chapter 5- materials Flashcards
What is density defined as?
A substance’s mass per unit volume
the equation for density is found on the data sheet. what do the symbols stand for?
ρ = m/v
ρ = m/v
ρ = density
m=mass (kg)
v= volume (m cubed)
Method for finding density of a regular solid?
- ruler - measure width, length and height
- top-pan balance - measure mass
- use ρ = m/v
Method for finding density of an irregular solid?
- top-pan balance - measure mass
- use a eureka can to measure water displaced by object
- volume of water = volume of object
- use ρ = m/v
What type of force is acting when a spring is squashed?
Compressive
What type of force is acting when a spring is lengthened?
Tensile
What is Hooke’s law?
The force needed to stretch a string is proportional to the extension of the spring from its natural length, provided the elastic limit isn’t exceeded
the equation for Hooke’s law is found on the data sheet. what do the symbols stand for?
F = kΔL
F = kΔL F= force (N) k= spring constant ( m per N) ΔL = change in length
What does it mean for the stiffness of the spring when spring constant is larger?
It is a stiffer spring
What happens to a spring when it is stretched beyond its elastic limit?
It doesn’t regain its initial length when the force applied is removed
How can the spring constant of springs ‘in parallel’ be calculated?
Multiply the spring constants together of the original springs
How can the spring constant of two springs ‘in series’ be calculated?
1/kₜₒₜₐₗ = 1/kₑₐ𝒸ₕ x number of springs
What is the area underneath a force-extension graph equal to?
Strain energy (as work done = force x distance)
One of the equations for strain energy is foudn on the data sheet. what do the symbols stand for? what is the other equation?
E=1/2 F△L
E=1/2 FΔL
E = 1/2kΔL²
E= energy joules F= force(N) k= spring constant ΔL= change in length
When does the equation Eₑ = 1/2kΔL² apply?
Only if the spring obeys Hooke’s law
In materials, what does the stiffness depend on?
Material, length and cross-sectional area
What is the definition of stress on a material?
The force acting per unit cross sectional area
The equation for tensile stress is found on the data sheet. what do the symbols stand for?
tensile stress= F/A
tensile stress= Force / cross sectional area (tensile stress= F/A)
What is stress measured in?
pascals (Pa)
What is the breaking stress of a material?
The maximum stress it can withstand without fracture
What can breaking stress also be referred to as?
Ultimate tensile stress
What happens when materials get a thinner section when they are stretched?
They break here as stress is increased here
What is strain defined as?
The extension produced per unit length
The equation for strain is found on the data sheet. what do the symbols stand for?
tensile strain= △L/L
tensile strain= Extension / length
What is strain measured in?
Has no units
How can the two separate equations for stress and strain be simplified?
- σ=F/A divided by ε=x/l
- F/A x L/ΔL = FL/AΔL
- F/ΔL is spring constant
- so E = kL/A
On a stress-strain graph showing a stiff and a flexible material, which material has the line with the steepest gradient?
The stiff material
What are materials that permanently deform described as?
Plastic
What are materials that return to their original shape after the stretching force is removed called?
Elastic
What two words can plastic materials also be described as?
- ductile - can be drawn into wires
* malleable - they can be hammered into sheets
Describe the force-extension graph of a metal wire.
- loading - the line starts straight, and curves as it surpasses the limit of elasticity
- unloading - the line doesn’t come back along the same line as when loading
- difference between loading and unloading lines = permanent extension of wire
Describe the force-extension graph of a rubber band.
- loading - the line is curved
- unloading - the line is curved, but doesn’t follow the same curve as the loading line
- unloading line finishes at the origin - rubber returns to its original shape
What is the opposite of a tough material?
A brittle material
What happens when you try to deform a malleable material e.g. lead?
It deforms plastically - gives way gradually, absorbing a lot of energy before it snaps
Do brittle materials deform plastically?
No
Do brittle materials absorb much energy before they break?
No, unlike plastic materials
What are force-extension graphs used for, vs stress-strain graphs?
- force-extension → usually for objects e.g. particular spring
- stress-strain → usually for materials (of any size)
What does the gradient represent on force-extension and stress-strain graphs?
- force-extension → spring constant (Nm⁻¹)
* stress-strain → Young Modulus, E (Nm⁻² or Pa)
What is the Young Modulus measured in?
Nm⁻² or Pa
What does the area represent on force-extension and stress-strain graphs?
- force-extension → work done (1/2kΔL²) in J
* stress-strain → work done per unit length (W/V) in Jm⁻³
What is work done per unit length measured in?
Jm⁻³
On a force-extension graph, what does it mean if the area of the unloading graph is smaller than that of the loading graph?
Some energy has been transferred
What is the reason for energy transference on a force-extension graph?
Some energy stored in the object (e.g. rubber band) becomes the internal energy of the molecules when the rubber band unstretches
On a force-extension graph, what does the area between the loading and unloading curve represent?
Difference between energy stored in the object when it is stretched and the useful energy recovered from it when it is unstretched
Brief explanation of experiment to find the Young Modulus of a wire?
- stress → wire with mass attached - measure mass using top-pan balance and use W=mg. measure diameter of wire using micrometer, then calculate area
- then stress = F/A
- strain → measure extension by measuring distance marker moves from original position, and length of wire. calculate strain
- vary mass for range of values - plot stress-strain graph
How to improve accuracy in the experiment to calculate the Young Modulus of a wire?
- use long thin wire and heavier weights → greater Δl so smaller % uncertainty
- measure diameter accurately using micrometer
- measure wire by holding ruler as close to the wire as possible
In an experiment to calculate the Young Modulus of a wire, how can kinks in the wire be avoided?
Weights are added at the beginning, before length measured
In an experiment to calculate the Young Modulus of a wire, how can we make sure there is no thermal expansion?
By comparing the test wire to a control wire
What is the elastic limit?
The point beyond which a wire will not return to its original length after weight has been added and then remove
What is density?
The mass of a material per unit volume.
What is the equation for density?
Density (kg/m³) = Mass (kg) / Volume (m³)
p = m / v
What is the symbol for density?
ρ - rho (looks like a ‘p’)
What are the units for density?
g/cm³ or kg/m³
Convert 1 g/cm³ to kg/m³.
1 g/cm³ = 1000 kg/m³
Is density affected by size or shape?
No, just the material.
What determines whether a material floats?
- The relative average densities.
* If a solid has a lower density than a fluid, it will float in the fluid
What is the density of water?
1 g/cm³ (which is 1000 kg/m³)
What is Hooke’s law?
- The extension of a stretched object (Δl) is proportional to the load (F) until the limit of proportionality.
- F = k x Δl
What is the equation for Hooke’s law?
Force (N) = Stiffness constant (N/m) x Extension (m)
F = k x Δl
What are the units for the spring constant, k?
N/m
What is k?
- The stiffness constant for a material being stretched
* With springs, it is usually called the spring constant
Describe the forces acting on a metal wire being stretched by a load.
- Load pulls down on the end of the wire with force F
- Support pulls up on the top of the wire with an equal reaction force R
- F = R
Does Hooke’s law only work for tensile forces?
No, it also works for compressive forces.
What things obey Hooke’s law?
• Springs
• Metal wires
• Most other materials
(Up to a point!)
What types of forces does Hooke’s law work for?
- Tensile (stretching)
* Compressive
Does Hooke’s law involve just one force?
- No, there must be two equal and opposite forces at the ends of the object.
- They can be tensile of compressive.
Is the value of k the same for tensile force as it is for compressive forces?
And it what materials?
- In springs - the same.
* In other materials (and some springs) - not always because some can’t compress
A material will only deform (stretch, bend, twist, etc.) there are …… acting on it
…there’s a pair of opposite forces acting on it.
Describe the forces acting on a fixed spring that has a compressive force acting on the base.
- The compressive force, F, pushes up onto the spring
- The support exerts an equal and opposite reaction force, R, down onto the spring
- F = R
How is Hooke’s law illustrated on a graph?
- Graph of force (y) against extension (x)
* Gradient of straight part is the value of k
When does Hooke’s law not work?
It stops working when the force is great enough (the limit of proportionality).
Why is a force-extension graph plotted with extension on the x axis?
So that the gradient gives k.
What is the limit of proportionality on a force-extension graph?
- The point at which the line starts to curve
* Hooke’s law works up to this point
How do you investigate extension?
Support the object being tested (the spring) from above with a clamp.
Measure it’s original length with a ruler - use a set square to make sure the ruler is parallel with the stand.
Add weights one at a time to the end of the object (the spring).
After each weight is added find new length then do: New length - original length = extension.
Make sure you can add a good number of weights before the object breaks to get a good picture of force-extension.
Plot a graph of load-extension.
How is Hooke’s law illustrated on a graph?
- Graph of force (y) against extension (x)
* Gradient of straight part is the value of k
How can you tell this material is elastic
It returns to its original length after extension (0,0)
What do X and Y represent?
What is tensile stress?
The force applied per unit cross-sectional area of a material.
What is tensile strain?
The change in length of a material divided by the original length when stretching.
How is the stretching of a material illustrated on a graph?
- Graph of stress (y) against strain (x)
* Gradient is straight part in the Young modulus
On a stress-strain graph, where is the breaking stress?
At the end of the line.
Why is the work done is stretching a material equal to 1/2 x F x d, even though W = F x d?
What happens when two springs in parallel share the load?
The force on each spring is shared.
(If the spring constant is the same for each spring the force on each spring is halved.)
The extension on each spring is shared.
(If the spring constant is the same for each spring the extension on each spring is halved.)
Overall the spring constant (for both springs combined) will increase.
(If the spring constant is the same for each spring the overall spring constant increases.)
What happens when two springs experience a force in series?
Both springs experience the same force.
Therefore they will both extend by the same amount.
Overall they will extend more.
If both springs are the same they will have an overall extension of 2x. x is extension of 1 wire.
Extends more so spring constant decreases
What is the equation for total spring constant in series?
1/k(T) = 1/k(1) + 1/k(2)
What happens if you have two identical springs in parallel and 1 spring below in series experiencing a force?
Group the top springs together and think of them as 1 big spring : kT = k1 + k2 = 2k.
There are now two springs in series: use the equation:
Describe an experiment to find the Young modulus of a wire.
1) Measure the diameter of a thin wire using a micrometer in several places and take an average.
2) Find the cross-sectional area of the wire using “A = πr²”.
3) Clamp the wire with a clamp at one end and over pulley at the other end, so that weights can be hung on the wire.
4) Align a ruler with the wire and attach a marker.
5) Start with the smallest weight to straighten the wire (but ignore this weight in calculations).
6) Measure the unstretched length of the wire from clamped end of the string to the marker.
7) Add 100g weights to the string and measure the extension.
8) Plot a stress (y) against strain (x) graph of your results. The gradient of the straight part is the Young modulus.
On a stress-strain graph, what does the area under the straight part of the line represent?
- The strain energy stored per unit volume.
* i.e. The energy stored per 1m³ of wire
Describe a typical stress-strain graph for a DUCTILE material being stretched, with all the important points.
- Straight line up until the limit of proportionality.
- Curves towards the x-axis slightly until the elastic limit
- Curves more towards the x-axis until the yield point
- After yield point, the line goes down slightly
- There may be a second peak before the breaking stress
- The UTS is the highest stress reached, usually on the second peak
Describe the stress-strain graph for a brittle material.
- Straight line through origin.
* Reaches breaking point without curving.
Describe the Force - Extension graph for a brittle material
Similar to stress - strain
Describe the loading-unloading force-extension graph for a metal wire stretched beyond its elastic limit.
- The loading line curves towards the x-axis until unloading starts.
- The unloading line is parallel to the loading line and crosses the x-axis at a positive extension value.
What does a stress strain graph look like for brittle, ductile and polymeric materials?
Brittle: High UTS, low strain (doesn’t move much before it breaks)
Ductile: Curved after elastic limit (region of plastic deformation)
Polymeric: Small force (and stress) but long extension (and strain)
