Potential energy, stability and energy in engineered systems

Lesson 5 introduced work, kinetic energy, gravitational potential energy and the conservation of mechanical energy, and showed how friction converts mechanical energy to heat. This lesson takes the energy method further. It explains which forces have a potential energy and why, shows how a potential energy curve reveals the force on a body, its equilibrium positions and whether they are stable, and develops the elastic potential energy of springs, which governs impact loads, bumpers and energy absorbers. It then widens the view to energy accounting in engineered systems: storage in pumped hydro and flywheels, the chain of efficiencies from electricity to useful work, and regenerative braking. Energy is often the quickest route to an answer, and it is always a powerful check.

Conservative and non-conservative forces

A force is conservative if the work it does on a body moving between two points depends only on the endpoints, not on the path. Equivalently, the work around any closed path is zero. Gravity and ideal spring forces are conservative: lifting a box up a ramp or straight up a ladder to the same height takes the same work against gravity, and that work is fully recovered when the box comes back down. Friction and drag are non-conservative: the longer the path, the more work they do, and that energy leaves the mechanical system as heat.

Only conservative forces have a potential energy U, defined so that the work done by the force from A to B is \(W = U_A - U_B\). Mechanical energy K + U is then conserved when only conservative forces do work, and in general

where \(W_{nc}\) is the work of the non-conservative forces (negative for friction, positive for an engine or a person pushing).

Force from potential energy

Since a conservative force does work equal to the decrease in potential energy, \(F\,dx = -dU\) in one dimension, so the force is minus the slope of the potential energy:

In three dimensions this becomes \(\mathbf{F} = -\nabla U\), the gradient of MATH-209. Checking: for gravity near the Earth, U = mgy gives F = −mg, downward; for a spring, \(U = \tfrac{1}{2}kx^2\) gives F = −kx, Hooke's law (Lesson 4). The force always points downhill on the potential energy curve, toward lower U.

Potential energy curves, equilibrium and stability

A graph of U(x) tells the whole story of one-dimensional motion under conservative forces. Where the slope is zero, the force is zero: these are equilibrium positions. At a minimum of U, a small displacement produces a force pushing the body back, so the equilibrium is stable; at a maximum, a small displacement produces a force pushing it farther away, so it is unstable; where the curve is flat over a range, the equilibrium is neutral. With total energy E, the body can move only where \(U(x) \le E\), since kinetic energy cannot be negative; the points where U = E are turning points, where the body momentarily stops and reverses.

Engineers use this reasoning for stability questions everywhere. A ship is stable when tilting it raises its centre of gravity relative to the centre of buoyancy, so its potential energy has a minimum in the upright position. A ball in a bowl is stable; a pencil balanced on its point is not. A slender column under load (MATH-201) loses stability when its potential energy changes from a minimum to a maximum as the load increases, which is buckling.

Elastic potential energy and impact

Stretching or compressing a spring by x stores elastic potential energy \(U = \tfrac{1}{2}kx^2\), the area under the force-extension line. This energy can be released as motion (a spring launcher, a bow) or must be absorbed when a moving mass is brought to rest (a bumper, a cable stopping a falling load, a car's crumple zone).

Energy gives a quick, important result for impact. A mass dropped onto a spring compresses it far more than the same mass resting on it, and the peak force is correspondingly larger.

Energy storage

Large-scale energy storage is one of the central challenges of electricity systems with high shares of wind and solar power. Mechanical storage stores energy as potential or kinetic energy. Pumped hydro storage, by far the largest form of grid storage worldwide, pumps water uphill when electricity is cheap and plentiful and releases it through turbines when it is needed; the stored energy is \(E = \rho Vgh\).

Flywheels store kinetic energy of rotation (Lesson 8) and can deliver it very quickly, which suits short-term frequency regulation and uninterruptible power supplies; springs and compressed air are used on smaller scales. Comparing the energy stored per kilogram shows why chemical storage dominates portable uses: a lithium-ion battery stores roughly 0.5 to 0.9 MJ/kg, gasoline about 46 MJ/kg of chemical energy, while lifting a kilogram of water 300 m stores only about 3 kJ.

Efficiency chains

In a system of several stages, each with its own efficiency, the overall efficiency is the product of the stage efficiencies. Every stage loses some energy as heat, so systems with many conversions waste a surprising share of their input.

The same accounting applies to time-varying loads. A pump that must deliver a large flow only for a few hours a day can be sized for that peak, or a smaller pump can fill a storage tank continuously and let the tank meet the peak: the energy delivered is the same, but the power rating, and with it the cost of the motor and electrical supply, is lower. Choosing between power and energy in this way is one of the commonest decisions in engineering design, from water systems to battery packs.

Regenerative braking

A conventional vehicle turns its kinetic energy into heat in the brakes every time it stops. Electric and hybrid vehicles can run their motors as generators to recover part of it. For a 1600 kg car stopping from 60 km/h (16.7 m/s), the kinetic energy is \(\tfrac{1}{2} \times 1600 \times 16.7^2 = 222\) kJ. If 60% is recovered, after losses in the motor, inverter and battery, about 133 kJ returns to the battery, enough to reaccelerate a good part of the way. In stop-and-go city driving, where braking is frequent, regeneration can extend range noticeably; on the highway, where drag (Lesson 4) dominates and braking is rare, it matters little. Energy accounting shows at once which improvements pay off in which situations.