materials

Density

mass per unit volume
Density is a SCALAR quantity

Units of density

kg m-3

Hooke's Law

extension is proportional to the force applied, up to the limit of proportionality.

Features of graph of force against extension confirming Hooke's Law

straight line, through the origin.

Units of spring constant

Nm-1

Springs in series

• Both springs experience the same force F.
• The total extension (of both springs together) is the sum of the extension of each spring individually.

(Identical) Springs in parallel

• The force F applied to the spring combination is shared across each of the springs individually (if there are two identical springs, each spring experiences a force of ½F).
• All springs have the same extension (and equals the extension for the spring combination).

Elastic limit

the maximum amount a material can be stretched by a force and still return to its original length when the force is removed.

Limit of proportionality

point beyond which force is no longer proportional to extension

Elastic behaviour

material will return to its original length (when force removed) with no permanent extension.

Plastic behaviour

material will be permanently extended (when force is removed).

Area under a force/extension graph

area under a graph of force against extension is work done on spring and hence the energy stored, as it is loaded.
or
area under a graph of force against extension is the work done by the spring, and hence energy released, as it is unloaded.

Area between the loading and unloading curves of an elastic band

internal energy retained, eg as heat, within the elastic band

Derivation of energy stored \= ½ FΔL from a graph of force against extension

ΔW\=FΔs, so area beneath line from origin to ΔL represents the work done to compress/extend spring.
• work done (on spring) equals the energy it stores.
• area under graph \= area of triangle \= ½ base x height, therefore energy stored \= ½ F x ΔL.

Derivation of
energy stored \= ½ FΔL

• Energy stored in a stretched spring \= work done stretching the spring.
• Work done \= Force x distance (moved in the direction of the force)
• As spring is stretched the force gets bigger (and so isn't constant).
• Force is proportional to Extension, so,
average force \= ( ), which \= ½ F.
• The work done \= average force x distance moved
• Energy stored \= work done \= ½ F ΔL
• This is the area under the graph of Force against Extension (½ base x height).

Tensile stress

tensile (stretching) force divided by its cross-sectional area

Units of stress

Pa or Nm-2

Tensile strain

extension of material divided by its original length

Units of strain

None (a strain can be given as a %, eg 0.3% is a strain of 0.003)

Breaking stress
(also known as ultimate tensile stress)

(tensile) stress needed to break a solid material

Description of stiffness

requires a large force (or stress) for a small deformation (or extension)

Description of fracture

Non-brittle fracture
Material necks (becomes narrower at its weakest point) which reduces the cross-sectional area so increases stress at that point until the wire breaks (at that point)
Brittle fracture
No plastic deformation, usually snaps suddenly without any noticeable yield (through crack propagation).

Description of brittle

a material that fractures without any plastic deformation

Description of ductile

material can be drawn into a wire (exhibits a lot of plastic deformation)

Description of strength (or weakness)

Material with a higher (or lower) breaking stress.

Young Modulus

ratio of tensile stress to tensile strain

Units of Young Modulus

Pa or Nm-2

Use of stress/strain curves to find Young Modulus

from a graph of stress against strain, Young Modulus is the gradient of the linear section of the graph (the region where stress and strain are directly proportional)

Area under a graph of stress against strain

energy stored per unit volume

One simple method of measuring Young Modulus

Measurements to make
• Original length of wire, L, with a ruler
• Diameter of wire with a micrometer
• Mass attached to end of wire
• Length of stretched wire with a ruler.
Reducing Uncertainty in each measurement
• Repeat measurements of length
• Repeat measurements of diameter of wire at different points
• Check for zero error on electronic scales
• Check for zero error on micrometer
How measurements are used to determine Young Modulus
• F\=weight\=mg
• Extension ΔL \= stretched length - original length
• Cross-sectional area of wire A \= πd2 / 4.
• Stress \= F/A; Strain \= ΔL/L
• Plot a graph of stress (y-axis) against strain (x-axis)
• Young Modulus is gradient of linear section of graph

Area under Force against Extension graphs

Area under graph equals energy stored.

• For an elastic band (shown), the area under the loading curve is the total energy stored in the elastic band when stretched.
• The area under the unloading curve is less, so not all energy stored in the band is released.
• The difference in energy between loading and unloading (area between the two curves) is lost as heat in the elastic band.

Area under Stress against Strain graphs

Area under graph equals energy stored per unit volume

Energy transfers in a compressed spring that is released

Elastic strain energy in spring is converted into kinetic energy, which in turn is converted into gravitational potential energy, as spring is thrown up in air.