11 - Materials

11.1 Density

Density of a substance is defined as its mass per unit volume , p=m/v

Measurements

Regular solid: measure mass using a top pan, measure dimensions using vernier calipers or micrometer and calculate volume (v=4/3 x π x r.cubed)

Liquid: Measure mass of empty cylinder, measure mass of cylinder with liquid and find difference

Irregular shape: Measure mass, immerse object in liquid in cylinder and observe increase in liquid level

Density of alloys

An alloy is a solid mixture of two or more metals

density, p = (p1V1/V)+(p2V2/V)

11.2 Springs

Hooke's law

States that the force needed to stretch a spring is directly proportional to the extension of the spring from its natural length

Force F= kΔL where k=sprinf constant and ΔL= extension from natural length

Greater k= stiffer spring, graph F against ΔL is a straight line with gradient k, If spring stretched beyond elastic limit it does not regain initial length

Springs in parallel

Weight supported by 2 springs, k = k1 + k2

Springs in series

Two springs joined end on end in series with each other. Tension is the same and equal with weight

1/k = 1/k1 + 1/k2

Energy stored in stretched spring

Elastic potential energy is stored in a stretched spring. If spring is suddenly released, stored elastic energy is suddenly transferred into kinetic energy in spring. Work done to stretch spring = 1/2FΔL

Elastic potential energy stored in a stretched spring, Ep = 1/2FΔL = 1/2kΔL.Squared

11.3 Deformation of solids

Force and solid materials

Force needed to stretch, twist or compress a material. The elasticity of a solid material is its ablility to regain shape after it has been deformed or distorted and the force used has been released

Deformation that stretches an object is tensile whereas deformation that compresses an object is compressive

A steel spring gives a straight line, in accordance to Hooke's law

Rubber band first extends easily however becomes fully stretched and is very difficult to stretch further

Polythene strip 'gives' and stretches easily after initial stiffness is overcome. After 'giving' easily, it extends little and then becomes difficult to stretch

Tensile stress and strain

Extension of a wire under tension may be measured using Searle's apparatus. Micrometer attached to control wire is adjusted so spirit level between control and test wire is horizontal. When test wire is loaded, it extends slightly, causing spirit level to drop on one side. Micrometer is then readjusted to make spirit level horizontal again. Change of micrometer reading is therefore equal to extension

Tensile stress in wire = σ=T/A (unit=pascal)

Tnsile strain in wire, ε=ΔL/L(ratio means no unit)

Young modulus E = tensile stress/tensile strain = σ/ε = T/A ÷ ΔL/L = TL/AΔL

Beyond elastic limit = yield point(wire weakens temporarily). Elastic limit is point beyond where wire is permanently stretched and suffers plastic deformation. Beyond max tensile stress, the ultimate tensile stress(breaking stress), the wire loses its strength, extends, and becomes narrower at weakest point.

Stress-strain curves for different materials

Stiffness of different materials can be compared using the gradient of stress-strain line, which is equal to Young modulus of material

Strength of material is ultimate tensile stress

Brittle material snaps without noticeable yield(glass)

Ductile material can be drawn into a wire. Copper more ductile than steel)

11.4 Stress and Strain

Loading and unloading of different materials

For metal wire, loading and unloading curves are the same, provided its elastic limit is not exceeded(returns to original shape when unloaded). Beyond elastic limit, unloading line is parallel to loading line. The wire will be slightly longer when unloaded (permanent extension)

For rubber band, change in length during unload is greater than during load for a given tension. It returns to same unstretched length, but unloading curve is below loading curve except at 0 and max extensions. Rubber band remains elastic as it regains initial length, But it has a low limit of proportionality

For polythene strip, extension during unloading is also greater than during loading. However, strip does no return to initial length when completely unloaded. Strip has a low limit proportionality and suffers plastic deformation

Strain energy

Area under line of force-extension graph= work done to stretch wire. Work done to deform object is strain energy

Metal wire or spring

Provided limit of proportionality is not exceeded, the work done=1/2TΔL. As elastic limit is not reached, work done is stored as elastic energy in wire: elastic energy stored in stretched wire = 1/2TΔL

Rubber band

Area between unloading and loading represents the difference between energy stored in rubber band when stretched and useful energy recovered when its unstretched. Difference occurs as some energy stored becomes internal energy of molecules

Polythene

As it does not regain initial length, area between loading and unloading represents work done to deform material permanently as well as internal energy retained when it unstretches