Lesson 9 of 9 · 17 min
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Must-know facts
22 facts
- 1Stress = restoring force per unit area = F/A; unit N m⁻² = Pa; dimensions [ML⁻¹T⁻²].
- 2Strain is a ratio of lengths or volumes: no unit, no dimensions.
- 3Three strains: longitudinal ΔL/L, shearing Δx/L = tan θ ≈ θ, volume ΔV/V.
- 4Hooke's law: stress ∝ strain for small deformations; the constant is the modulus of elasticity.
- 5Stress-strain curve: O-A proportional (Hooke's law), B yield point or elastic limit (σy), C permanent set if unloaded, D ultimate tensile strength (σu), E fracture.
- 6D and E close: brittle. D and E far apart: ductile.
- 7Elastomers (rubber, aorta tissue): large elastic region, no Hooke's law over most of it, no well-defined plastic region.
- 8Y = FL/(AΔL); G = F/(Aθ); B = −p/(ΔV/V); all three have the unit Pa.
- 9Y and G apply only to solids; B applies to solids, liquids and gases.
- 10For most materials G ≈ Y/3.
- 11Y: steel 2.0 × 10¹¹, copper 1.1 × 10¹¹, bone 9.4 × 10⁹ N m⁻² (values used in NCERT's examples).
- 12Same 0.1% strain in a 0.1 cm² wire: steel 2000 N, copper 1100 N, brass 900 N, aluminium 690 N; steel is the most elastic.
- 13Compressibility k = 1/B; a gas squeezes roughly a million times more easily than a solid.
- 14Bulk modulus of water 2.2 × 10⁹ N m⁻²; air at STP 1.0 × 10⁵ N m⁻² (1.0 × 10⁻⁴ GPa).
- 15Water at 3000 m ocean depth (g = 10) is compressed by 1.36%.
- 16Poisson's ratio = lateral strain/longitudinal strain; unitless; steels 0.28-0.30, aluminium alloys about 0.33.
- 17Elastic energy U = ½ × stress × strain × volume; per unit volume u = ½σε = ½Y(strain)².
- 18Beam sag δ = Wl³/(4bd³Y): depth counts as d³, breadth only as b.
- 19Crane rope for 10 t: A ≥ Mg/σy = 3.3 × 10⁻⁴ m² (radius about 1 cm); with a safety factor of 10, radius about 3 cm, made of braided thin wires.
- 20Mountain height limit: hρg = 30 × 10⁷ N m⁻² with ρ = 3 × 10³ kg m⁻³ gives about 10 km.
- 21A wire hung from a ceiling with weight F at its end has tension F, not 2F, at every section, so the stress is F/A.
- 22Stress is not a vector: no single direction can be assigned to it.
Common traps
Where marks are lost
Saying rubber is more elastic than steel because it stretches more.
Using the diameter as the radius when finding A = πr².
Leaving areas in mm² or cm² in Y = FL/(AΔL).
Taking the tension in a hanging wire as 2F because the ceiling also pulls with F.
Thinking a body loaded beyond the proportional limit A is permanently deformed.
Giving bulk modulus a negative value because of the minus sign in B = −p/(ΔV/V).
Applying Young's or shear modulus to a liquid.
Widening a beam to stop it sagging.
Treating Poisson's ratio or strain as having units.
Formulas
12 to know
Stress
σ = F/A
Unit N m⁻² = Pa; [ML⁻¹T⁻²].
Strains
longitudinal ΔL/L; shearing Δx/L = tan θ ≈ θ; volume ΔV/V
All dimensionless.
Hooke's law
stress = k × strain
Valid only in the linear part OA.
Young's modulus
Y = σ/ε = (F/A)/(ΔL/L) = FL/(AΔL)
So ΔL = FL/(AY).
Shear modulus
G = (F/A)/(Δx/L) = FL/(AΔx) = F/(Aθ); σs = Gθ
G ≈ Y/3 for most materials.
Bulk modulus
B = −p/(ΔV/V)
Positive; applies to solids, liquids and gases.
Compressibility
k = 1/B = −(1/Δp)(ΔV/V)
Largest for gases.
Poisson's ratio
(Δd/d)/(ΔL/L) = (Δd/ΔL)(L/d)
Pure number; steels 0.28-0.30.
Elastic energy
U = ½ × stress × strain × volume = ½ Y A l²/L
Per unit volume u = ½σε.
Beam sag
δ = Wl³/(4bd³Y)
Beam supported near its ends, load W at the centre.
Minimum rope area
A ≥ Mg/σy
Keeps the rope within its elastic limit.
Mountain height limit
hρg = elastic limit of rock
30 × 10⁷ N m⁻², ρ = 3 × 10³ kg m⁻³, g = 10 m s⁻² give h = 10 km.
Key terms
15 terms
- Elasticity
- The tendency of a body to recover its original size and shape once the deforming force is removed.
- Plasticity
- The behaviour of a body that stays deformed after the force is removed, as putty and mud do.
- Stress
- Internal restoring force per unit area of a deformed body; equal in size to the applied force per unit area.
- Strain
- Change in a dimension divided by the original dimension; a pure number.
- Hydraulic stress
- Uniform normal stress from a surrounding fluid; equal to the fluid pressure and changing only volume.
- Elastic limit (yield point)
- Point B of the stress-strain curve, the largest stress after which the body still recovers fully.
- Permanent set
- The strain left in a body after unloading from beyond its yield point.
- Ultimate tensile strength
- The highest stress on the stress-strain curve, point D; past it the wire keeps stretching under a smaller force until it breaks.
- Brittle
- A material whose fracture point lies close to its ultimate strength.
- Ductile
- A material that strains a long way between its ultimate strength and fracture.
- Elastomer
- A substance such as rubber or aorta tissue that can take large strains and recover, without following Hooke's law.
- Modulus of rigidity
- Another name for the shear modulus G.
- Compressibility
- Reciprocal of bulk modulus; fractional volume change per unit rise in pressure.
- Poisson's ratio
- Lateral strain divided by longitudinal strain for a stretched wire.
- Buckling
- Sideways bending of a deep, thin bar under a load that is not exactly placed.