Answer key · Unit 12

Materials and water.

Each lesson’s practice questions, their correct answers and the mistakes they spot. Answers read off the Scene use the same setting the learner sees.

Misconception catalogue

Answer key 3

Lesson 1 of 3 · About 25 minutes

Natural materials

How wood is stronger than steel per kilogram though it is made of sugar: strength along the grain, levels within levels, nacre that stops cracks, and strength for weight.

Open the lesson

The 5 steps and their answers

Step 1 of 5
Open this step
Strong along the grain
Practice · Calculate
Question
Raise Push until the spruce strip breaks. At what stress, on the stretched face, does it break?

Correct answer
75 MPa ± 3%
Setup, choices and feedback
The Scene at this step
Material
Woodfixed
Grain
Alongfixed
Push
3 mm
Zoom to the molecules
Sample
Accepted within
± 3%
Computed from the Scene at this step’s setting
Breaking stress for this material and grain/nodes/bench/outputs/strength
Any unit of the same quantity is accepted; it is converted before checking.
Feedback for a right answer
75 MPa, at a push of about 6.3 mm. That is like hanging 7.5 tonnes on every square centimetre, from a material made mostly of sugar chains.

Hint
The chart beside the Scene shows the stress at each push, and the stress at which the strip broke.

Worked solution
1. Raise the push gradually: the stress on the underside climbs.
2. At about 6.3 mm the strip breaks.
3. The chart shows it broke at 75 MPa, spruce’s strength along the grain.
Step 2 of 5
Open this step
Across the grain
Practice · Predict, then look
Question
At the same push, 3 mm, switch Grain to Across. What happens to the force needed to bend the wood?

Correct answer
- Correct It falls a lot, to about a fifteenth
Setup, choices and feedback
The Scene at this step
Material
Woodfixed
Grain
Alongfixed
Push
3 mmfixed
Zoom to the molecules
Sample
Choices
- Correct It falls a lot, to about a fifteenth
- Wrong It stays the same: it is the same wood
  Misconception: Thinks a material’s strength is fixed by what it is made of, whatever way it is arranged or loaded.
  Feedback from the catalogue
  The same material can be strong one way and weak another. In wood, along the grain the pull falls on the cell walls, made of cellulose, and across it only on the glue between them: it takes fifteen times less force. Natural materials build strength by arranging weak ingredients well.
- Wrong It grows: across the grain there is more glue holding on
  Feedback
  The glue, lignin, is much weaker than the cell walls. Across the grain the pull falls on it, not on the cellulose.
Once the prediction is locked in, the Scene changes
Grain
Acrossfixed
The Scene then shows: 1.23 N · Force on the nose; 0 once broken/nodes/bench/outputs/force
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
The force drops from about 18 N to about 1.2 N. Across the grain the wood also breaks at a twentieth of the stress, 4 MPa. A tree spends no material on strength in a direction that carries no load.

Worked solution
1. Along the grain the pull is carried by the cell walls, made of stiff cellulose.
2. Across the grain it only stretches the lignin between the cells, and the walls crosswise.
3. The same material, arranged differently: the force is about fifteen times smaller.
Step 3 of 5
Open this step
Levels within levels
Practice · Choose
Question
Bone holds tiny crystals of mineral and fibres of protein, collagen. Which does what?

Correct answer
- Correct The stiff crystals carry the load, and the soft protein passes it from crystal to crystal and stops cracks
Setup, choices and feedback
The Scene at this step
Material
Bone
Grain
Along
Push
3 mm
Zoom to the molecules
Nano
Choices
Shown in a random order.
- Correct The stiff crystals carry the load, and the soft protein passes it from crystal to crystal and stops cracks
- Wrong The protein carries all the load, and the mineral only fills space
  Feedback
  Protein on its own is too soft: a bone with its mineral removed bends like rubber. The crystals give it its stiffness.
- Wrong The mineral does everything, and the protein just sticks it together
  Feedback
  Mineral on its own is brittle, like chalk. Without the protein to pass the load and spread cracks, one crack would run through the whole bone.
Feedback for a right answer
Every natural material is built this way: the stiff part carries, the soft part passes on and absorbs. Bone’s crystals are so thin, a few nanometres, that they hold almost no flaws, so they are nearly as strong as a perfect crystal.

Worked solution
1. The crystals are stiff but brittle.
2. The protein is soft, but stretches without breaking.
3. In the hierarchy each small crystal carries part of the load, and the protein passes it on and stops cracks at the borders between the parts.
Step 4 of 5
Open this step
Stopping cracks
Practice · Choose
Question
Nacre is 95% mineral, yet it takes roughly a thousand times more energy to break than a plain crystal of the same mineral. Why?

Correct answer
- Correct The crack cannot cut through the bricks: it zigzags round them, pulling them out of the mortar and stretching strands of protein
Setup, choices and feedback
The Scene at this step
Material
Nacrefixed
Grain
Along
Push
10 mm
Zoom to the molecules
Micro
Choices
Shown in a random order.
- Correct The crack cannot cut through the bricks: it zigzags round them, pulling them out of the mortar and stretching strands of protein
- Wrong Its bricks are stronger than any other crystal
  Misconception: Thinks a strong material is bound to be tough, hard to break once it has a crack.
  Feedback from the catalogue
  Strength is the stress a sound material can take; toughness is how hard it is to break once it has a crack. Glass is strong but not tough: a crack runs straight through it. Nacre is made of brittle bricks, yet it is a thousand times tougher than they are, because a crack has to zigzag round them, absorbing energy.
- Wrong It is soft, so it bends instead of breaking
  Feedback
  Nacre is nearly as stiff as mineral, and does not bend much. Its toughness comes from how the crack is held up, not from softness.
Feedback for a right answer
In the Scene the crack zigzags between the bricks. Every brick pulled out and every protein loop opened absorbs energy, and the crack has a far longer path to travel. That is how a brittle material becomes tough.

Worked solution
1. In a plain crystal a crack, once started, runs straight, breaking very few bonds on the way.
2. In nacre the crack meets a brick and has to go round it through the mortar.
3. At each detour bricks pull out and protein strands stretch, and all of that absorbs energy.
Strength and toughness are different properties: glass is strong but not tough.
Step 5 of 5
Open this step
Light as well as strong
Practice · Calculate
Question
Spruce breaks along the grain at 75 MPa, and its density is 0.45 g/cm³. What is its strength per kilogram, in kN·m/kg (megapascals per g/cm³)?

Correct answer
166.7 kN·m/kg ± 3%
Setup, choices and feedback
The Scene at this step
Material
Woodfixed
Grain
Alongfixed
Push
3 mm
Zoom to the molecules
Sample
Accepted within
± 3%
Computed from the Scene at this step’s setting
Strength per kilogram, strength ÷ density (1 kN·m/kg is 1 MPa per g/cm³)/nodes/bench/outputs/specificStrength
Any unit of the same quantity is accepted; it is converted before checking.
Wrong answers it recognises
- 75 kN·m/kg ± 3%
  Misconception: Compares materials by their strength alone, leaving out how much they weigh.
  Feedback from the catalogue
  For a bridge, a wing or a tree, what matters is how much strength each kilogram buys. Divide the strength by the density: steel is six times stronger than spruce but seventeen times heavier, so per kilogram the wood is three times stronger. Carbon fibre is 3.5 times stronger than steel, and 17 times per kilogram.
Feedback for a right answer
About 167 kN·m/kg, roughly three times mild steel’s 55. Steel is six times stronger than wood, but seventeen times heavier.

Hint
Divide the strength by the density.

Worked solution
1. Strength: 75 MPa; density: 0.45 g/cm³.
2. $\dfrac{75}{0.45} \approx 167$ ⁠.
3. Compared with mild steel: $\dfrac{430}{7.85} \approx 55$ ⁠, so the wood is about three times better.
If you answered $75$ ⁠, you took the strength alone. For a tree, a wing or a bridge, what matters is how much strength each kilogram buys.

Lesson 2 of 3 · About 25 minutes

Engineered materials

A tensile test on steel, aluminium, carbon fibre and a shape-memory alloy: Hooke's law, yielding and the stretch that stays, a brittle but strong material, specific strength, and an alloy that springs back.

Open the lesson

The 5 steps and their answers

Step 1 of 5
Open this step
Stress and strain
Practice · Calculate
Question
Mild steel has E = 200 GPa. What is the stress in the specimen when it is stretched by 0.1%?

Correct answer
200 MPa ± 3%
Setup, choices and feedback
The Scene at this step
Material
Mild steelfixed
Stretch
0.1 %fixed
Grips
Hold
Zoom inside
Specimen
Accepted within
± 3%
Computed from the Scene at this step’s setting
Engineering stress on the specimen now; 0 once let go or broken/nodes/test/outputs/stress
Any unit of the same quantity is accepted; it is converted before checking.
Feedback for a right answer
200 MPa: on the 40 mm² cross-section that is a pull of 8 kN, the weight of eight hundred kilograms, and the specimen has stretched by only a tenth of a per cent.

Hint
0.1% = 0.001, and 200 GPa = 200 000 MPa.

Worked solution
1. Strain: $\varepsilon = 0.1\% = 0.001$ ⁠.
2. Hooke’s law: $\sigma = E\varepsilon = 200\,000 \times 0.001$ ⁠ MPa.
3. $\sigma = 200$ ⁠ MPa, below the yield at 275 MPa, so the law still holds.
Step 2 of 5
Open this step
Past the yield
Practice · Predict, then look
Question
The steel has been stretched by 12%, far past its yield. What happens when the grips let go?

Correct answer
- Correct It springs back only a little, and stays almost 12% longer
Setup, choices and feedback
The Scene at this step
Material
Mild steelfixed
Stretch
12 %fixed
Grips
Holdfixed
Zoom inside
Specimen
Choices
- Correct It springs back only a little, and stays almost 12% longer
- Wrong It returns to its original length, like a spring
  Misconception: Expects a metal stretched past its yield to return to its length, like a spring.
  Feedback from the catalogue
  A metal springs back only by its elastic part, up to its yield strength. Beyond it, layers of atoms in the grains slip over each other and stay in their new places. Steel stretched by 12% comes back by only about 0.2%. Superelastic alloys such as NiTi are the exception: in them the crystal switches structure instead of slipping.
- Wrong It comes about halfway back
  Feedback
  The steel comes back only by the elastic part, the stress over E: about 0.2%. All the rest is grains that slipped, and that does not come back.
Once the prediction is locked in, the Scene changes
Grips
Let gofixed
The Scene then shows: 11.79 · Stretch the specimen keeps once let go, % of its gauge length/nodes/test/outputs/kept
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
The specimen stays about 11.8% longer. It comes back by only about 0.2%, the elastic part; the rest is layers of atoms that slipped in the grains and stayed in their new places.

Worked solution
1. At 12% the stress is about 412 MPa.
2. On release only the elastic part comes back: $\dfrac{412}{200\,000} \approx 0.002$ ⁠, that is 0.2%.
3. What stays: 12% − 0.2% ≈ 11.8%.
Past its yield a metal changes shape for good. That is how bars are bent and car panels pressed.
Step 3 of 5
Open this step
Brittle but strong
Practice · Choose
Question
Carbon fibre snaps all at once at about 1.1% stretch, and mild steel stretches 28% before it breaks. Which is stronger?

Correct answer
- Correct The carbon fibre: it takes over three times the stress
Setup, choices and feedback
The Scene at this step
Material
Carbon fibre
Stretch
1 %
Grips
Hold
Zoom inside
Specimen
Choices
Shown in a random order.
- Correct The carbon fibre: it takes over three times the stress
- Wrong The steel: it stretches much further before it breaks
  Misconception: Takes a brittle material, one that snaps without stretching, to be weak.
  Feedback from the catalogue
  Brittle means a material breaks without stretching first, not that it is weak. Carbon fibre snaps at 1.1% stretch, but takes 1500 MPa, 3.5 times mild steel’s strength. Its drawback is that there is no warning before it breaks, so it is designed with a wide safety margin.
- Wrong The same strength: both break in the end
  Feedback
  Strength is the highest stress the material can take, not the fact that it breaks. Compare the top of each curve.
Feedback for a right answer
Carbon fibre takes about 1500 MPa, and mild steel about 430 MPa. Carbon fibre is strong and stiff, but brittle: it never yields and gives no warning before it breaks.

Hint
Strength is the highest stress on the curve, not how far it reaches.

Worked solution
1. Tensile strength is the top of the stress–strain curve.
2. Carbon fibre: about 1500 MPa. Mild steel: about 430 MPa.
3. The carbon fibre is about 3.5 times stronger.
How far a material stretches before breaking is another property, ductility: steel is ductile, and carbon fibre brittle.
Step 4 of 5
Open this step
Strength for weight
Practice · Calculate
Question
Carbon fibre’s tensile strength is 1500 MPa, and its density 1.6 g/cm³. What is its specific strength, in kN·m/kg?

Correct answer
937.5 kN·m/kg ± 3%
Setup, choices and feedback
The Scene at this step
Material
Carbon fibrefixed
Stretch
1 %
Grips
Hold
Zoom inside
Specimen
Accepted within
± 3%
Computed from the Scene at this step’s setting
Tensile strength per unit density (1 kN·m/kg is 1 MPa per g/cm³)/nodes/test/outputs/specificStrength
Any unit of the same quantity is accepted; it is converted before checking.
Wrong answers it recognises
- 1500 kN·m/kg ± 3%
  Misconception: Compares materials by their strength alone, leaving out how much they weigh.
  Feedback from the catalogue
  For a bridge, a wing or a tree, what matters is how much strength each kilogram buys. Divide the strength by the density: steel is six times stronger than spruce but seventeen times heavier, so per kilogram the wood is three times stronger. Carbon fibre is 3.5 times stronger than steel, and 17 times per kilogram.
Feedback for a right answer
About 940 kN·m/kg, roughly seventeen times mild steel’s 55. That is why aircraft wings, racing bicycles and wind-turbine blades are made of carbon fibre.

Hint
Divide the tensile strength by the density.

Worked solution
1. Strength: 1500 MPa; density: 1.6 g/cm³.
2. $\dfrac{1500}{1.6} \approx 938$ ⁠.
3. Compared with mild steel, $\dfrac{430}{7.85} \approx 55$ ⁠: about 17 times.
If you answered $1500$ ⁠, you took the strength alone, without the weight.
Step 5 of 5
Open this step
Springing back
Practice · Predict, then look
Question
The NiTi specimen has been stretched by 5%, more than twenty times what steel can take without yielding. What happens when the grips let go?

Correct answer
- Correct It goes right back to its original length
Setup, choices and feedback
The Scene at this step
Material
NiTifixed
Stretch
5 %fixed
Grips
Holdfixed
Zoom inside
Inside
Choices
- Correct It goes right back to its original length
- Wrong It stays stretched, like steel that has yielded
  Feedback
  In steel, layers of atoms slipped, and nothing brings them back. In NiTi no atom has slipped: the crystals only switched to another structure, and switch back when the pull goes.
- Wrong It snaps
  Feedback
  Letting go takes the pull to zero, so nothing can break because of it.
Once the prediction is locked in, the Scene changes
Grips
Let gofixed
The Scene then shows: 0 · Stretch the specimen keeps once let go, % of its gauge length/nodes/test/outputs/kept
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
The specimen goes all the way back: 0% is kept. Every row of atoms switches back to the cubic structure. This is superelasticity, and it is why NiTi makes spectacle frames that bend and spring back, and stents that open inside blood vessels.

Worked solution
1. Up to about 6%, NiTi stretches by switching its crystal structure, with no atoms slipping.
2. At 5% it is still in that range.
3. When the pull goes, the crystals switch back to cubic, and the specimen to its original length.

Lesson 3 of 3 · About 25 minutes

Desalination

Why it takes so much pressure to make seawater drinkable: osmosis, osmotic pressure, reverse osmosis and the brine that pushes back, and a membrane that lets water through and stops ions.

Open the lesson

The 5 steps and their answers

Step 1 of 5
Open this step
Water flows the wrong way
Practice · Predict, then look
Question
Drop Push on the salt water to None. What happens?

Correct answer
- Correct Fresh water crosses into the salt water, and its level rises
Setup, choices and feedback
The Scene at this step
Push on the salt water
27 barfixed
Salt in the water
35 g/Lfixed
Show the molecules
Off
Choices
- Correct Fresh water crosses into the salt water, and its level rises
- Wrong Salt crosses into the fresh water until they even out
  Misconception: Thinks osmosis moves salt from the salty side into the fresh water.
  Feedback from the catalogue
  The membrane stops the ions, so salt cannot cross. What flows is the water, from the fresh side into the salty side: on the salty side some of the space beside the membrane is taken by ions, so fewer water molecules cross from it than arrive.
- Wrong Nothing: with no push there is no flow
  Feedback
  Even with no push from outside, water flows: fewer water molecules cross from the salty side than arrive from the fresh side.
Once the prediction is locked in, the Scene changes
Push on the salt water
0 barfixed
The Scene then shows: -27.3 · Bar: applied pressure minus the starting osmotic pressure; above 0 fresh water comes out/nodes/cell/outputs/netPressure
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
The fresh side drains, and the salt water rises, lifting the piston. This is osmosis: left alone, the sea draws fresh water in rather than giving any up.

Worked solution
1. The membrane passes water and stops ions, so salt cannot cross.
2. Water molecules cross both ways, but on the salty side some of the space beside the membrane is taken by ions.
3. More water arrives from the fresh side than goes back: water flows into the salt water.
Step 2 of 5
Open this step
Osmotic pressure
Practice · Reach the target
Question
Find the Push on the salt water that stops the flow, so the levels stay still, then press Check.

Goal
- 0 ± 0.5
  At the start of the step: -27.3
Setup, choices and feedback
The Scene at this step
Push on the salt water
0 bar
Salt in the water
35 g/Lfixed
Show the molecules
Off
Read from the Scene at
Bar: applied pressure minus the starting osmotic pressure; above 0 fresh water comes out/nodes/cell/outputs/netPressure
One setting that reaches it
Push on the salt water
27 bar
Feedback for a right answer
About 27 bar: the osmotic pressure of seawater. That is the pressure at the bottom of 270 metres of water.

Feedback for a wrong answer
Not yet. If water still flows into the salt side, push harder; if it flows out, less.

Hint
On the gauge: the black needle is the push, and the orange mark the osmotic pressure. Bring them together.

Worked solution
1. Each gram of salt per litre adds about 0.78 bar.
2. In seawater, 35 g/L: $35 \times 0.78 \approx 27.3$ ⁠ bar.
3. At 27 bar the flow all but stops.
Step 3 of 5
Open this step
Saltier water
Practice · Calculate
Question
Brine holds 70 grams of salt per litre, and each gram per litre adds about 0.78 bar. What is its osmotic pressure, in bar?

Correct answer
54.6 ± 3%
Setup, choices and feedback
The Scene at this step
Push on the salt water
0 bar
Salt in the water
70 g/Lfixed
Show the molecules
Off
Accepted within
± 3%
Computed from the Scene at this step’s setting
Bar: osmotic pressure of the starting seawater/nodes/cell/outputs/osmoticPressure
Feedback for a right answer
About 55 bar, twice seawater’s: twice the dissolved particles, twice the pressure. Find the balance point in the Scene to check.

Hint
Multiply the amount of salt by 0.78.

Worked solution
1. Osmotic pressure is proportional to the number of dissolved particles.
2. $70 \times 0.78 \approx 54.6$ ⁠.
3. About 55 bar.
Step 4 of 5
Open this step
Reverse osmosis
Practice · Calculate
Question
The pump pushes at 60 bar on seawater, 35 g/L. What share of the seawater comes out fresh before the flow stops? Give it as a fraction, such as 0.3.

Correct answer
0.545 ± 0.03
Setup, choices and feedback
The Scene at this step
Push on the salt water
60 barfixed
Salt in the water
35 g/Lfixed
Show the molecules
Off
Accepted within
± 0.03
Computed from the Scene at this step’s setting
Fraction of the starting seawater that has crossed as fresh water at balance (negative when water flows in)/nodes/cell/outputs/recovery
Wrong answers it recognises
- 1 ± 0.05
  Misconception: Expects a push above the osmotic pressure to squeeze all the water out of the seawater.
  Feedback from the catalogue
  The salt stays behind, so the water under the piston grows saltier as water is taken out, and its osmotic pressure rises. When it matches the pump’s push, the flow stops. At 60 bar that happens when only about half of the seawater has come out fresh.
Feedback for a right answer
About 0.55: a little over half. Real plants stop at about that point too, and pipe the brine back to sea, spread out so it does not settle on the seabed.

Hint
The flow stops when the osmotic pressure of the brine left behind reaches 60 bar. How salty is it then?

Worked solution
1. The flow stops when the osmotic pressure of what is left equals 60 bar: $\dfrac{60}{0.78} \approx 77$ ⁠ grams of salt per litre.
2. The salt does not cross, so it is packed into less water: $\dfrac{35}{77} \approx 0.45$ ⁠ of the water is left.
3. Out fresh: $1 - 0.45 \approx 0.55$ ⁠.
If you answered $1$ ⁠, you assumed all the water can be squeezed out. But the more water comes out, the harder the brine left behind pushes back.
Step 5 of 5
Open this step
The membrane
Practice · Choose
Question
Why do water molecules get through the membrane, but not sodium and chloride ions?

Correct answer
- Correct Each ion carries a shell of water molecules clinging to its charge, which makes it too big and too tightly bound to slip between the polymer chains
Setup, choices and feedback
The Scene at this step
Push on the salt water
60 bar
Salt in the water
35 g/L
Show the molecules
On
Choices
Shown in a random order.
- Correct Each ion carries a shell of water molecules clinging to its charge, which makes it too big and too tightly bound to slip between the polymer chains
- Wrong The membrane has holes smaller than a grain of salt, like a sieve
  Misconception: Pictures a desalination membrane as a sieve whose holes are too small for grains of salt.
  Feedback from the catalogue
  Water holds no grains of salt: salt comes apart into separate ions, about as small as a water molecule. Nor does the membrane have holes you could see. Water molecules dissolve into the polymer and wander between its chains, while each ion carries a shell of water clinging to its charge, which makes it too big and too tightly bound to follow.
- Wrong The ions are too heavy for the pump to push
  Feedback
  The pressure pushes the whole solution, ions and water together. The difference is in getting through the polymer, not in weight.
Feedback for a right answer
Water molecules dissolve into the polymer and wander through the tiny gaps between its chains. An ion is wrapped in water held tightly by its charge, so it bounces off. A good membrane holds back more than 99.5% of the salt.

Worked solution
1. In water salt comes apart into ions, and each ion pulls water molecules to it by its charge.
2. The ion with its shell of water is bigger than a single water molecule, and bound to its shell.
3. A lone water molecule gets into the polymer and through it; the wrapped ion does not.
The membrane has no holes you could see: there are no grains of salt in the water at all, only separate ions.

Across the unit · 8 misconceptions

Misconception catalogue

Each common mistake this unit recognises, in a few words. Open one for what the learner reads when an answer matches it, and where it is asked.

Index

Natural materials

Strong along the grain

Question

Correct answer

The Scene at this step

Hint

Worked solution

Across the grain

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

Levels within levels

Question

Correct answer

The Scene at this step

Choices

Worked solution

Stopping cracks

Question

Correct answer

The Scene at this step

Choices

Worked solution

Light as well as strong

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

Engineered materials

Stress and strain

Question

Correct answer

The Scene at this step

Hint

Worked solution

Past the yield

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

Brittle but strong

Question

Correct answer

The Scene at this step

Choices

Hint

Worked solution

Strength for weight

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

Springing back

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

Desalination

Water flows the wrong way

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

Osmotic pressure

Question

Goal

The Scene at this step