Physics 3.3 - 3.6 - Energy transfers and conservation

Systems, stores and pathways

Physics starts an energy account by choosing a system: the object or group of interacting objects being studied. The boundary is an imaginary line around that system. Energy can be transferred between stores inside the boundary, and it can cross the boundary into or out of the system.

This lesson needs four stores most often:

A transfer pathway is different from a store. It describes how energy is moved between stores or across a system boundary. In this lesson, forces can transfer energy by mechanical working, an electric current can transfer energy by electrical working, and a temperature difference can transfer energy by heating. A store is shown at a particular time; a pathway happens during a time interval.

For example, choose a trolley as the system. A person pushes it and the push causes it to accelerate. Energy crosses the trolley's boundary by mechanical working, so the trolley's kinetic store increases. The force causes the acceleration; energy is the quantity being accounted for, not the cause of the motion.

The arrows and boxes used later are models. An energy arrow is not a material pipe, and energy is not a fluid travelling along a visible track.

Drawing and reading energy-transfer diagrams

A qualitative energy-transfer diagram separates stores from pathways:

[starting store] -- transfer pathway --> [final store]

The box labels tell you where energy is stored at the chosen start and end instants. The arrowhead shows the direction of transfer, and the arrow label names how the transfer occurs. For a car using friction brakes, a simple account is:

[kinetic store of car] -- mechanical working and heating --> [thermal stores of brakes, road and surroundings]

That line is bookkeeping, not a picture of the car's path. To draw a sound scientific diagram:

1. State the system and the start and end instants.
2. Put only stores in boxes.
3. Draw arrows in the direction energy is transferred and label their pathways.
4. Include every important final store or transfer across the boundary.
5. Check that the energy account balances.

A Sankey diagram adds quantity. Each arrow's width is proportional to the energy it represents. A single input arrow can split into several output arrows, but the combined output widths must equal the input width because energy is conserved.

Worked Sankey construction

During one interval, 1200 J is transferred electrically to a kettle system. Of this, 900 J increases the thermal store of the water and 300 J increases the thermal stores of the kettle body and nearby surroundings. Use the scale $1\,\mathrm{mm}:100\,\mathrm{J}$.

1. Check the account: $900\,\mathrm{J} + 300\,\mathrm{J} = 1200\,\mathrm{J}$, so all the input is represented.
2. Input width: $\frac{1200\,\mathrm{J}}{100\,\mathrm{J\,mm^{-1}}}=12\,\mathrm{mm}$.
3. Water branch width: $\frac{900\,\mathrm{J}}{100\,\mathrm{J\,mm^{-1}}}=9\,\mathrm{mm}$.
4. Kettle-and-surroundings branch width: $\frac{300\,\mathrm{J}}{100\,\mathrm{J\,mm^{-1}}}=3\,\mathrm{mm}$.
5. Draw: start with a 12 mm-wide arrow. Split it into a 9 mm branch and a 3 mm branch, keeping both arrowheads visible and labelling each destination store.
6. Sense-check the geometry: $9\,\mathrm{mm} + 3\,\mathrm{mm} = 12\,\mathrm{mm}$. The branches are narrower, but no energy has disappeared.

[DIAGRAM: asset_name: Physics 3.3-3.6 Sankey account; asset_slug: phy_b_sankey_1200_900_300; recommended_method: matplotlib; description: Exact input arrow of width 12 units splits into a 9-unit water branch and 3-unit kettle-and-surroundings branch, labelled 1200 J, 900 J and 300 J. The energy widths are 12:9:3; arrowheads are direction indicators, not measured widths.]

To interpret a Sankey diagram, read the arrow labels before judging the widths. A wide arrow represents a larger transfer than a narrow arrow drawn to the same scale. Add all branches before deciding whether the account conserves energy.

You can also compare transfers over a common time. If the kettle receives $1200\,\mathrm{J}$ in $60\,\mathrm{s}$, the mean input rate is $1200/60=20\,\mathrm{J\,s^{-1}}$. The two destination rates are $15\,\mathrm{J\,s^{-1}}$ and $5\,\mathrm{J\,s^{-1}}$, which also add to the input rate. This compares transfers on the same energy and time scale.

Five changes, one accounting rule

For each required change, first describe what physically causes the change. Then compare the stores at the start and end. The precise account depends on the system boundary and on whether resistive effects are included.

The upward-projectile and braking accounts can look like opposites: one changes kinetic storage mainly into gravitational storage, while the other changes kinetic storage mainly into thermal storage. Conservation does not predict which change occurs. Forces, current and temperature differences explain the mechanism; conservation checks whether the energy account is possible and complete.

For a collision, avoid the shortcut kinetic energy becomes sound. Sound is a wave that carries some energy away from the collision region. A wider system eventually accounts for that energy in the motion and thermal stores of matter in the surroundings.