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, 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 often track kinetic, gravitational potential, elastic potential and thermal stores. In nuclear processes, mass-energy must also be included because changes in mass are no longer negligible.

That is what makes energy methods so useful. Rather than following every tiny 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 vary continuously, so kinematic equations can become awkward or invalid. Energy often lets us solve the problem using only 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 is not accepted because one perfect experiment proved it. It is accepted because many different experiments, each with uncertainty, fit the same accounting rule when all relevant energy transfers are included.

Uncertainty does not stop laws being developed. Instead, the claim has to be tested against ranges of possible values, repeated measurements, different methods and carefully stated assumptions. If an apparent energy loss is larger than experimental uncertainty, a physicist does not first abandon conservation. They look for an unmeasured transfer: heating, sound, deformation, rotation, electrical work, or energy carried away from the system.

Links across physics

Conservation laws run through 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 is useful to ask, “What is the system, and what can cross its boundary?”

Travelling waves provide a helpful 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, transfer energy to neighbouring particles, and then return to their equilibrium positions.

Equilibrium can be described in energy terms too. 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 the transfer of energy when a force acts through a displacement of its point of application. Physics uses the word more tightly 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 gets transferred. When an engine lifts a load, chemical energy in the fuel is transferred, via 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 transfer for the object you are looking at. Positive work increases the relevant energy store of the object or system; negative work takes energy away 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 showing energy or power transfers. It uses arrows, and the width of each arrow is proportional to the amount transferred. The input usually comes in from the left. Useful output carries on to the right, while wasted or degraded transfers branch off, often upward or downward.

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

When you construct a Sankey diagram, pick a scale and use it consistently. If the input is 100 J, an output of 25 J should be drawn with one quarter of the input arrow width. If the input is a power, such as 200 W, the same rule applies to power flows rather than energy amounts.

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

The component of force along the displacement

For a constant force, only the part of the force parallel to the displacement does work:

[W = Fs\cos heta]

where (W) is the work done by the force (J), (F) is the magnitude of the constant force (N), (s) is the magnitude of the displacement (m), and ( heta) is the angle between the force and the displacement (degrees or radians).

Here the angle has to be the angle between the two vectors: force and displacement. If the force acts in the same direction as the displacement, ( heta = 0^\circ), so the work is positive and maximum. If the force is perpendicular to the displacement, ( heta = 90^\circ), so the work is zero. A force component opposite to the displacement gives negative work.

This explains the standard circular-motion case. 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 useful for springs, collisions and real resistive forces, where the force may change during the motion. In practical graph work, check 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:

[W_{ ext{net}} = \Delta E_{ ext{system}}]

where (W_{ ext{net}}) is the work done by the resultant force on the system (J), and (\Delta E_{ ext{system}}) is the change in the energy of the system (J).

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

So two calculations can look similar but 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 increased another energy store.

When several forces act, don't add their magnitudes and call it energy. Calculate the work done by each force, including 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, for example in a spring.

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

[E_{ ext{mech}} = E_k + E_p + E_H]

where (E_{ ext{mech}}) is the total mechanical energy of the system (J), (E_k) is kinetic energy (J), (E_p) is gravitational potential energy (J), and (E_H) is elastic potential energy (J).

Conservative and non-conservative forces

A conservative force does work between two points that does not depend on 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 whether the path is vertical or sloped, as long as the start and end heights are the same.

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

With no frictional or resistive forces, and only conservative forces doing work, total mechanical energy is conserved:

[\Delta E_{ ext{mech}} = 0]

where (\Delta E_{ ext{mech}}) is the change in total mechanical energy (J).

When non-conservative forces do work, treat the change in total mechanical energy as the work done by those forces:

[\Delta E_{ ext{mech}} = W_{ ext{nc}}]

where (W_{ ext{nc}}) is the work done on the system by non-conservative forces (J).

This approach is usually the neatest one for real situations. A skier going down 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 hasn’t disappeared; it has been transferred away from the mechanical store.

Using energy instead of kinematics

Energy methods are often safer than kinematics when acceleration is not constant. On a curved slide, a pendulum arc, or a roller-coaster track, the direction changes and the force components change too. If friction is negligible, you can find the speed at a lower point using energy conservation, without working out the detailed acceleration at every point.

In the laboratory, a falling mass can be used to pull a cart along a track. As the falling mass loses gravitational potential energy, 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 suggests friction, pulley rotation, sound, heating or measurement uncertainty.

Translational kinetic energy

For an object moving in a way where rotation does not need to be considered, the kinetic energy of translational motion is

[E_k = \frac{1}{2}mv^2 = \frac{p^2}{2m}]

where (m) is the mass of the object (kg), (v) is its speed (m s(^{-1})), and (p) is its linear momentum (kg m s(^{-1})).

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

For a change in speed,

[\Delta E_k = \frac{1}{2}m\left(v_f^2 - v_i^2\right)]

where (\Delta E_k) is the change in kinetic energy (J), (v_f) is final speed (m s(^{-1})), and (v_i) is initial speed (m s(^{-1})).

Watch the order carefully: final speed squared minus initial speed squared. It is not the square of the change in speed. That algebra slip is common, and it gives different physics.

The work link is straightforward. 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 treated using this idea.

Near-surface gravitational potential energy

Near Earth’s surface, a change in gravitational potential energy is found using

[\Delta E_p = mg\Delta h]

where (\Delta E_p) is the change in gravitational potential energy (J), (g) is the gravitational field strength near Earth’s surface (N kg(^{-1}), equivalent to m s(^{-2})), and (\Delta h) is the change in vertical height (m).

Use this formula for a change, not an absolute value. You first choose a reference height; after that, only the height difference matters. If an object rises, (\Delta h) is positive and its gravitational potential energy increases. If it falls, (\Delta h) is negative and gravitational potential energy decreases.

Gravity near Earth’s surface doesn’t care about the path taken. 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 distance involved, 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 works well.

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

Energy conservation still works in both cases. Only the expression used for the gravitational potential energy changes.

Springs and area under the force–extension graph

For a spring that obeys Hooke’s law,

[F = k\Delta x]

where (k) is the spring constant (N m(^{-1})), and (\Delta x) is the extension or compression from the spring’s natural length (m).

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 giving 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:

[E_H = \frac{1}{2}k(\Delta x)^2]

The square term is worth noticing. If the extension increases from (\Delta x) to (2\Delta x), the total stored elastic potential energy becomes four times as large, 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 cannot fly towards anyone. For each extension, measure the maximum height reached several times, then average the height.

If air resistance and other losses are small, the elastic potential energy stored at release becomes gravitational potential energy at the top of the flight. So (\frac{1}{2}k(\Delta x)^2 = mgh), where (h) is the maximum height reached above the release point (m). A graph of (h) against ((\Delta x)^2) should be linear, and its gradient can be used to find the spring constant.

Use repeated readings to estimate the uncertainty in height, add uncertainty bars, and draw a best-fit line with reasonable maximum- and minimum-gradient lines. Comparing this spring constant with the value found from a direct Hooke’s-law force–extension graph tests both the method and its assumptions.

Rate of energy transfer

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

[P = \frac{\Delta W}{\Delta t} = Fv]

where (P) is power (W), (\Delta W) is the work done or energy transferred during a time interval (J), and (\Delta t) is the time interval (s). The expression (Fv) applies when the force is in the direction of motion; otherwise use the component of force along the velocity.

One watt is one joule per second. Two machines can do the same total work, but the machine that does it 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 delivers a fixed useful power, the available driving force is smaller at higher speed. Drag generally increases with speed as well, so eventually all the engine power is used overcoming resistive forces and the vehicle reaches a maximum speed.

Rates elsewhere in physics

Rates of change appear all through 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

[\eta = \frac{ ext{useful work out}}{ ext{total work in}} = \frac{ ext{useful power out}}{ ext{total power in}}]

where (\eta) is efficiency, written either as a decimal with no unit or as a percentage.

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%. The same idea applies to power: 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. For the same ball, that energy is proportional to height, so the ratio of rebound height to previous drop height gives the bounce efficiency, provided air resistance and measurement uncertainty are treated sensibly.