Unit 1.2: Materials
What is Strokes law (viscous *DRAG* equation)
*F = 6πηrv* F is the viscous DRAG (N), *η (eta) is the viscosity of the fluid (Nsm-2 or Pa s)*, r is the radius of the object (m) and v is the speed the object is moving at (ms-1).
What are tensile and compressive forces and how do they apply to Hooke's law
*Tensile forces* create *tension* in a *stretched spring*. *Compressive forces* create *compression* in a *squashed spring*. Tensile or compressive forces in the spring *act in the opposite direction* to the tensile or compressive *forces stretching or squashing it.* *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*
How is tensile (or compressive) strain calculated
*Tensile strain*, e, is defined as the *change in length* (i.e. the extension), over *original length* of the material: *ε = ∆x/x* *Tensile strain has no units,* it is a *ratio* usually written as a number or % e.g. extending a 0.5 m wire by 0.02 m would produce a strain of (0.02 ÷ 0.5) × 100 = 4%.
What is the equation(s) for elastic strain energy
*∆Eel = 1/2Fx* since F = kx: *∆Eel = 1/2kx2* (because 1/2bh = the area of a triangle under a graph)
What is Hooke's law and equation?
the extension of a stretched wire, ∆x, is proportional to the change in load or force, ∆F: *∆F = k∆x* where k = stiffness constant (N/m) and ∆x is in meters
What is a elastic deformation
If a deformation is *elastic*, the material *returns to its original shape* once the *forces are removed* *(before the elastic limit)*
What is plastic deformation
If a deformation is plastic, the material is *permanently stretched* and will *not return to its original shape* *(above the elastic limit)*
What is the difference between tensile stress and strain and compressive stress and strain
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.* (negative change in extension)
CORE PRACTICAL 3: Determine the Young modulus of a material Procedure
1) The *test wire* should be *long and thin*. The longer and thinner the wire, the *more it extends* for the same force — this reduces the percentage uncertainty in your measurements. 2) First you need to find the *cross-sectional area of the wire*. Use a *micrometer* to measure the *diameter* of the wire in *several places* and take an *average*. By *assuming that the cross-section is circular*, you can use the *formula* for the area of a circle: *πr^2* 3) *Clamp* the *wire to the bench* (as shown in the diagram) so you can *hang weights off one end* of it. 4) Measure the *distance between the fixed end* of the wire and the *marker* using a *meter rule* - this is your *unstretched length*. 5) Then if you *increase the weight*, the *wire stretches and the marker moves*. 6) *Increase the weight in equal intervals* (e.g. 1N), *recording the marker reading each time* — the *extension = new length - original length*. Because you *can't take repeat readings* (the wire might snap or be permanently stretched), you should take *more readings than usual*. 7) You can use your *results* from this experiment to *calculate the stress and strain* of the wire and plot a *stress-strain curve*.
How does elastic deformation work (in terms of atoms)
1) When the material is put under *tension*, the *atoms* of the material are *pulled apart* . 2) Atoms can *move slightly* relative to their *equilibrium positions*, without changing position in the material. 3) Once the *load is removed*, the *atoms return* to their *equilibrium distance* apart
What do different stress - strain graphs show
- *Stiff materials* have a *large Young's modulus (high gradient)* (difficult to stretch) - *Weak materials snap at a low stress level* - A material can be both stiff and weak - *Strong materials snap at high stress levels*
How can a stress - strain graph be presented differently in an exam
- The strain axis can be written as a % to find the gradient, you will have to convert the % to a number between 0 and 1 e.g. 0.3% = 0.003 (divide by 100) - The stress axis may be in MPa, you have to convert to Pa for calculations (x 10^6)
How does plastic deformation work (in terms of atoms)
1) Some *atoms* in the material *move position* relative to one another. 2) When the *load is removed*, the *atoms don't return to their original positions.*
When do materials stop obeying Hooke's law (limit of proportionality and elastic limit)
1) The *first part* of the graph (up to point P) shows *Hooke's law being obeyed* — there's a *straight-line* relationship between force and extension. 2) When the *force* becomes *great enough*, the graph starts to *curve*. Metals generally *obey Hooke's law up to the limit of proportionality, P.* 3) The point marked E on the graph is called the *elastic limit*. If you *exceed the elastic limit*, the material will be *permanently stretched*. When all the *force is removed*, the material will be *longer than at the start*
What is Young's modulus and the equation
A measure of *stiffness*. It is used by engineers to make sure the materials they are using can withstand sufficient forces. . It is calculated by: *Young modulus, E = stress/strain = σ/ε* The *unit of Young's modulus, E, is Pa or Nm-2* (the same as stress as strain has no unit)
What is breaking stress and ultimate tensile stress (UTS)
As you increase the force on a wire, the *tensile stress increases.* This *pulls the atoms apart* until the stress is so great it *pulls apart the atoms completely* and the *MATERIAL BREAKS* - this stress is called the *BREAKING SRESS.* The *ultimate tensile stress (UTS)* is the *MAXIMUM STRESS* a material can withstand Both UTS and B depend on *temperature*
What is the shape and sections of a stress - strain graph (6)
Before a, the material obeys Hooke's Law and the *gradient is constant and = Young's modulus* a - *limit of proportionality*, *after* this point the material *stops obeying Hooke's law* but still *returns to its original shape (still elastic)* b - Elastic limit - material *behaves plastically and no longer returns to its original shape* c - Yield point The peak of the graph shows the UTS (maximum stress the material can withstand) d - Fracture/Breaking point - the material snaps at this point. *Different materials* have different *breaking points*
What is the equation for density
Density (kg/m^3) = mass (kg) /volume (m^3) ρ = m/V
How is elastic strain energy found from a graph
Elastic Strain Energy (J) is the Area under a Force-Extension Graph This is because *work needs to be done* to stretch a material which is stored as *elastic strain energy* If the force-extension graph is *non-linear*, you'll need to *estimate the area by counting squares* or dividing the curve into *trapeziums*.
How does the breaking stress and ultimate tensile strength help engineers
Engineers have to *consider the UTS and breaking stress of material*s when *designing a structure* — e.g. they need to make sure the stress on a material won't reach the UTS when the conditions change. For example a rope climbing manufacturer needs to be able to ensure customers their ropes can hold them without snapping
What is laminar flow and when does it occur
Laminar flow is a flow pattern where all the parts of the fluid are flowing in the *same direction* — the *layers* in the fluid *do not mix*. (*smooth parallel layers of fluid*) Laminar flow usually occurs when a fluid is *flowing slowly or when an object is moving slowly through a fluid*
Example: Elastic strain energy A metal wire is 55.0 cm long. A force of 550 N is applied to the wire, and the wire stretches. The length of the stretched wire is 56.5 cm. Calculate the elastic strain energy stored in the wire.
Remember to convert cm to m 55 cm = 0.55 m E el = 1/2 x 550 x (0.565-0.55) E el = 4.125 = 4.1 J (2 s.f.)
How can submarines use the upthrust principle to sink/float
Submarines make use of upthrust to dive underwater and return to the surface. To *sink*, *large tanks are filled with water* to *increase the weight* of the submarine so that it *exceeds the upthrust*. To *rise* to the surface, the tanks are filled with *compressed air* to *reduce the weight* so that it's *less than the upthrust*
How is tensile (or compressive) stress calculated
Tensile stress, s, is defined as the force applied, F over the cross-sectional area, A: *σ = F/A* where σ is measured in N/m2 or Pa
What is upthrust and what is it equal to
The *upwards force* exerted by a fluid on a *fully or partially submerged* object It's caused because the top and bottom of a submerged object are at different depths (so experience different pressures - (p = ρhg)) upthrust (N) = weight of fluid displaced (N)
What is turbulent flow and when does it occur
Turbulent flow is a different flow pattern. You don't get nice layers like you do with laminar flow, *all the parts of the fluid get mixed up* Turbulent flow usually occurs when a *fluid is flowing quickly*, or an *object is moving quickly* through a fluid.
What does viscosity depend on (gases and liquids)
Viscous drag depends on the *viscosity* (or "thickness") of a *fluid*, η. Viscosity is *temperature-dependent* — liquids get *less viscous as temperature increases*, but *gases* (like dry air) get *more viscous as temperature increases*.
What is viscous drag
When an object *moves* through a *fluid*, or a fluid moves past an object, you get *friction between the surface of the object and the fluid*. This is viscous drag.
What is the yield point of the stress-strain curve?
When he *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 *atoms suddenly change place* during the yield point to *reduce stress*
How are stress and strain related?
When you apply a load on a material, there is *tensile stress and strain* Up to the *limit of proportionality,* *stress and strain are proportional to each other*
What is hysteresis?
Where the extension under a certain load would be different to past loads at that extension. e.g. *elastic bands do not obey Hooke's law* and show hysteresis when *loading and unloading* weights
How can you use your results from the experiment determine the Young modulus of a material to calculate stress and strain
You can calculate stress with stress = F/A and strain with strain = ∆x/x. *Then plot a graph of stress against strain* *The gradient of the graph = the young's modulus* The area under the graph = *strain energy (or energy stored) per unit volume*, i.e. the energy / m3 of wire
How can you calculate the viscosity from the practical results
You can then calculate the *viscosity, η,* of the liquid. The ball bearings are falling at *terminal velocity*, so the sum of the *forces acting on the ball is zero*: *viscosity η = 2r^2(ρ - σ)𝑔/(9𝑣)* where r is the radius of the ball bearing, g is the gravitational field strength due to gravity, *ρ is the density of the ball bearing, σ is the density of the liquid* and v is the average terminal velocity between the elastic bands. • *Average the viscosity* values to find *mean viscosity*
What does strokes law apply for (3)
applies only to: - *small spherical objects* - moving at *low speeds* - with *laminar flow* (or no turbulent flow)
What is the safety in the practical: use a falling-ball method to determine the viscosity of a liquid (2)
• *Spilled liquid* can make it easier to *slip* on floors so *mop up any spills* • Wear *goggles* to *avoid eye splashes*
CORE PRACTICAL 3: Determine the Young modulus of a material Safety
• *Wire snaps* and can recoil due to large amount of energy stored due to extension - wear *safety glasses* • *Do not stand directly under the masses* (can fall)
CORE PRACTICAL 2: Use a falling-ball method to determine the viscosity of a liquid (METHOD)
• *Zero a mass balance* with a *250cm3 measuring cylinder* on top • Pour *washing up liquid* up to the 200cm3 mark, record the mass and determine the density of the liquid using: D = m/V • Measure the *mass of the ball bearing* using a top pan balance. Measure *diameter* of the ball *bearing* with *micrometer* (at several positions and find *average*). *Halve the diameter to get radius*. Find volume of ball: 𝑉 = 4/3 π𝑟3 • Calculate *density of the ball bearing* (D = m/V) • Place 3 *elastic bands* along the measuring cylinder *10cm apart,* measured with a *ruler* the *highest* should be high enough to ensure the ball is travelling at *terminal velocity when reaching it* • *Drop the ball into the cylinder* (use *forceps* to hold it securely) and *start the stopwatch* when the *ball passes the top* of the *1st elastic band*, *lap* when the *bottom of the ball* just *passes each of the other rubber bands* • *Record* these *2 times (t1 and t2)* • *Repeat 3 more times* with the same radius sphere to *reduce the effect of random error*. Then *repeat* procedure for *ball bearing with different radii* • You can use a *magnet* to *remove bearings*. • For each radius, *find average t1 and t2*, *calculate velocities, v1 and v2* and *average* to find *Vavg for each radius*
What are some evaluation points for the practical: use a falling-ball method to determine the viscosity of a liquid
• Keep *temp* roughly the *same* as it may affect the viscosity of the liquid • Ensure the lap timer is hit for *constant parts of the ball* (i.e. at the bottom of the ball) • *Larger distance* between elastic bands will *lower percentage uncertainty*, but there will still be a high *uncertainty* in time due to *human reaction time*. • *Light gates* and data loggers can be used to *eliminate uncertainty* due to reaction time • *Strong magnet* could be used to *remove ball bearings* from the tube • If *ball falls close to wall*, *repeat reading* since the flow will *no longer be laminar* • If *velocity* at second band *higher* than first band, ball bearing might not have reached terminal velocity when you started timing, so *move bands further down tube and try again*
CORE PRACTICAL 3: Determine the Young modulus of a material Evaluation
• Use a *large distance* between the paper tape at the start, to *reduce uncertainty* • *Wait for necking to finish before taking final length measurements* • *Area* of the wire may not be constant so take *several measures* and find *mean* • For more *precise reading*, use *smaller masses (0.5N)* • *Small extension hard to measure accurately*; gives *large percentage uncertainty* • use a reference marker to *avoid parallax error* when measuring extension;
