DENISTY / HOOKE’S LAW / STRESS & STRAIN / YOUNG · PDF file12 What is the equation for Hooke’s Law? • Force ... line relationship between load and extension and it goes....

PHYSICS ASUNIT 2SECTION 2: MATERIALS

DENISTY / HOOKE’S LAW / STRESS & STRAIN / YOUNG MODULUS / STRESS CURVES / BRITTLE MATERIALS

# Question DENSITY

1 What is density? • A measure of compactness of a substance • It is a property of all materials, different materials have different densities 2 What does density relate? • Relates the mass of a substance with its volume 3 Define density • The density of a substance is defined as its mass per unit volume 4 What is the equation for density? • 𝜌 = 𝑚 𝑣 5 What are the units of density? • Kgm-3 6 What does density depend on? • What the object is made of • The material of an object 7 What doesn’t density depend on? • Doesn’t depend on its size or shape 8 What does the average density of an object determine? • Whether an object floats or sinks 9 What makes something float? • An object will float on a fluid if it has a lower density than the fluid 9.1 How would you measure the density of a regular solid? • Measure its mass using a top pan balance • Measure its dimensions using vernier calipers or a micrometer and calculate its volume using

the appropriate formula • Calculate its density from mass/volume 9.2 How would you measure the density of a liquid? • Measure the mass of an empty measuring cylinder • Pour some of the liquid into the measuring cylinder and measure the volume of the liquid

directly • Use as much liquid as possible to reduce the percentage error in your measurement • Measure the mass of the cylinder and liquid to enable the mass of the liquid to be calculated • Calculate the density from mass/volume 9.3 How would you measure the density of an irregular solid? • Measure the mass of the object • Immerse the object on a thread in liquid in a measuring cylinder • Observe the increase in liquid level, this is the volume of the object • Calculate the density of the object from its mass/volume 10 How would you measure the mass of an alloy? • An alloy is a mixture of two or more metals • For example brass is an alloy of copper and zinc and has good corrosion resistance • If the volume of the metal 𝐴 = 𝑉!, then the mass of metal 𝐴 = 𝜌!𝑉!

• If the volume of metal 𝐵 = 𝑉!, then the mass of metal 𝐵 = 𝜌!𝑉! • Therefore the mass of the alloy = 𝑚 = 𝜌!𝑉! + 𝜌!𝑉! 10.1 How would you measure the density of an alloy? • Hence, the density of the alloy, 𝜌 = !

!= !!!!!!!!!

!= !!!!

!+ !!!!

!

SPRINGS 11 What is Hooke’s Law? • 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 12 What is the equation for Hooke’s Law? • Force 𝐹 = 𝑘∆𝐿 13 What does each symbol represent in Hooke’s Law equation? • 𝐹 is the force, in Newtons • 𝑘 is the spring constant (sometimes referred to as the stiffness constant) (Nm-1) • ∆𝐿 is the extension of the spring from its natural length 14 What would a graph for Hooke’s Law look like?

15 How would you find the extension of springs in parallel? • The extension ∆𝐿 of each spring is the same 15.1 How would you find the spring constant of springs in parallel? • Therefore, the force needed to stretch spring one, P = 𝐹! = 𝑘!∆𝐿 • The force needed to stretch spring two, Q = 𝐹! = 𝑘!∆𝐿 • Since the weight W is supported by both springs, W = 𝐹! + 𝐹! = 𝑘!∆𝐿 + 𝑘!∆𝐿 = 𝑘∆𝐿 • Where 𝑘 = 𝑘! + 𝑘!

16 How would you find the extension of springs in series? • The tension in each spring is the same and is equal to the weight W • The extension of spring 𝑃,∆𝐿! =

𝑊𝑘𝑃

• The extension of spring 𝑄,∆𝐿! =𝑊𝑘𝑄

• Therefore the total extension ∆𝐿 = ∆𝐿! + ∆𝐿! =𝑊𝑘𝑃+ 𝑊𝑘𝑄= 𝑊

𝑘

How would you find the spring constant of springs in series? • If ∆𝐿 = ∆𝐿! + ∆𝐿! =

𝑊𝑘𝑃+ 𝑊𝑘𝑄= 𝑊

𝑘

• Then !!= !

!!+

!!!

17 What is extension or compression proportional to? • The extension/compression of a spring is proportional to the force applied to it 18 Draw a labeled graph of Load against extension for a spring

• What are the 4 features on the line?

• 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 the stiffness constant, k • When the load becomes great enough, the graph starts to curve

• The point marked E is the elastic limit • If you increase the load past the elastic limit, the material will be permanently stretched • When all of the force is removed, the material will be longer than at the start • Point P is the limit of proportionality 19 What is the limit of proportionality?

• What is it also known as? • This is a point very near the elastic limit • It is the point beyond which the force is no longer proportional to extension • The limit of proportionality is also described as the point beyond which the load-extension graph

is no longer linear (straight) 20 What is the elastic limit? • The point beyond which a wire/material is permanently stretched 21 What energy is stored in a stretched spring? • Elastic potential energy is stored in a stretched spring 22 What happens to the energy stored in the spring if it’s suddenly released? • If the spring is suddenly released, the elastic energy stored in it is suddenly converted to kinetic

energy of the spring 23 What is the equation for the elastic potential energy stored in a stretched spring? • Elastic potential energy stored in a stretched spring, 𝐸! =

!!𝐹∆𝐿 = !

!𝑘∆𝐿!

23.1 How would you derive the energy stored in a stretched spring formula? • The work done on the spring is stored as elastic potential energy • Therefore, the elastic potential energy !

!𝐹∆𝐿

• Also, since 𝐹 = 𝑘∆𝐿, where 𝑘 is the spring constant, then 𝐸! =!!𝑘∆𝐿!

24 How would you find the energy stored in a stretched spring from a force vs extension graph?

• The area under the line is the energy stored !!𝐹∆𝐿

25 What is the plastic region? • If a deformation is plastic, the material is permanently stretched (after force has been removed) • 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 26 How would you investigate the extension of a spring under load? • The object under test should be supported at the top e.g. using a clamp • A measurement of the objects original length should be 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 trail investigation if you can to test the range and size of the weights needed • You want to be able to add the same size and weight each time and add a large number of

weights before the point the object breaks to get a good picture of how the extension of the object varies with the force applied

• 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, then using: • Extension = new length – original length • Finally, plot a graph of load against extension to show the results

27 What is a loading/unloading curve? • As you load forces onto a material, it will be stretch according to Hooke’s Law • This will create a loading curve • When you remove the masses, the force on a material is reduced and the material either

returns to its original length of will stay deformed • This is an unloading curve 28 When is a loading curve the same as an unloading curve? • For a metal wire, its unloading curve and the loading curve are the same straight line • Provided its elastic limit is not exceeded, the wire, when unloaded, returns to the same length

as it had before it was loaded

29 When is a loading curve the different to its unloading curve? • If the wire is stretched beyond its elastic limit, the unloading line is parallel to the loading line • In this case, when it is completely un-stretched, the wire is slightly longer so it has a permanent

extension

30 What is a loading/unloading curve for an elastic band (rubber)? • For a rubber band, the change of length during unloading for a given change in tension is

greater than during loading • The rubber band returns to the same un-stretched length, but the unloading curve is below the

loading curve • Except at zero extension and maximum extension, where it is the same as the loading curve • The rubber band remains elastic as is regains its initial length, but has a lower limit of

proportionality

31 what is a loading/unloading curve for polythene? • For a polythene strip, the extension during unloading is also greater than during loading • However, the strip does not return to the same initial length when it is completely unloaded • The polythene strip has a low limit of proportionality and suffers plastic deformation

32 What is strain energy? • The area under a line of force against extension graph is equal to the work done to stretch a

wire • The work done to deform an object is referred to as the strain energy 33 What is the strain energy for a metal wire (or string)? • Provided the limit of proportionality is not exceeded, to stretch a wire to an extension of ∆𝐿, the

work done =!!𝑇∆𝐿

• Where T is the tension in the wire at this extension • Because the elastic limit is not reached, the work done is stored as elastic energy in the wire • Therefore, the elastic energy stored in a stretched wire =!

!𝑇∆𝐿

• Because the graph of tension against extension is the same for unloading as for loading, all the energy stored in the wire can be recovered when the wire is unloaded

34 What is the strain energy for a rubber band? • The work done to stretch the rubber band is the area under the blue curve • The work done by the rubber band when it’s unloaded is represented under the red curve • The area between the two curves therefore represents the difference between energy stored in

the rubber band when it is stretched and the useful energy recovered from it when it is un-stretched

• The difference occurs because some of the energy stored in the rubber band becomes internal energy of the molecules when the rubber band un-stretched

35 What is the strain energy for polythene? • As it does not regain its initial length, the area between the loading and unloading curve

represents work done to deform the material permanently, as well as internal energy retained by the polythene when it un-stretches

STRESS AND STRAIN 36 What is tensile stress? • Defined as the force applied, F, divided by the cross sectional area, A • 𝑠𝑡𝑒𝑠𝑠 = !

!

• Units are Nm-2 or Pascal’s, Pa

37 What is tensile strain? • A stress causes a strain • Tensile strain is defined as the change in length i.e. the extension, divided by the original length

of the material • 𝑠𝑡𝑟𝑎𝑖𝑛 = ∆!

!

• Strain has no units because it’s a ratio

38 What is the difference between compressive and tensile forces? • Tensile will stretch a material and are thought of as positive • Compressive forces will squash the material and are thought of as negative 39 How can you measure stress and strain?

• Using Searle’s apparatus

• We actually use two wires of equal lengths attached to a rigid support. • Although the support is rigid it to can 'give' slightly under the forces applied. • This can affect results. • By using two wires, spurious strain can be eliminated from the measurements. • One wire acts as a control wire. • We can accurately measure extension of the other (test) wire. • Both control and test wires are attached to the other ends by a horizontal bar supporting a spirit

level. • The bar is hinged to the control wire so that when the test wire is extended due to the addition

of weights on the side of the test wire, the spirit level is tilted by a small amount. • We can remove any tilt of the spirit level and restore it to the horizontal position by turning the

screw of a micrometer, which is positioned on the test wire side and making the bar mounted spirit level travel in the desired direction.

• Step 1: Attach equal weights both wires to make them equally taut. • Step 2: Measure the initial length of the wire several times to obtain the average value of lo • Step 3: Measure the diameter of the wire at several points along the wire and the average value

of the diameter (d) and then calculate the circular cross-sectional area (!!𝜋𝑑!)

• Step 4:Adjust the spirit level so that it is in the horizontal position by turning the micrometer. Record the micrometer reading to use it as the reference reading.

• Step 5:Load the test wire with a further weight. Wait while the wire is being stretched to the equilibrium position and the spirit level is maximally tilted.

• Step 6: Adjust the micrometer screw to restore the spirit level into the horizontal position. • Step 7: Subtract the first micrometer reading from the second micrometer reading to obtain the

extension (e) of the test wire. • Step 8: Calculate stress and strain from the formulae 𝑠𝑡𝑒𝑠𝑠 = !

! and 𝑠𝑡𝑟𝑎𝑖𝑛 = ∆!

!

• Step 9: Repeat steps 4,5,6 to obtain more values of stresses and strains • Step 10: Plot the above values on stress strain graph; it should be a straight line. Determine the

value of the gradient to find the Young Modulus and the area under the curve is the strain energy per unit volume

Safety points • It is possible that a wire under tension can snap suddenly and damage eyes. • Wear safety glasses. • It is also possible that weights attached to the wires could fall down and land on your feet or

other part of the body. • So be careful and wear closed toed shoes 40 What would a typical graph of stress against strain look like?

Label points A to F A) Limit of proportionality B) Elastic limit C) Yield point D) Y2 E) Ultimate tensile stress F) Breaking point

41 What is the Young Modulus? • The value of stress/strain • 𝑌𝑜𝑢𝑛𝑔 𝑀𝑜𝑑𝑢𝑙𝑢𝑠,𝐸 = 𝑠𝑡𝑟𝑒𝑠𝑠, 𝜎𝑠𝑡𝑟𝑎𝑖𝑛, 𝜀 =

𝑇𝐿𝐴∆𝐿

• The units for Young Modulus are Nm-1 or Pascal’s, the same as stress because strain has no units

42 When is the Young Modulus valid for a material? • 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 (A), the stress and strain of a material are

proportional to each other • So below this limit for a particular material, stress divided by strain is a constant • This is called the Young Modulus 43 What are the elastic limit and the yield point? • Beyond A, the limit of proportionality, the line curves and continues beyond the elastic limit, B,

to the yield point, C • This is where the wire weakens temporarily • The elastic limit is the point beyond which the wire is permanently stretched and suffers plastic

deformation 44 What happens beyond point D on the stress/strain graph? • Beyond D, a small increase in the stress causes a large increase in strain as the material of the

wire undergoes plastic flow • Beyond maximum stress, the ultimate tensile stress (UTS), the wire loses its strength, extends

and becomes narrower at its weakest point • Increase of stress occurs due to the reduced area of cross-section at this point until the wire

breaks at point F • The ultimate tensile stress is sometimes called the breaking stress 45 How can you compare the stiffness of different materials? • The stiffness of different materials can be compared using the gradient of the stress-strain line

which is equal to the Young Modulus of the material • The higher the Young Modulus (steeper the gradient) the stiffer the material 46 How is the strength of a material found from a stress/strain graph? • The strength of a material is its ultimate tensile strength (UTS) which is its maximum stress 47 What is the area under a stress/strain line? • The strain energy per unit volume 48 What is a ductile material?

• A material that can be drawn into a long thin wire • This would be represented by a shallow line that will be put under a lot of strain for relatively

small stresses 49 What is a brittle material? • If you apply a force to a brittle material, it won’t deform plastically, but it will suddenly snap when

the stress gets to a certain size • Brittle materials can also be quite weak if they have cracks in them 50 What would a stress/strain graph look like for a brittle material?

• No plastic deformation 51 What is the molecular structure of brittle materials (ceramics)? • Ceramics are made by melting certain materials and then letting them cool • The arrangements of atoms in a ceramic can be crystalline or polycrystalline – where there are

many regions (or grains) of crystalline structure • The atoms in each grain line up in a different direction • However they’re arranged, the atoms in a ceramic are bonded in a giant rigid structure • The strong bonds between the atoms make them stiff, while the rigid structure means that

ceramic are very brittle • When stress is applied to a brittle material, any tiny cracks at the material’s surface get bigger

and bigger until the material breaks completely • This is called a brittle fracture • The cracks in brittle materials are able to grown because these materials have a rigid structure • Other materials, like most metals, are not brittle because the atoms within them can move to

prevent any cracks getting bigger