Mechanical properties of matter — OCR A-Level Physics
Test yourself on Mechanical properties of matter with OCR A-Level practice questions.
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Mechanical properties of matter explained
A force–extension graph plots the applied force on the vertical axis against the extension (or compression) on the horizontal axis.
Read the full explanation
For a material obeying Hooke's law the line is straight through the origin, so force is proportional to extension. The work done in stretching or compressing the material equals the area under the graph, because work is the product of force and the displacement it produces. For a straight line from the origin to a point (x, F), that area is a triangle: work done = ½ × base × height = ½Fx. If the graph curves, you cannot use ½Fx; you must estimate the area, for example by counting squares or splitting it into strips. Extension is the change in length, not the total length, and the same reasoning applies to compression.
(b) elastic potential energy; E = 1/2 Fx; E = 1/2 kx²
Elastic potential energy is the energy stored in a material when it is stretched or compressed elastically, and it is recovered when the deforming force is removed. For a material obeying Hooke's law, the stored energy equals the work done, which is the area under the force–extension graph: E = ½Fx, where F is the final force and x is the extension. Since F = kx, substituting gives E = ½kx², where k is the force constant in N m⁻¹. Both expressions apply only while the limit of proportionality is not exceeded. For example, a spring of force constant 200 N m⁻¹ stretched by 0.10 m stores E = ½ × 200 × (0.10)² = 1.0 J.
(c) stress, strain and ultimate tensile strength
Stress is the force acting per unit cross-sectional area, σ = F/A, measured in pascals (Pa), where 1 Pa = 1 N m⁻². Strain is the fractional change in length, ε = x/L, where x is the extension and L the original length; it has no unit. Stress and strain allow materials of different sizes and shapes to be compared. Ultimate tensile strength is the maximum stress a material can withstand before it breaks when being stretched. On a stress–strain graph, the ultimate tensile strength is the highest stress value reached. For example, a wire of cross-sectional area 2.0 × 10⁻⁶ m² carrying a force of 10 N experiences a stress of 5.0 × 10⁶ Pa.
(d)
This row is a guided-reading item with no supplied statement text beyond the label (d), so it is not an assessed answer target. Use it to teach how to read the specification and select appropriate study material for section 3.4.2 Mechanical properties of matter. Learners should locate the official OCR A Level Physics A specification, find section 3.4.2, and read the learning outcomes listed there. They should identify the physics content on force–extension graphs, elastic potential energy, stress, strain and ultimate tensile strength, and check which practical skills and mathematical requirements are linked. They should then choose a textbook or revision guide that covers these outcomes, and use the specification wording to check their notes for completeness.
(i) Young modulus = tensile stress / tensile strain, E = σ/ε
The Young modulus E quantifies stiffness: how much a material stretches elastically per unit of applied stress. Tensile stress σ is the force F applied perpendicular to a specimen's cross-sectional area A, so σ = F/A, measured in pascals (Pa), where 1 Pa = 1 N m⁻². Tensile strain ε is the fractional extension: ε = ΔL/L, where ΔL is the extension and L the original length; strain is dimensionless, though it may be quoted as a percentage. Combining these gives E = σ/ε = (F/A)/(ΔL/L) = FL/(AΔL), in Pa. For example, a wire of area 1.0 × 10⁻⁶ m², original length 2.00 m, extending 1.0 mm under 50 N: σ = 5.0 × 10⁷ Pa, ε = 5.0 × 10⁻⁴, so E = 1.0 × 10¹¹ Pa. E applies only within the elastic limit, where stress is proportional to strain.
(ii) techniques and procedures used to determine the Young modulus for a metal
A practical determination of the Young modulus for a metal wire uses a long, thin specimen clamped at one end and loaded at the other over a pulley. A metre rule or scale measures the original length L between fixed marks, and a micrometer measures the wire diameter d at several places and orientations; the mean diameter gives cross-sectional area A = πd²/4. A mass hanger adds known weights, so force F = mg using g = 9.81 m s⁻². Extension ΔL is measured with a vernier scale or travelling microscope, often against a fiducial mark, after tapping the wire to remove kinks. Plotting F against ΔL gives a straight line through the origin within the elastic limit; its gradient k = F/ΔL, so E = kL/A. Alternatively, E = FL/(AΔL) from a single consistent set of readings.
(e) stress–strain graphs for typical ductile, brittle and polymeric materials
Stress–strain graphs reveal how materials deform. A ductile material such as copper or mild steel shows an initial straight elastic region obeying Hooke's law, then yields and undergoes large plastic strain before breaking; it may show a yield point and ultimate tensile stress. A brittle material such as glass or cast iron is stiff and strong but fractures with little plastic strain, so its graph is almost linear to breaking. A polymeric material such as rubber shows a curved, non-linear response with a low Young modulus, large elastic strain and hysteresis: loading and unloading curves differ, and the area between them represents energy dissipated per unit volume. The gradient of the initial linear region gives the Young modulus; the area under the curve up to a strain gives energy stored per unit volume.
(f) elastic and plastic deformations of materials.
Elastic deformation is reversible: when the deforming force is removed, the material returns to its original shape and size. Atoms are displaced slightly from equilibrium positions but return, and the material obeys Hooke's law up to the elastic limit, where stress is proportional to strain. Plastic deformation is permanent: the force exceeds the elastic limit, atomic planes slip past one another, and the material does not return to its original shape when unloaded. A wire stretched beyond its elastic limit remains longer; a ductile metal may neck and eventually fracture. The elastic limit, yield point and breaking point mark transitions on a stress–strain graph. Energy stored elastically is recoverable; energy used in plastic deformation is dissipated, often as heat.
Your focus
- Interpret a force–extension or force–compression graph, identifying extension as change in length.
- Explain why the area under the graph represents work done on the material.
- Calculate work done from a straight-line graph using the triangle area, and describe how to estimate area for a curved graph.
Show all 24 objectives
- State the expressions for elastic potential energy stored in a stretched or compressed material.
- Use E = ½Fx and E = ½kx² to calculate stored energy for Hooke's law behaviour.
- Explain the condition under which these equations are valid.
- Define stress and strain and state their units.
- Calculate stress and strain from force, area, extension and original length.
- Identify ultimate tensile strength from a stress–strain graph and describe its meaning.
- Locate section 3.4.2 of the OCR A Level Physics A specification and identify its learning outcomes.
- Select study material that covers force–extension graphs, elastic potential energy, stress, strain and ultimate tensile strength.
- Check personal notes against the specification outcomes to identify gaps.
- Define tensile stress and tensile strain with correct symbols and units.
- Apply E = σ/ε to calculate the Young modulus, stress, strain, force, area or extension.
- Convert between SI prefixes and area units accurately when solving Young modulus problems.
- Describe a valid procedure to determine the Young modulus of a metal wire.
- Explain how measurements of length, diameter, force and extension combine to give E.
- Evaluate sources of uncertainty and describe steps that reduce them.
- Interpret stress–strain graphs for ductile, brittle and polymeric materials.
- Relate graph features such as gradient, yield point and breaking point to material properties.
- Explain hysteresis and energy dissipation in polymeric materials using stress–strain curves.
- Distinguish between elastic and plastic deformation using correct terminology.
- Explain elastic and plastic behaviour in terms of atomic displacement and slip.
- Interpret stress–strain graphs to identify the elastic limit, yield point and breaking point.
Mechanical properties of matter exam tips
Marking Points
- Work done in stretching or compressing equals the area under the force–extension (or force–compression) graph.
- For a straight-line graph through the origin, the area is a triangle, giving work done = ½Fx.
- Extension means the change in length from the natural length, not the total length of the sample.
- A curved graph requires area estimation, such as counting squares or dividing the region into strips.
- Elastic potential energy is the energy stored when a material is elastically stretched or compressed.
- E = ½Fx applies to a straight-line force–extension graph through the origin, where F is the final force and x the extension.
- E = ½kx² follows from substituting F = kx, with k the force constant in N m⁻¹.
- The expressions apply only up to the limit of proportionality; beyond it the stored energy is not given by these equations.
- Stress is force per unit cross-sectional area, σ = F/A, measured in pascals.
- Strain is extension divided by original length, ε = x/L, and is dimensionless.
- Ultimate tensile strength is the maximum stress a material can withstand before breaking.
- Stress and strain allow comparison of materials independently of sample dimensions.
- Defines tensile stress as force per unit cross-sectional area, σ = F/A, with units Pa or N m⁻².
- Defines tensile strain as extension divided by original length, ε = ΔL/L, and recognises it as dimensionless.
- States and applies E = σ/ε, equivalently E = FL/(AΔL), within the elastic limit.
- Uses consistent SI units and converts mm to m, mm² to m² and GPa to Pa correctly before substituting.
- Measures original length L between fixed reference marks with a metre rule, avoiding parallax.
- Measures wire diameter with a micrometer at several positions and orientations, then averages to find A = πd²/4.
- Applies known loads and records extension ΔL using a vernier scale or travelling microscope, tapping the wire to remove kinks.
- Plots a graph of force F against extension ΔL and uses the gradient k with E = kL/A, or substitutes into E = FL/(AΔL).
- Identifies safety and reliability measures such as a safety cushion or G-clamp, eye protection, and keeping stress within the elastic limit.
- Identifies the initial linear region as elastic behaviour where the gradient equals the Young modulus.
- Describes ductile behaviour: yield point, significant plastic deformation and a clear breaking point.
- Describes brittle behaviour: little or no plastic deformation, fracture soon after the elastic limit.
- Describes polymeric behaviour: non-linear curve, low gradient, large elastic strain and hysteresis between loading and unloading.
- Interprets the area under a stress–strain graph as energy per unit volume and the area between hysteresis loops as energy dissipated.
- Defines elastic deformation as reversible, with the material returning to its original dimensions when the load is removed.
- Defines plastic deformation as permanent, with the material retaining a changed shape after unloading.
- Links elastic behaviour to Hooke's law and the elastic limit, and plastic behaviour to atomic plane slip.
- Interprets the elastic limit, yield point and breaking point on a stress–strain graph.
- Distinguishes recoverable elastic strain energy from energy dissipated during plastic deformation.
Examiner Tips
- 💡Check whether the graph is a straight line through the origin before choosing the triangle-area method.
- 💡State clearly that work done is represented by the area under the graph, then show the area calculation.
- 💡If the graph is curved, describe a valid area-estimation method rather than forcing ½Fx.
- 💡Write down the equation, substitute values with units, then evaluate to avoid order-of-operation errors.
- 💡Check that the extension is in metres before squaring it in E = ½kx².
- 💡If the material has been stretched beyond the limit of proportionality, state that these equations no longer apply.
- 💡Convert area to square metres and length to metres before substituting into stress and strain equations.
- 💡Check that strain has no unit and stress is in pascals when reporting answers.
- 💡On a stress–strain graph, identify the highest point as the ultimate tensile strength.
- 💡Open the official specification and read section 3.4.2 before choosing study material.
- 💡Make a checklist of the learning outcomes and tick each one as you revise it.
- 💡Use the specification wording to write your own summary notes rather than copying text.
- 💡Write the defining equation, then substitute values with units shown, so method marks are visible even if arithmetic slips.
- 💡Check unit conversions first: 1 mm² = 1 × 10⁻⁶ m² and 1 GPa = 1 × 10⁹ Pa.
- 💡Sanity-check the magnitude: metals typically have E of order 10¹⁰–10¹¹ Pa, so an answer of 10⁵ Pa signals an error.
- 💡Describe the procedure in a logical sequence: set up, measure L, measure d, load, measure ΔL, analyse.
- 💡State how you reduce uncertainty: repeat diameter readings, use a long wire, use a travelling microscope, tap the wire.
- 💡Show the graph method clearly, labelling axes F and ΔL and explaining that the gradient equals k.
- 💡Sketch the three curve shapes and label elastic region, yield point, plastic region and breaking point.
- 💡Use the gradient of the linear portion to compare stiffness, and the breaking stress to compare strength.
- 💡For polymers, mention hysteresis and explain that the enclosed area represents energy dissipated per unit volume.
- 💡Use the words reversible and permanent explicitly when defining elastic and plastic deformation.
- 💡Refer to a stress–strain graph and mark the elastic limit, yield point and breaking point to support your explanation.
- 💡When discussing energy, state whether it is stored elastically and recovered or dissipated during plastic flow.
Common Mistakes
- Using the total length instead of the extension when reading the horizontal axis; the correction is to plot or read the change in length from the natural length.
- Assuming ½Fx always applies; the correction is that it holds only for a straight-line graph through the origin, and a curved graph needs area estimation.
- Treating the gradient as the work done; the correction is that the gradient gives the stiffness (force per unit extension) while the area gives the work done.
- Using the full force throughout the extension; the correction is that the average force is ½F, which is why the factor ½ appears.
- Forgetting to square the extension in E = ½kx²; the correction is to square x before multiplying.
- Applying the equations beyond the limit of proportionality; the correction is that they hold only for straight-line Hooke's law behaviour.
- Using the total length instead of the original length in the strain calculation; the correction is to divide the extension by the original length.
- Confusing stress with force; the correction is that stress is force divided by cross-sectional area and has units of pascals.
- Treating ultimate tensile strength as the breaking stress on a force–extension graph; the correction is that it is the maximum stress on a stress–strain graph.
- Treating the label (d) as a complete specification statement; the correction is to consult the official specification for the full learning outcome.
- Studying only the equations without the graph and material-property context; the correction is to cover force–extension graphs, energy, stress, strain and ultimate tensile strength together.
- Relying on a single revision guide without checking it against the specification; the correction is to cross-check coverage against the official learning outcomes.
- Using the final length instead of the original length in the strain denominator; the correction is to divide extension by the original length L.
- Forgetting to convert area from mm² to m² by multiplying by 10⁻⁶; the correction is to express A in m² before computing stress.
- Treating strain as having units of metres; the correction is to recognise strain is a ratio of two lengths and is dimensionless.
- Applying E = σ/ε beyond the elastic limit; the correction is to use the equation only where the material obeys Hooke's law.
- Measuring diameter once only; the correction is to take repeat readings at different places and orientations and average them.
- Using the total length of wire including the clamped portion; the correction is to measure only the length between the fixed marks that actually extends.
- Ignoring the weight of the hanger or adding masses without converting to force; the correction is to compute F = mg for each load.
- Plotting extension against force and then using the gradient directly as E; the correction is to use the gradient k = F/ΔL in E = kL/A.
- Assuming all materials have a straight-line stress–strain graph; the correction is to recognise polymers curve and brittle materials may fracture before much strain.
- Confusing strength with stiffness; the correction is to note that strength relates to breaking stress while stiffness relates to the gradient, the Young modulus.
- Thinking the area under a stress–strain graph is force; the correction is that it represents energy per unit volume because stress times strain is energy per unit volume.
- Treating the breaking point as the same as the elastic limit; the correction is that the elastic limit is where plastic deformation begins, while the breaking point is where fracture occurs.
- Believing elastic deformation means the material does not change shape at all; the correction is that it changes shape but returns to the original shape when unloaded.
- Thinking plastic deformation occurs before the elastic limit; the correction is that plastic deformation begins once the elastic limit is exceeded.
- Assuming all energy put into deforming a material is recovered; the correction is that energy used in plastic deformation is dissipated, often as heat.
- Confusing the elastic limit with the breaking point; the correction is that the elastic limit is where permanent deformation starts, while the breaking point is where the material fractures.