materials

limit of proportionality

• the point after which extension is no longer directly proportional to the force applied

hooke’s law

• force applied is directly proportional to extension up to the limit of proportionality

• on a graph - passes through the origin and is a straight line

elastic material

returns to its original length when the force is removed

tensile stress $\sigma$

• the force per unit perpendicular cross sectional area

tensile strain $\char"0190$

• the extension per original unit length

young’s modulus

• stress / strain

• provided that the limit of proportionality has not been exceeded

• $$E=\frac{TL}{A\Delta L}$$

• wire of uniform diameter

brittle

material snaps without any noticeable / significant yield (after little to no plastic deformation)

ductile

noticeable yield before breaking - the material can be drawn into a wire

explain how the formula for energy stored in a spring can be dervived

• work done = force x distance

• area beneath line of force extension graph

springs in series and in parallel

series

• effective spring constant (1/k₁ + 1/k₂)^-1

• same force in both springs

parallel

• effective spring constant k₁+k₂

• same extension in both springs

graph of tensile stress against tensile strain for metal wire

• from origin to limit of proportionality, tensile stress is proportional to tensile strain - stress / strain is gradient - also youngs modulus - constant

• beyond this, graph curves and continues beyond elastic limit - wire permanently stretched - plastic deformation

• yield point - wire temporarily weakens

• beyond this - small increase in the tensile stress causes a large increase in tensile strain - plastic flow

• ultimate tensile stress - breaking stress

stress- strain curves for different materials

• strength of a material is its ultimate tensile stress

• stiffness is its young’s modulus - greater gradient means stiffer material

loading and unloading curves (force against extension)

• metal wire - loading and unloading curves are the same - provided elastic limit has not been reached - wire returns to its original length when unloaded. beyond elastic liimit, unloading line is parallel to loading line - wire permanently extended

• rubber band - change of length during unloading is greater - unloading curve below loading curve except at 0 and max extension - remains elastic - very low limit of proportionality.

• polythene - the extension during unloading is greater than during loading - strip does not return to same initial length - plastic deformation- low limit of proportionality

work done to deform and strain energy

• total work done to deform object is total area under loading curve

• strain energy is the energy that is stored in the material’s elastic potential store - area under unloading curve

• work done to permanently deform is the energy that is stored internally in the material’s molecules when deformation continues past elastic limit - this energy cannot be recovered - area enclosed between loading and unloading curve

• metal wire: work done = ½ TdeltaL - provided limit of proportionality not exceeded - energy stored in elastic potential as elastic limit not reached - all energy stored in wire can be recovered when unloaded

• rubber band : work done = area under loading curve. area between the loading curve and the unloading curve - difference in energy stored in stretched rubber band and the useful energy recovered in unloading - some energy becomes stored in interernal energy of the molecules when the rubber band unstretches

• rubber band: area under the loading curve and the unloading curve = work done to deform the material permanently, as well as interanl energy retained when it unstretches

energy stored in a stretched spring

Ep = ½ F delta L

since F = k delta L

Ep = ½ k (delta L)²

uses average force since force is not constant - variest linearly with extension

density

the mass of a substance per unit volume

density measurements

regular solid

• measure its mass using a top pan balance

• measure dimensions using vernirer callipers or a micrometer

• calculate volume using suitable equation

• calculate ddensity from mass / volume

liquid

• measure mass of an empty measuring cylinder

• pour some liquid into into measuring cylinder and easure volume - use as much as possible to reduce percentage error in measurement

• measure mass of cylinder and liquid and subtract mass of cylinder

• calculate density by mass / volume

irregular solid

• measure mass of solid with top pan balance

• immerse object on a thread in liquid in measuring cylinder

• observe the increase in liquid level - this is the volume of the object

• calculate density by mass / volume

stress in a uniform cable with non negligible mass

• stress increases linearly with distance from the bottom

• extra mass added due to mass per unit length of cable $\mu$

• Tension at bottom = mg

• tension at top $\operatorname{mg+\mu gL}$