, reverse thrust, and brakes to stop. At 35 m/s (half speed), KE falls to one quarter: 24.5 MJ — which is why landing groundspeed and rejected-take-off energy are so sensitive to speed.

The work–energy theorem states that the net work done on a body equals its change in kinetic energy: W_net = ΔKE = KE_final − KE_initial. Accelerating from rest, the net work done equals the final KE.

Gravitational Potential Energy

Gravitational potential energy relative to a chosen reference height is:

PE = m g h

where h is height above the reference (m). Unit joules. Only changes in PE matter for energy accounting: raising mass m by Δh stores m g Δh; lowering it releases the same amount (to KE or to work against other forces).

Worked example. A 25 kg tool bag raised 1.5 m onto a stand: ΔPE = 25 × 9.81 × 1.5 ≈ 368 J. If the bag falls freely from that height, PE converts to KE: at ground level, ½ m v² = 368 ⇒ v² = 2 × 368 / 25 ≈ 29.4 ⇒ v ≈ 5.4 m/s (same result as v² = u² + 2 a s with a = g).

Aircraft potential energy in a climb is supplied by excess engine power (beyond that needed for level flight). In a descent, PE converts to KE unless drag (or thrust reduction) balances the conversion — energy methods explain why speed builds in a steep dive if not managed.

Total Mechanical Energy and Conservation

Total mechanical energy (ideal case):

E_mech = KE + PE = ½ m v² + m g h

If only conservative forces do work (gravity, ideal spring forces), E_mech is conserved: it stays constant while KE and PE trade. Friction, drag, inelastic deformation, and heat generation are non-conservative: they reduce mechanical energy while total energy of the isolated system (including thermal energy) is still conserved.

Worked example — conservation (no drag). A glider of mass 500 kg dives from height 200 m to 100 m above the same datum, starting from rest at the top (idealised).

Loss in PE = m g Δh = 500 × 9.81 × 100 = 490 500 J
This equals gain in KE if no drag: ½ m v² = 490 500 ⇒ v² = 1 962 ⇒ v ≈ 44.3 m/s

With drag, final speed is lower; the “missing” mechanical energy has become thermal energy in the air and structure (skin friction heating is usually small at low speed; the main effect is kinetic energy not gained).

Efficiency

Real machines never convert all input energy into the desired form:

Efficiency η = (useful energy output) / (energy input)
or η = (useful power output) / (power input)

Often expressed as a percentage: η × 100%. Losses appear as heat (friction, electrical resistance, incomplete combustion), sound, or vibration.

Worked example. An electric hydraulic pump draws 2.0 kW electrical power and delivers 1.4 kW of hydraulic power to the system. η = 1.4 / 2.0 = 0.70 = 70%. The 0.6 kW difference heats the fluid and motor.

Worked example — mechanical advantage context. If a machine lifts a 2 000 N load through 0.5 m while the operator applies 500 N through 2.5 m: useful work out = 2 000 × 0.5 = 1 000 J; work in = 500 × 2.5 = 1 250 J; η = 1 000/1 250 = 80%. Velocity ratio and mechanical advantage were covered with simple machines; efficiency links energy in to energy out.

Heat as a Form of Energy

Heat is energy transferred because of a temperature difference (detailed in the thermodynamics chapter). For dynamics, remember:

• Friction and inelastic collisions convert mechanical energy into internal (thermal) energy; temperature of brakes, tyres, and oil rises.
• Energy is conserved overall; it is not destroyed when a aircraft stops — KE becomes heat in brakes and tyres plus some sound.
• “Work done against friction” is a common way to compute energy dissipated: W_friction = f_k × d (kinetic friction force times sliding distance).

Worked example — brake energy. A 12 000 kg aircraft lands at 50 m/s. KE = ½ × 12 000 × 2 500 = 15 MJ. If brakes and rolling resistance remove essentially all of this over the ground roll, that order of energy appears as heat in the brake packs and tyres — reason for cool-down and fuse-plug awareness after heavy braking.

Formula Checklist

Always convert mass to kg, height to metres, and force to newtons before substituting. Keep KE’s v² dependence in mind for any speed-related energy or braking question.