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what is density
 density is a measure of the "compactness" of a substance . it relates the mass of a substance to how much space it takes up . the density of a material is its mass per unit volume :
 p = m/v
 p = density in kg m^3
 m = mass in kg
 v = volume in m^3
 however if you are given the mass in g and the volume in cm^3 you can work out the density of an object in in kg cm^3

the density of an object depends on
what it's made of . Density of a material doesn't vary with shape or size

the average density of an object determines whether
it sinks or floats

A solid object will float on a fluid if it has
a lower density than the fluid

1 g cm^3 = ...... kg m^3
1000

why does an empty bottle float on water
the average density of the bottle and the air inside it combined is lower than the density of the water

you can work out the volume of a sphere using
V = 4/3 x Pi x r^3

how can you measure the density of regular solids
 find the mass
 find the volume by using length , width and height measurements
 use the formula : density = mass /volume

how can you measure the density of irregular solids
 fill a displacement can and let the water run out of the spout
 place the object in the can and measure the volume of water displaced
 find the mass
 use the formula : density = mass/volume

alloys are
 mixtures of metals
 e.g. brass is a mixture of copper and zinc

to calculate the mass of an alloy use the formula
M = PaVa + PbVb

to calculate the density of an alloy use the formula
 P = PaVa + PbVb
 
 V

what is Hooke's law
if a metal wire of original length L is supported at the top and then a weight attached to the bottom , it stretches . The weight pulls down with force F , producing an equal and opposite force at the support

draw a diagram to show a supported metal wire extending by ΔL when a weight is attached

Robert Hooke discovered in 1676 that the extension of a stretched wire , ΔL , is
proportional to the load or force , F . This relationship is now called Hooke's law .

Hooke's law can be written as
 F = KΔL
 F = force in newtons
 k = the stiffness constant Nm^1
 ΔL = extension in m
 k is a constant that depends on the object being stretched

an object's stiffness constant is the
force needed to extend it by 1m . It depends on the material that it's made from , as well as its length and shape .

a metal spring also changes length when you apply a pair of opposite forces . the extension or compression of a spring is proportional to the
force applied  so Hooke's law applies

draw a diagram to show metal springs with tensile and compressive forces acting on them

if two things are proportional , it means
that if one increases , the other increases by the same proportion

a tensile force
stretches something

a compressive force
squashes it

for springs , K , in the formula F=KΔL is usually called the
spring stiffness or spring constant

Hooke's law works just as well for compressive forces as tensile forces . For a spring , k , has the same value whether the forces are
tensile or compressive (though some springs and many other objects can't compress)

don't be put off if you're asked a question that involves 2 or more springs . the formula F=KΔL still applies ...
for each spring

there's a limit to the force you can apply for Hooke's law to stay true . Draw a diagram to show load against extension for a typical metal wire

explain the graph on the previous slide
 the first part of the graph shows Hooke's law being obeyed  there's a straightline relationship between load and extension and it goes straight through the origin . The gradient of the straight line is stiffness constant , k .
 when the load becomes great enough , the graph starts to curve . The point marked E on the graph is called the elastic limit . If you increase the load past the elastic limit , the material will be permanently stretched . When all the force is removed , the material will be longer than at the start
 metals generally obey Hooke's law up to the limit of proportionality , marked P on the graph , which is very near the elastic limit . The limit of proportionality is the point beyond which the force is no longer proportional to the extension . The limit of proportionality is also known as the Hooke's law limit

there are some materials such as rubber that only obey Hooke's law for
really small extensions

you may be asked to give two features of a graph which show that the material under test obeys Hooke's law . Just remember , the graph of a material which obeys Hooke's law will start with
a straight line through the origin

the limit of proportionality can be described as
the point beyond which the loadextension graph is no longer linear

draw a diagram to show the experimental set up you could use in the lab to investigate how the extension of an object varies with the force used to extend it
and explain how you would use the equipment
 the object under test should be supported at the top, e.g. using a clamp and a measurement of its original length taken using a ruler . Weights should then be added one at a time to the other end of the object .
 the weights used will depend on the object being tested  you should do a trial investigation if you can to work out the range and size of weights needed . You want to be able to add the same size weight each time and add a large number of weights before the point the object breaks to get a good point of how the extension of the object varies with the force applied to it.
 after each weight is added , the extension of the object should be calculated . This can be done by measuring the new length of the object with a ruler and then using : extension = new length  original length
 finally a graph of load against extension should be plotted to show the results

if a deformation is elastic , what happens to the material once the forces are removed
the material returns to it's original shape so it has no permanent extension

annotate a graph to show loading and unloading of an elastic material

loading just means
increasing the force on the material

unloading means
reducing the force on the material

what happens when an elastic material is put under tension
the atoms of the materials are pulled apart from one another . Atoms can move small distances relative to their equilibrium positions without actually changing position in the material . Once the load is removed , the atoms return to their equilibrium distance apart .

for a metal , elastic deformation happens as long as
Hooke's law is obeyed

if a deformation is plastic , what happens to the material once the forces are removed
the material is permanently stretched . some atoms in the material move position relative to one another . When the load is removed , the atoms don't return to their original positions . A metal stretched past its elastic limit shows plastic deformation

for springs in series what are the equations
 extension a = f/ka
 extension b = f/kb
 total extension = extension a + extension b
 total extension = f/ka + f/kb
 the effective spring constant :
 1/k = 1/ka + 1/kb

the important thing to remember about springs in series
each spring experiences the same force

remember that weight =
 mass x gravitational field strength
 kg x 9.81 m/s^{2}

draw a diagram to show springs in series

draw a diagram to show two springs in parallel

for springs in parallel what are the equations
 force needed to stretch a > fa = ka x extension
 force needed to stretch a when there is weight bar or a weight equally between the two springs is half the force on the weight bar
 force needed to stretch b > fb = kb x extension
 force needed to stretch b when there is a weight bar or a weight equally between both springs is half the force on the weight bar
 NB the extensions are only the same if there is a weight bar or the weight is placed equally between two springs
 total force = fa + fb = (ka x extension) + (kb x extension)
 the effective spring constant = ka + kb

weight and force are equivalent in this topic
:)

two samples of the same material with different dimensions will stretch different amounts under the same force . Stress and strain are
measurements that take into account the size of the sample , so the stress strain graph is the same for any sample of a particular material

a material subjected to a pair of opposite forces may
deform i.e. change shape .

if the forces stretch the material , they're
tensile

if the forces squash the material they're
compressive

tensile stress is defined as
the force applied , F , divided by the crosssectional area , A : stress = F/A

the units of stress are
Nm^{2 }or Pascals , Pa


tensile strain is defined as
the change in length , i.e. the extension , divided by the original length of the material : strain = extension/L

what are the units of strain
strain doesn't have units it's just a number

NB it doesn't matter whether the forces producing the stress and strain are tensile or compressive  the same equations apply . the only difference is that you tend to think of tensile forces as positive and compressive forces as negative
:)

as a greater tensile force is applied to a material , the stress on it
increases

draw a diagram to show a stressstrain graph showing the ultimate tensile strength and breaking stress of a material

the effect of stress is to
start to pull the atoms apart from one another . Eventually stress becomes so great that atoms separate completely and the material breaks . This is shown by point B on the previous graph . the Stress at which this occurs is called the breaking stress  the stress that's big enough to break the material .

the point marked UTS on the recent graph is called the
ultimate tensile stress . This is the maximum stress that the material can withstand . Engineers have to consider the UTS and breaking stress of materials when designing a structure

1MPa is the same as
1 x 10^{6 }Pa

when a material is stretched , work has to be done to in stretching the material . Before the elastic limit , all the work done in stretching is stored as potential energy in the material . This stored energy is called
elastic strain energy

on a force against extension the , the elastic strain energy is given by the
area under the graph

draw a graph with a title "the area under a force extension graph for a stretched material is the elastic strain energy stored by it"

provided a material obeys Hooke's law , the potential energy stored inside it can be calculate quite easily using a formula . This formula can be derived using
a force extension graph and work done

the energy stored by a stretched material is equal to the .... on the material in stretching it . so on a force extension graph , the area underneath the straight line from the origin to the extension represents the ...... or the ...........
 work done
 energy stored or the work done

work done is equal to
force x displacement

on a force displacement graph the force acting on the material is not constant . therefore you need to
work out the average force acting on the material , from zero to F which is 1/2F . So it's the area underneath the graph : work done = 1/2 F x extension

and so the elastic energy is
E = 1/2 F x extension

because Hooke's law is being obeyed
F = k x extension which means F can be replaced in the equation to give
E = 1/2 K x extension x extension

if the material is stretched beyond the elastic limit , some work is done
changing the positions of atoms . This will not be stored as strain energy and so isn't available when the force is released

to calculate the elastic strain energy the force has to be in ... and the extension has to be in .........

the young modulus is a measure of
how stiff a material is , it is really useful for comparing the stiffness of different materials , for example if you're trying to find out the best material for making a particular product

when you apply a load to stretch a material , it experiences a
tensile stress and a tensile strain

up to a point called the limit of proportionality the stress and strain are
proportional to each other .

below this limit of proportioaniliity , for a particular material , stress divided by strain is a constant . this constant is called
the young modulus , E

E =
E = tensile stress / tensile strain
 E = F/A
 
 EXTENSION/L
 E = FL
 
 A X Extension
 where
 F = force in newtons
 A = cross sectional area in m^{2 }
 L = initial length in m
 and extension is also in m
 the units of young modulus are the same as stress (Nm^{2 }or pascals) , since strain has no units

draw and label suitable apparatus required for measuring the Young Modulus of a material in the form of a long wire

list the measurements you would make using the apparatus on the previous slide
 length of the wire between clamp and mark
 diameter of wire
 extension of wire for known weight

describe how the measurements on the previous slide would be carried out
 length measured by metre ruler
 diameter using a micrometer at several positions and mean taken
 known weight added and extension measured by noting the difference between the marker reading and unstretched length
 repeat readings for increasing loads

explain how you would calculate the young modulus from your measurements
 graph of force against extension
 gradient gives F/extension
 so gradient x unstretch length /cross sectional area of wire = young modulus
cross sectional area can be worked out using A = pi(diameter/2) ^{2}

the gradient of a stress strain graph gives the
young modulus , E

the area under a stress strain graph gives the
strain energy (or energy stored) per unit volume

the stress strain graph for a material is a
 straight line provided that Hooke's law is obeyed , so you can calculate the energy per unit volume as
 energy per unit volume = 0.5 x stress x strain

steel has a high young modulus , which means
under huge stress there's only a small strain . This makes it an ideal building material for things like bridges

remember when using the gradient to work out the young modulus
you can only use it up to the limit of proportionality , after then the stress and strain are no longer proportional

don't forget to convert any lengths to ... and areas to ... when working out the young modulus ^{
}

draw a typical stress strain graph

explain the graph on the previous slide
 Before point P , the graph is a straight line through the origin . this shows the material is obeying Hooke's law . The gradient of the line is constant and represents the Young modulus
 Point P is the limit of proportionality  after this , the graph is no longer a straight line , but starts to bend . At this point , the material stops obeying Hooke's law but would still return to its original shape if the stress was removed
 Point E is the elastic limit p at this point the material starts to behave plastically . From point E onwards , the material would no longer return to its original shape once the stress was removed
 Point Y is the yield point  here the material suddenly starts to stretch without any extra load . The yield point or yield stress is the stress at which a large amount of plastic deformation takes place with a constant or reduced load
 the area under the first part of the graph gives the energy stored in the material per unit volume

plastic deformation is useful if
you dont want a material to return to its original shape , e.g. drawing copper into wires or gold into gold foil

draw a stressstrain graph of a brittle material
the graph starts with a straight line through the origin . so brittle materials obey Hooke's law . However , when the stress reaches a certain point , the material snaps  it doesn't deform plastically

examples of brittle materials
 a chocolate bar is an example of a brittle material  you can break chunks of chocolate off the bar without the whole thing changing shape
 ceramics (glass and pottery) are brittle too  they tend to shatter

the structure of brittle materials
 atoms in ceramics are bonded in a giant rigid structure . the strong bonds between the atoms make them stiff , while the rigid structure means that ceramics are very brittle .


when stress is applied to a brittle material any
tiny cracks at the metals surface are made bigger and bigger until the material breaks completely . this is called brittle fracture . the cracks in brittle materials are able to grow because these materials have a rigid structure . other materials , like most metals aren't brittle because the atoms within them can move to prevent any cracks getting bigger

