Density

Density describes how much mass is contained in a given volume. A material with a higher density has more mass in the same volume than a material with a lower density.

For a body of mass $$m$$ and volume $$V$$, its density $$\rho$$ (rho) is mass divided by volume:

The SI unit of density is kilograms per cubic metre, written as $$\mathrm{kg,m^{-3}}$$. It tells us the mass in kilograms per cubic metre of volume.

For example, a body with a mass of $$6,\mathrm{kg}$$ and a volume of $$0.002,\mathrm{m^3}$$ has a density of:

Finding mass or volume from density

The density relationship can be rearranged to find mass when density and volume are known, or volume when mass and density are known:

Hooke’s law

When a force stretches a spring, the spring’s length increases. Its extension is the increase from its original length, not its total length.

For a spring obeying Hooke’s law, the applied force is directly proportional to the extension: doubling the force doubles the extension. This relationship is written as:

Here, $$F$$ is the applied force in newtons, $$\Delta L$$ is the extension in metres, and $$k$$ is the spring constant, or stiffness, in $$\mathrm{N,m^{-1}}$$.

The spring constant is the force per unit extension. A spring with a larger $$k$$ needs a greater force to produce the same extension, so it is stiffer.

Hooke’s law on a force–extension graph

In the region where Hooke’s law holds, a graph of force against extension is a straight line through the origin. If force is on the vertical axis and extension is on the horizontal axis, the gradient is the spring constant $$k$$.

The limit of proportionality is the point beyond which force and extension are no longer directly proportional. The graph is no longer a straight line through the origin, so Hooke’s law no longer describes that relationship.

Proportionality and elastic deformation

Elastic deformation is reversible: when the force is removed, the spring returns to its original length. Beyond the elastic limit, removing the force leaves some permanent deformation.

The limit of proportionality and the elastic limit are distinct. A spring can pass its limit of proportionality but still return to its original length when unloaded: its behaviour is then elastic, even though force and extension are no longer proportional.

Tensile stress and strain

When a specimen is pulled, it experiences a stretching force and may extend. Tensile stress describes the force relative to the specimen’s original cross-sectional area. Tensile strain describes the extension relative to its original length.

Tensile stress

Tensile stress is the stretching force per unit original cross-sectional area. The cross-section is the surface that would be exposed by cutting across the specimen, perpendicular to the direction of the force.

Here, $$\sigma$$ is tensile stress, $$F$$ is the stretching force and $$A$$ is the original cross-sectional area. For the same force, a smaller original area gives a greater tensile stress.

Stress is measured in pascals ($$\mathrm{Pa}$$). One pascal is one newton per square metre: $$1,\mathrm{Pa} = 1,\mathrm{N,m^{-2}}$$.

Tensile strain

Tensile strain is the fractional extension: the increase in length compared with the original length.

Here, $$\varepsilon$$ is tensile strain, $$\Delta L$$ is the extension and $$L_0$$ is the original length. For the same extension, a shorter original length gives a greater strain.

Strain has no unit: it is a ratio of two lengths measured in the same unit, so the units cancel. By using original area and original length, stress and strain describe loading and deformation while accounting for the specimen’s starting dimensions.

Deformation and fracture graphs

A stretched material changes length as the pulling force increases. A force–extension graph shows force against extension, while a stress–strain graph shows tensile stress against strain. Tensile stress is the pulling force per unit of original cross-sectional area; strain is the extension relative to the original length. Both graphs show how a specimen deforms as it is stretched.

Linear and elastic regions

A straight-line region through the origin shows that the plotted quantities are proportional: for example, doubling the force doubles the extension within that region.

Deformation is elastic if the specimen returns to its original length when the load is removed. Elastic behaviour is not necessarily linear: a specimen can deform elastically even when its graph is curved. A straight loading graph alone does not show how the specimen behaves on unloading.

Plastic deformation and fracture

If the specimen does not return to its original length after unloading, it has undergone plastic deformation. The extension remaining at zero load is its permanent extension. On a stress–strain graph, the strain remaining at zero stress is the permanent strain.

Fracture occurs when the material breaks. It marks the end of the loading curve. The tensile stress at that point on a stress–strain graph is the breaking stress.

A brittle material fractures after little plastic deformation. It therefore has little permanent deformation before failure, even though it may have deformed elastically while loaded. A material that undergoes more plastic deformation can retain a larger extension or strain if it is unloaded before fracture.

Comparing materials using graphs

Graphs can be used to compare the deformation and failure of materials: look for proportional regions, the deformation remaining after unloading, and the point of fracture. On stress–strain graphs, the stress at each fracture point allows the materials’ breaking stresses to be compared. A greater total extension or strain before fracture does not by itself show how much of that deformation is permanent.

Elastic strain energy

When an object is stretched or otherwise deformed, work is done on it. If the deformation is elastic, the object returns to its original shape when the deforming force is removed. Energy stored during the deformation can then be recovered. This stored energy is called elastic strain energy.

Work done in extending a spring

The work done in extending a spring is represented by the area under its force–extension graph. The force usually changes as the spring is extended, so the work done is not generally the final force multiplied by the final extension.

For a spring obeying Hooke’s law, force is directly proportional to extension. Its force–extension graph is a straight line through the origin. From zero extension to a final force $$F$$ and extension $$\Delta L$$, the area under the graph is a triangle. Its area, and hence the elastic strain energy stored, is:

Here, $$E$$ is the energy stored in joules, $$F$$ is the final force in newtons, and $$\Delta L$$ is the extension in metres.

Permanent deformation

If an object is permanently deformed, it does not return completely to its original shape when the force is removed. In that case, not all the work done in deforming it is recoverable as elastic strain energy.

Energy conservation in deformation

A deformed spring can store energy as elastic strain energy. When the spring returns towards its original shape, this energy can be transferred to a moving object as kinetic energy, to gravitational potential energy if the object rises, or to a combination of both.

The energy is not necessarily transferred completely into these forms. Some may remain stored in the spring, while some may be transferred into permanent deformation or other non-recoverable forms. An energy balance accounts for all these changes.

Energy changes during motion

An object of mass $$m$$ moving at speed $$v$$ has kinetic energy $$E_k = \tfrac{1}{2}mv^2$$. Its kinetic energy change is the final kinetic energy minus the initial kinetic energy.

If the object changes height by $$\Delta h$$, its gravitational potential energy changes by $$\Delta E_g = mg\Delta h$$, where $$g$$ is the gravitational field strength. The change is positive when the object rises and negative when it falls.

Energy balance for deformation

For a system with no additional energy supplied from outside, the initial energy equals the final energy plus energy transferred into non-recoverable forms during the change:

Here $$E_{\mathrm{el}}$$ is elastic strain energy, and the subscripts $$\mathrm{i}$$ and $$\mathrm{f}$$ mean initial and final. Gravitational potential energies are measured from the same chosen reference height. An energy term is zero only if the conditions justify it: for example, the final kinetic energy is zero if the object is momentarily at rest.

Non-recoverable energy includes energy transferred into permanent deformation and other forms that cannot return as elastic energy. It must be included even though it is not available to produce motion or raise the object.

For example, suppose a spring releases $$10,\mathrm{J}$$ of elastic strain energy. If an object gains $$6,\mathrm{J}$$ of kinetic energy and $$3,\mathrm{J}$$ of gravitational potential energy, the remaining $$1,\mathrm{J}$$ must have been transferred into non-recoverable forms.

Energy conservation in ethical transport design

When a moving vehicle collides and slows down, its kinetic energy decreases. Energy cannot disappear: it is transferred to other stores or processes. Transport designers must decide how to manage these transfers, knowing that their choices can affect people’s safety and use resources.

Energy transfers in a collision

During a collision, forces deform the vehicle and anything it strikes. Some kinetic energy is transferred through deformation, while some is transferred to thermal energy and sound. Deformation can help manage the vehicle’s loss of kinetic energy, but it can also damage structures.

A designer might allow part of a vehicle to deform while keeping the space around its occupants more secure. This can reduce the risk to occupants, but the outcome depends on the collision and on what or whom the vehicle strikes. Conservation of energy sets a limit on the design: the vehicle’s lost kinetic energy must go somewhere; no design can make it vanish.

Weighing the consequences of transport design

Energy conservation helps designers assess what a design can achieve. Ethical evaluation asks whether its consequences are acceptable: who benefits, who bears the risks, and how those risks compare with other impacts.

For example, a design that offers greater protection may require more material and other resources. Designers must weigh that resource use against the safety benefits, while also considering risks to people outside the vehicle as well as those inside it. Energy conservation informs this decision, but it does not by itself decide how competing consequences should be valued.