Solids
Overview: Study of internal restoring forces in solids, elasticity, stress, strain, and Hooke's Law.
1. Stress and Strain
Stress (σ)
Restoring force per unit area. Unit: N/m² or Pascal (Pa).
σ = F / A
- Longitudinal Stress: Tensile (stretching) or Compressive.
- Shearing Stress: Force tangential to surface.
- Hydraulic Stress: Force perpendicular to surface (pressure).
Strain (ε)
Ratio of change in dimension to original dimension. Unitless.
ε = ΔL / L
2. Hooke's Law
For small deformations, stress is directly proportional to strain.
Stress ∝ Strain ⇒ σ = E ε
Where E is the Modulus of Elasticity.
Stress-Strain Curve
- Proportional Limit: Hooke's Law obeys.
- Yield Point: Permanent set begins.
- Ultimate Tensile Strength: Maximum stress before breaking.
- Fracture Point: Material breaks.
3. Modulii of Elasticity
Young's Modulus (Y)
Ratio of Longitudinal Stress to Longitudinal Strain.
Y = (F/A) / (ΔL/L) = FL / AΔL
Shear Modulus (G)
Ratio of Shearing Stress to Shearing Strain.
G = (F/A) / θ
Bulk Modulus (B)
Ratio of Hydraulic Stress to Volumetric Strain.
B = - P / (ΔV/V)
Compressibility (k): Reciprocal of Bulk Modulus (k = 1/B).
4. Elastic Potential Energy
Work done in stretching a wire is stored as potential energy.
U = ½ × Stress × Strain × Volume
U = ½ kx² (for spring)
5. Poisson's Ratio (ν)
Ratio of lateral strain to longitudinal strain.
ν = - (Δd/d) / (ΔL/L)
Theoretical theoretical range: -1 to 0.5. Practical range: 0 to 0.5.
Numericals
Stress Calculation
Q1. A load of 4 kg is suspended from a ceiling wire of radius 2 mm. Calculate the
stress.
Stress = F / A
F = mg = 4 × 9.8 = 39.2 N
A = πr² = 3.14 × (2×10-3)²
A = 12.56 × 10-6 m²
σ = 39.2 / 12.56 × 106
σ = 3.12 × 106 N/m²
Young's Modulus
Q2. A steel wire of length 2.5 m and area 2.5 mm² stretches by 1.5 mm under a
load of 50 N. Find Y.
Y = (F L) / (A ΔL)
F=50, L=2.5, A=2.5e-6, ΔL=1.5e-3
Y = (50 × 2.5) / (2.5×10-6 ×
1.5×10-3)
Y = 125 / (3.75 × 10-9)
Y = 3.33 × 1010 N/m²
Energy Density
Q3. A wire is stretched by a force. Stress is S, Strain is e. Find Energy per unit
volume.
u = Energy / Volume
u = ½ × Stress × Strain
u = ½ Se
Elongation
Q4. Two wires of same material have lengths 1:2 and radii 2:1. Ratio of elongation for
same load?
ΔL = (FL) / (AY)
ΔL ∝ L / r² (Since F, Y same)
Ratio = (L1/L2) × (r2/r1)²
Ratio = (1/2) × (1/2)²
Ratio = 1/2 × 1/4 = 1/8
Bulk Modulus
Q5. Pressure on a sphere increases by 80 atm, volume decreases by 0.01%. Find Bulk
Modulus.
B = ΔP / (ΔV/V)
ΔP = 80 × 105 Pa
ΔV/V = 0.01/100 = 10-4
B = 80 × 105 / 10-4
B = 8 × 1010 Pa
Breaking Force
Q6. Breaking stress of a wire is S. If radius is doubled, what is new breaking stress
and force?
Breaking Stress depends on material. It remains S.
Breaking Force F = Stress × Area
F ∝ r²
New Force = 4 times original force.
Thermal Stress
Q7. A steel rod of length L is clamped. Temp rises by ΔT. Find Thermal Stress.
ΔL = L α ΔT (Thermal exp)
Strain = ΔL/L = α ΔT
Stress = Y × Strain
Stress = Y α ΔT
Shear Modulus
Q8. For most materials, what is the relation between Y and G (approx)?
Young's Modulus Y is typically larger than Shear Modulus G.
Approx relation: Y ≈ 3G (for isotropic materials)
(Just a factual check often asked)
Self Weight Elongation
Q9. Elongation of a hanging wire due to its own weight W?
Force varies from 0 at bottom to W at top.
Average force F = W/2
ΔL = (W/2 L) / (AY)
ΔL = WL / 2AY
Compressibility
Q10. Compressibility of water is 4.5 × 10-10 Pa-1. Find
density change at depth where P=100atm.
ΔV/V = k P
ΔV/V = 4.5e-10 × 100 × 105 = 4.5 ×
10-3
ρ = M/V → Δρ/ρ = ΔV/V
Density increases by 0.45%
Equations & Formulas
| Concept | Formula |
|---|---|
| Stress | σ = F / A |
| Strain | ε = ΔL / L |
| Hooke's Law | σ = E ε |
| Young's Modulus | Y = FL / AΔL |
| Shear Modulus | G = F / Aθ |
| Bulk Modulus | B = - PV / ΔV |
| Poissons Ratio | ν = -(Δd/d)/(Δl/l) |
| Elastic Potential Energy | U = ½ × stress × strain × Vol |
| Thermal Stress | σ = Y α ΔT |
| Relation Y, B, G | 9/Y = 1/B + 3/G |
50 NEET Facts
Key points for Mechanical Properties of Solids.
1. Elasticity
Property to regain original shape after removal of deforming force. Steel is more elastic than rubber.
2. Elastic limit
Maximum stress up to which material shows perfect elastic behavior.
3. Plasticity
Permanent deformation after force removal. Putty and Clay are nearly perfectly plastic.
4. Stress vs Pressure
Pressure is external force per unit area. Stress is internal restoring force per unit area.
5. Young's Modulus
Property of material, not dimension. Independent of length or area.
6. Temperature Effect on Y
As temperature increases, Y decreases (interatomic forces weaken).
7. Ductile Materials
Have large plastic range (Gold, Copper). Can be drawn into wires.
8. Brittle Materials
Very small plastic range (Glass, Ceramics). Break immediately after yield point.
9. Elastomers
Materials that can be stretched to large strains (e.g., Rubber, Aorta). Non-linear stress-strain.
10. Modulus of Rigidity
Solids have Shear Modulus. Fluids do not (G=0 for fluids).
11. Bulk Modulus
Solids > Liquids > Gases. Solids are least compressible.
12. Isothermal vs Adiabatic B
For gases, Badi = γP and Biso = P.
13. Poisson's Ratio Signs
Usually positive. Negative for auxetic materials (rare).
14. Volumetric Strain
The ratio of change in volume to original volume.
15. Steel vs Rubber
For same stress, strain in rubber > strain in steel. Since Y = Stress/Strain, Ysteel >
Yrubber.
16. Quartz Fiber
Used in galvanometers because of very low elastic after-effect (returns to zero quickly).
17. Elastic Fatigue
Loss of strength due to repeated stress cycles (e.g., bridge wires weakening).
18. Potential Energy Density
U = 1/2 * stress * strain. Area under Stress-Strain curve.
19. Breaking Stress
Constant for a material. Breaking force depends on Area.
20. Safety Factor
Ratio of Yield Stress to Working Stress. Must be > 1.
21. Spring Constant (k)
F = kx. k depends on geometry and modulus of rigidity G.
22. Cutting a Spring
If spring is cut into n equal parts, k of each part becomes nk (stiffer).
23. Springs in Series
1/keq = 1/k1 + 1/k2.
24. Springs in Parallel
keq = k1 + k2.
25. Hollow vs Solid Shaft
Hollow shaft is stronger than solid shaft of same mass (higher polar moment of inertia).
26. Strain Energy
Stored internally. Released when unloaded.
27. Hysteresis Loop
Area represents energy dissipated as heat during loading/unloading cycle (e.g., rubber tires get hot).
28. Hooke's Law Validity
Only valid up to Proportional Limit in linear region.
29. Invar Steel
Used for pendulums because its expansion coefficient & Modulus change with temp is negligible.
30. Malleability
Ability to be hammered into sheets (Gold, Silver).
31. Tenacity
Ability to withstand tension without breaking.
32. Impact of Impurities
Adding carbon to iron increases strength and brittleness (Steel).
33. Annealing
Heating and slow cooling increases ductility.
34. Hammering
Increases elasticity and strength (work hardening).
35. Bulk Modulus of Ideal Gas
Isothermal = Pressure P.
36. Perfectly Rigid Body
Y, B, G are infinity. No strain for any stress.
37. Dimensions of Moduli
Same as pressure [M L-1 T-2].
38. Strain Dimension
Dimensionless [M0 L0 T0].
39. Maximum Poisson Ratio
Upper limit 0.5 (Rubber approx 0.5, Cork approx 0).
40. Cantilever Depression
δ = WL3 / 3YI.
41. Girder Shape
I-shape used to maximize Moment of Inertia I, reducing depression for given weight.
42. Sag in Bridge
Increases with L3. Hence bridges have pillars at short intervals.
43. Buckling
Ideally vertical pillar bends sideways under load.
44. Mountain Height Limit
Limited by the crushing strength of rocks at the base (approx 10km on Earth).
45. Elastic After Effect
Delay in regaining original shape. Glass has large effect. Phosphor bronze has small.
46. Rubber Band Heat
When stretched adiabatically, rubber band gets warm.
47. Relation density & strain
Density decreases on stretching (volume usually increases unless ν=0.5).
48. Speed of Sound
v = sqrt(Y/density) in solids. Depends on elasticity.
49. Seismic Waves
S-waves (Shear waves) propagate only in solids (because fluids have G=0).
50. Earth Core
S-waves don't pass through outer core, implying it is liquid.
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