Deformation of solids

A-Level Physics 9702 · Topic 6 revision notes · AS

Springs and wires under load: Hooke's law, stress, strain and the Young modulus, elastic and plastic behaviour, and the energy stored.

11 definitions to learn word for word, 7 equations in their own column, 5 diagrams and graphs. Read it, cover it, test yourself.

Definitions (mark-scheme wording)EquationsExam traps

6.1aHooke's law and the spring constant

Hooke's law: the extension of a spring (or wire) is directly proportional to the applied force, provided the limit of proportionality is not exceeded.
Spring constant: the force per unit extension (k = F/x).
Limit of proportionality: the point beyond which force (stress) is no longer proportional to extension (strain).
Tensile / compressive force: a pair of forces that stretches / squashes an object.

Force-extension graph: straight line through the origin up to the limit of proportionality.
Gradient of F against x = k; a stiffer spring has a steeper line.
Extension x = stretched length - original length (not the total length).
Compression works the same way: x is the decrease in length.
k depends on the sample (material, length, thickness), not only on the material.

Force-extension graph: linear to P (limit of proportionality), still elastic to E, plastic beyond.

Using the total length instead of the extension in F = kx.
Assuming the limit of proportionality and the elastic limit are the same point; E is usually just beyond P.

Tensile stress: the force per unit cross-sectional area (acting normal to the area).
Tensile strain: the extension per unit original length.
Young modulus: the ratio of tensile stress to tensile strain (within the limit of proportionality).

Young modulus is a property of the material, independent of sample size and shape.
Strain has no unit; stress and Young modulus are in Pa.
Stress-strain graph: gradient of the linear region = E.
Wire area: A = πd²/4; measure d with a micrometer at several places and orientations, take the mean.
Experiment: long, thin wire (large, measurable extension); measure L with a metre rule, x with a vernier/travelling microscope.
Plot F against x: E = gradient × L / A.

Measuring E: a long wire loaded over a pulley; a marker against a fixed scale gives the extension.

Stress-strain graphs: gradient in the linear region = Young modulus. Steel is stiffer than copper.

Using the diameter instead of the radius (or forgetting to square) in A = πr².
Forgetting unit conversions: mm to m, mm² to m² (× 10^-6). E for metals is ~10¹¹ Pa.

Elastic deformation: deformation in which the object returns to its original shape and size when the load is removed.
Plastic deformation: deformation in which the object does not return to its original shape when the load is removed (permanent deformation).
Elastic limit: the maximum force (stress) beyond which the object no longer returns to its original length when the load is removed.
Elastic potential (strain) energy: the energy stored in a deformed object because of its change in shape.

Beyond the elastic limit, unloading leaves a permanent extension; the unloading line is parallel to the loading line.
Strain energy = area under the F-x graph (work done to deform it).
Linear region only: area is a triangle, E_P = ½Fx = ½kx².
Non-linear graph: find the area by counting squares or splitting into shapes.
Plastic deformation: area between loading and unloading curves = energy dissipated as thermal energy.
Doubling the extension (within the limit) quadruples the stored energy.

Strain energy stored = shaded triangle under the F-x line = ½Fx.

Loaded past the elastic limit: unloading runs parallel to the loading line and leaves a permanent extension.

½Fx and ½kx² apply only to the linear region; beyond it use the area under the curve.
Using the total length instead of the extension, or forgetting the ½.