Energy as a conserved quantity

Energy is a scalar quantity that describes the capacity of a system to produce changes by transfer between stores or between systems. Scalar matters here: energy has a size and a unit, but no direction. The SI unit is the joule, $\text{J}$. One joule is the energy transferred when a force of one newton acts through one metre in the direction of the force.

The principle of conservation of energy is a law of physics that states that the total energy of an isolated system remains constant because energy cannot be created or destroyed, only transferred or transformed. In ordinary mechanics, we usually keep track of kinetic, gravitational potential, elastic potential and thermal stores. In nuclear processes, mass-energy has to be included as well, because changes in mass are no longer negligible.

That is what makes energy methods so useful. Rather than following every small detail of the motion, we can compare the energy at the start with the energy at the end. Take a curved path: the acceleration may change continuously, so the kinematic equations can become awkward or may not apply. An energy approach often solves the same problem from just the initial and final states.

Evidence, laws and uncertainty

A law of physics is an empirical statement that summarizes a repeated pattern in measurements without necessarily explaining why the pattern occurs. Conservation of energy was not accepted because one perfect experiment proved it. It was accepted because many different experiments, each with uncertainty, matched the same accounting rule when all relevant energy transfers were included.

Uncertainty does not stop laws from being developed. It tells us to test the claim against ranges of possible values, repeated measurements, different methods and clearly stated assumptions. If an apparent energy loss is larger than the experimental uncertainty, a physicist does not usually throw out conservation straight away. They first look for an unmeasured transfer: heating, sound, deformation, rotation, electrical work, or energy carried away from the system.

Links across physics

Conservation laws turn up across the course. Charge is conserved in circuits, momentum is conserved in isolated collisions and particle interactions, and energy is conserved in mechanical, thermal, electrical and nuclear systems. Before writing any conservation statement, it helps to ask, “What is the system, and what can cross its boundary?”

Travelling waves give a neat contrast with moving objects. A travelling wave is a disturbance that transfers energy through a medium or field while the oscillating particles of a material medium have no net displacement after the wave has passed. The particles move locally, pass energy to neighbouring particles, and then return to their equilibrium positions.

Equilibrium can also be described using energy. A stable equilibrium is a state in which a small displacement increases the total energy of the system, so the system tends to return. In a gas under a piston, or in a star, the familiar force balance can often be rephrased as the system settling into an energy condition compatible with the constraints.

What physicists mean by work

Work is an energy transfer that happens when a force acts through a displacement of its point of application. Physics uses the word more narrowly than everyday speech. You might feel tired holding a heavy bag still, but if the bag does not move, no mechanical work is done on the bag.

Work done by a force is not a separate kind of energy. It’s the way energy is transferred. When an engine lifts a load, chemical energy in the fuel is transferred, by mechanical work, into gravitational potential energy of the load. When a braking force slows a bicycle, mechanical energy is transferred mainly into internal energy of the brakes, tyres and surroundings.

The sign of work shows the direction of the transfer for the object being considered. Positive work increases the relevant energy store of the object or system; negative work takes energy from it. Resistive forces, such as friction and drag, usually do negative work on moving objects.

A useful boundary question is: “work done on what?” The same interaction can be described in different ways, depending on the chosen system. A road can do forward static friction work on a car’s tyres, while the engine transfers chemical energy through the drivetrain. Choosing the system clearly keeps the energy accounting honest.

Reading a Sankey diagram

A Sankey diagram is a scaled flow diagram that shows energy or power transfers with arrows. The arrow widths are proportional to the amounts transferred. The input usually comes in from the left. Useful output carries on to the right, while wasted or degraded transfers branch away, often upward or downward.

“Wasted” does not mean destroyed. It means the energy has been transferred into stores that are not useful for the intended purpose, commonly internal energy of the surroundings. For example, a lamp may transfer only a small fraction of its input energy as visible light; most becomes heating of the lamp and air.

When you construct a Sankey diagram, choose a scale and use it consistently. If the input is $100\ \text{J}$, an output of $25\ \text{J}$ should have one quarter of the input arrow width. If the input is a power, such as $200\ \text{W}$, apply the same rule to power flows rather than energy amounts.

Sankey diagrams are good for checking conservation. At any junction, the total width entering must equal the total width leaving, allowing for all useful and wasted transfers. If the diagram does not balance, the energy accounting is incomplete.

The component of force along the displacement

For a constant force, the work done depends only on the part of the force that acts parallel to the displacement:

Here the angle must be the angle between the two vectors: force and displacement. If the force points in the same direction as the displacement, $\theta = 0^\circ$, so the work is positive and maximum. If the force is perpendicular to the displacement, $\theta = 90^\circ$, so the work is zero. If the force has a component opposite the displacement, the work is negative.

This deals with a classic circular-motion question. For a body moving in a circular trajectory under a centripetal force, the instantaneous displacement is tangential while the centripetal force is radial. They are perpendicular, so the centripetal force does no work. It changes the direction of the velocity, not the speed or kinetic energy.

Force–distance graphs

For motion in one dimension, the work done by a force is the area under a graph of force against displacement in the direction of the force. With a constant force, that area is a rectangle. With a varying force, estimate or calculate the area under the curve. Areas below the displacement axis represent negative work.

This graph method is especially useful for springs, collisions and real resistive forces, where the force may change during the motion. In practical graph work, use the units on both axes: one square on the graph represents a definite number of joules because $N\,m$ is J.

Work done by the resultant force

A resultant force is the single force with the same overall effect as all the forces acting together on a system. The work done by the resultant force gives the net energy change of the system:

For the simplest particle model, net work changes kinetic energy. For a more extended system, that energy change may include kinetic energy of translation, rotation, elastic deformation, internal energy and potential energy, depending on what you have put inside the system boundary.

So two calculations that look almost the same can mean different things. The work done by a lift force on a rising helicopter is not just the gain in gravitational potential energy if the helicopter also gains kinetic energy. The extra work has gone into increasing another energy store.

When several forces act, don’t add their magnitudes and call that energy. Calculate the work done by each force, with signs, and add the works; or first find the resultant force in the direction of motion and use that in the work calculation.

The mechanical energy store

Kinetic energy is the energy store a body has because it is moving. Gravitational potential energy is the energy store linked to the position of a mass in a gravitational field. Elastic potential energy is the energy store linked to reversible deformation of an object such as a spring.

Mechanical energy means the total of the kinetic energy, gravitational potential energy and elastic potential energy in a system:

Conservative and non-conservative forces

A conservative force does work between two points that is independent of the path taken. Gravity is the standard school example near Earth’s surface. If you lift an object from one height to another, the change in gravitational potential energy is the same for a vertical path or a sloped path, provided the start and end heights are the same.

A non-conservative force does work that depends on the path taken and usually transfers mechanical energy into non-mechanical stores. Friction and air resistance are the usual examples. Drag won’t give the mechanical energy back when the motion is reversed; most of it has gone into internal energy of the surroundings.

If there are no frictional or resistive forces, and only conservative forces do work, total mechanical energy is conserved:

If non-conservative forces do work, the change in total mechanical energy is treated as the work done by those forces:

This gives a clean way to deal with real situations. A skier descending a slope gains less kinetic energy than the gravitational potential energy lost, because resistive forces have done negative work on the skier. The “missing” mechanical energy has not vanished; it has been transferred away from the mechanical store.

Using energy instead of kinematics

Energy methods are often safer than kinematics when the acceleration is not constant. On a curved slide, a pendulum arc, or a roller-coaster track, the direction changes and so do the components of force. If friction is negligible, energy conservation can give the speed at a lower point without finding the detailed acceleration at every point.

In the laboratory, a falling mass can pull a cart along a track. The falling mass loses gravitational potential energy, while the cart, string, pulley and falling mass gain kinetic energy. To test conservation, reduce friction by slightly tilting the track until the cart moves at constant speed without the hanging mass, measure masses and final speed, repeat trials, and compare energy lost with energy gained. Any shortfall points to friction, pulley rotation, sound, heating or measurement uncertainty.

Translational kinetic energy

For an object moving without considering rotation, the kinetic energy of translational motion is

Because the speed is squared, doubling the speed makes the kinetic energy four times larger. This is why high-speed vehicles need much more energy to accelerate, and a much longer distance to stop.

For a change in speed,

Look carefully at the structure: final speed squared minus initial speed squared. It is not the square of the change in speed. That algebra slip is very common, and it changes the physics.

The work link is direct. Positive resultant work on a moving object increases its translational kinetic energy. Negative resultant work decreases it. Braking, impact stopping distances and acceleration from rest can all be handled this way.

Near-surface gravitational potential energy

Near Earth’s surface, the change in gravitational potential energy is

Use this formula for changes, not for an absolute value. First choose a reference height; after that, only the difference in height counts. If an object rises, $\Delta h$ is positive and its gravitational potential energy increases. If it falls, $\Delta h$ is negative and its gravitational potential energy decreases.

For gravity near Earth’s surface, the path taken doesn’t matter. Lift a crate vertically through a height, or push the same crate up a smooth ramp to the same height, and it gains the same gravitational potential energy. The ramp changes the force and the distance, but not the energy change due to gravity.

Why the formula is only local

The equation $\Delta E_p = mg\Delta h$ assumes that $g$ stays constant, so the gravitational field is effectively uniform over the height change. For ordinary heights near Earth’s surface, that approximation is good.

Far from Earth, the field lines spread out and the gravitational field strength decreases with distance from Earth’s centre. In that situation $g$ is no longer constant over the motion, so the simple near-surface expression has to be replaced by the more general gravitational potential energy treatment from gravitation.

Energy conservation still works in both cases. The expression used for the gravitational potential energy is the part that changes.

Springs and area under the force–extension graph

For a spring that obeys Hooke’s law,

A stiffer spring has a larger spring constant. It therefore takes a larger force to produce the same extension. On a force–extension graph, Hooke’s law appears as a straight line through the origin, with the gradient equal to the spring constant.

The elastic potential energy stored in the spring is the work done to stretch or compress it. On the force–extension graph, that work is the triangular area under the line:

The square term is the key detail. If the extension increases from $\Delta x$ to $2\Delta x$, the total stored elastic potential energy becomes four times larger. So the extra work done over that interval is three times the original stored energy.

Measuring a spring constant energetically

A neat practical method is to stretch a spring by measured amounts and launch it vertically. Wear eye protection, and make sure the spring can’t fly towards anyone. For each extension, measure the maximum height several times, then average the height.

If air resistance and other losses are small, the elastic potential energy stored at the start becomes gravitational potential energy at the top of the flight. This gives

A graph of $h$ against $(\Delta x)^2$ should be linear. The spring constant can then be found from the gradient.

Use repeated readings to estimate the uncertainty in height. Add uncertainty bars, then draw a best-fit line and reasonable maximum- and minimum-gradient lines. Comparing this spring constant with one found from a direct Hooke’s-law force–extension graph gives a useful test of the method and its assumptions.

Rate of energy transfer

Power tells us how fast work is done or how fast energy is transferred. It is defined by

The expression $Fv$ works when the force acts in the direction of motion; if it doesn’t, use the component of the force along the velocity.

One watt is one joule per second. Two machines might do the same total work, but the machine that finishes in less time has the greater power.

Two machines transfer the same energy, but the shorter time gives the larger power.

The expression $P = Fv$ helps explain why constant-power motion usually doesn’t give constant acceleration. If a vehicle’s engine provides a fixed useful power, a higher speed means a smaller available driving force. Drag generally increases with speed as well, so eventually all the engine power goes into overcoming resistive forces and the vehicle reaches a maximum speed.

Rates elsewhere in physics

Rates of change show up all over physics. Velocity is the rate of change of displacement, acceleration is the rate of change of velocity, current is the rate of flow of charge, activity is the rate of radioactive decay, and induced emf is linked to the rate of change of magnetic flux. Power fits the same pattern: it is the rate of energy transfer.

Useful output compared with total input

Efficiency is a dimensionless ratio comparing the useful energy or power output of a process with the total energy or power input. It is given by

The definition works for both energy and power because, in a single process, the useful and total energy transfers take place over the same time interval. If a motor receives 1000 J and 250 J becomes useful lifting energy, its efficiency is 0.25 or 25%. Likewise, if it receives 1000 W and delivers 250 W of useful mechanical power, the efficiency is 25%.

No real process is perfectly efficient. Friction, electrical heating, turbulence, deformation and sound commonly transfer energy to internal energy. These transfers may be unwanted, but they still obey conservation of energy.

Efficiency measurements often come from data. For a bouncing ball, the useful output after a bounce can be represented by the gravitational potential energy at the rebound height. Since that energy is proportional to height for the same ball, the ratio of rebound height to previous drop height gives the bounce efficiency, provided air resistance and measurement uncertainty are treated sensibly.

Energy per unit volume

Energy density means the energy a fuel releases per unit volume of that fuel. Its SI unit is $\text{J}\,\text{m}^{-3}$, although fuel data are often given in $\text{MJ}\,\text{m}^{-3}$ or $\text{GJ}\,\text{m}^{-3}$.

Fossil fuels have been used so widely partly because of their high energy density: a small volume can store a lot of transferable energy. Nuclear fuels are in a different league, with vastly larger energy densities, because nuclear processes release energy from changes in nuclear binding rather than from chemical rearrangement of atoms.

Typical volumetric energy densities; gases are at about 1 atm.

Energy density by itself doesn’t tell you which fuel is “best”. Device efficiency, availability, safety, environmental impact, storage pressure or temperature, cost, and the useful form of the output energy all matter too.

In calculations, start by finding the total input energy required, including efficiency if the question gives it. Then divide by the energy released per unit volume. Watch the powers of ten: this topic jumps quickly between joules, megajoules and gigajoules.