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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
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the density of an object depends on
what it's made of . Density of a material doesn't vary with shape or size
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the average density of an object determines whether
it sinks or floats
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A solid object will float on a fluid if it has
a lower density than the fluid
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1 g cm^-3 = ...... kg m^-3
1000
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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
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you can work out the volume of a sphere using
V = 4/3 x Pi x r^3
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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
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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
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alloys are
- mixtures of metals
- e.g. brass is a mixture of copper and zinc
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to calculate the mass of an alloy use the formula
M = PaVa + PbVb
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to calculate the density of an alloy use the formula
- P = PaVa + PbVb
- ---------------
- V
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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
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draw a diagram to show a supported metal wire extending by ΔL when a weight is attached
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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 .
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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
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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 .
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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
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draw a diagram to show metal springs with tensile and compressive forces acting on them
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if two things are proportional , it means
that if one increases , the other increases by the same proportion
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a tensile force
stretches something
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a compressive force
squashes it
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for springs , K , in the formula F=KΔL is usually called the
spring stiffness or spring constant
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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)
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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
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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
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explain the graph on the previous slide
- the first part of the graph shows Hooke's law being obeyed - there's a straight-line 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
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there are some materials such as rubber that only obey Hooke's law for
really small extensions
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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
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the limit of proportionality can be described as
the point beyond which the load-extension graph is no longer linear
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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
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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
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annotate a graph to show loading and unloading of an elastic material
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loading just means
increasing the force on the material
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unloading means
reducing the force on the material
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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 .
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for a metal , elastic deformation happens as long as
Hooke's law is obeyed
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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
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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
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the important thing to remember about springs in series
each spring experiences the same force
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remember that weight =
- mass x gravitational field strength
- kg x 9.81 m/s2
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draw a diagram to show springs in series
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draw a diagram to show two springs in parallel
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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
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weight and force are equivalent in this topic
:)
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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
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a material subjected to a pair of opposite forces may
deform i.e. change shape .
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if the forces stretch the material , they're
tensile
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if the forces squash the material they're
compressive
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tensile stress is defined as
the force applied , F , divided by the cross-sectional area , A : stress = F/A
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the units of stress are
Nm2- or Pascals , Pa
-
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tensile strain is defined as
the change in length , i.e. the extension , divided by the original length of the material : strain = extension/L
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what are the units of strain
strain doesn't have units it's just a number
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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
:)
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as a greater tensile force is applied to a material , the stress on it
increases
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draw a diagram to show a stress-strain graph showing the ultimate tensile strength and breaking stress of a material
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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 .
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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
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1MPa is the same as
1 x 106 Pa
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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
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on a force against extension the , the elastic strain energy is given by the
area under the graph
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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"
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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
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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
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work done is equal to
force x displacement
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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
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and so the elastic energy is
E = 1/2 F x extension
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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
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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
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to calculate the elastic strain energy the force has to be in ... and the extension has to be in .........
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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
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when you apply a load to stretch a material , it experiences a
tensile stress and a tensile strain
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up to a point called the limit of proportionality the stress and strain are
proportional to each other .
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below this limit of proportioaniliity , for a particular material , stress divided by strain is a constant . this constant is called
the young modulus , E
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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 m2
- 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
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draw and label suitable apparatus required for measuring the Young Modulus of a material in the form of a long wire
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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
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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
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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
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the gradient of a stress strain graph gives the
young modulus , E
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the area under a stress strain graph gives the
strain energy (or energy stored) per unit volume
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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
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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
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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
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don't forget to convert any lengths to ... and areas to ... when working out the young modulus
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draw a typical stress strain graph
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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
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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
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draw a stress-strain 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
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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
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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 .
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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
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