Class 11 : Physics (In English) – Lesson Chapter 8: Mechanical Properties of Solids

EXPLANATION & SUMMARY

🔶 1. Introduction
Solids resist deformation when external forces are applied. This resistance and response are studied under mechanical properties of solids. The chapter deals with how solids change shape, stretch, compress, or twist under forces.

🔶 2. Interatomic Forces in Solids
Solids have atoms/molecules closely packed due to strong intermolecular forces.
When force is applied, atoms displace from equilibrium but tend to restore their position.
This restoring tendency gives elastic properties to solids.

🔶 3. Elasticity
Elasticity is the property of a body by which it tends to regain its original shape and size after removal of the deforming force.
Perfect elasticity: Body regains shape completely.
Plasticity: Body does not regain shape.
Most real materials show elastic behavior for small deformations.

🔶 4. Stress
Stress is the restoring force per unit area set up inside the material.
Formula:
Stress = Force / Area = F / A
Unit: N/m² or Pascal (Pa)
It is a tensor quantity, but generally treated as scalar in this context.
Types:
Normal stress: Perpendicular to surface
Tensile (stretching)
Compressive (squeezing)
Shearing stress: Parallel to surface

🔶 5. Strain
Strain is the fractional change in dimension of the body.
It has no unit (dimensionless).
Types:
Longitudinal strain = Change in length / Original length = ΔL / L
Volumetric strain = Change in volume / Original volume = ΔV / V
Shear strain = Relative displacement / Distance between layers = x / h

🔶 6. Hooke’s Law
Within the elastic limit, stress is directly proportional to strain.
Mathematically:
Stress ∝ Strain
Or, Stress = E × Strain
Where E is the modulus of elasticity (a constant for the material).

🔶 7. Elastic Moduli
Elastic modulus = Stress / Strain
Types:
Young’s Modulus (Y)
For longitudinal strain
Y = (F × L) / (A × ΔL)
Shear Modulus or Modulus of Rigidity (η)
For shear strain
η = Shearing stress / Shear strain
Bulk Modulus (B)
For volume change under pressure
B = -ΔP / (ΔV / V)
Negative sign because volume decreases with increased pressure.

🔶 8. Poisson’s Ratio (σ)
It is the ratio of lateral strain to longitudinal strain.
σ = Lateral strain / Longitudinal strain
It is dimensionless.
Typical range: 0.2 to 0.5 for solids.

🔶 9. Stress-Strain Curve
Graph between stress and strain shows how a material behaves when stretched.
Key Points on Curve:
Proportional Limit (A): Stress ∝ Strain holds (Hooke’s law valid).
Elastic Limit (B): Material returns to original shape after load is removed.
Yield Point (C): Beyond this, plastic deformation begins.
Ultimate Stress (D): Maximum stress the material can handle.
Breaking Point (E): Material breaks.
Area under curve = energy stored in body = strain energy

🔶 10. Elastic and Plastic Behavior
Elastic behavior: Reversible deformation
Plastic behavior: Permanent deformation after elastic limit
Example: Metal wire stretches elastically for small load but shows plasticity under high stress.

🔶 11. Applications of Elasticity
Bridge design: Materials with high Young’s modulus are used.
Spring balance: Works on Hooke’s law.
Crane wires: Must resist large stress without permanent deformation.

🔶 12. Energy Stored in a Stretched Wire
Strain energy (U) = (1/2) × stress × strain × volume
U = (1/2) × (F × ΔL)
Also,
U = (1/2) × Y × strain² × volume
This is elastic potential energy.

🔶 13. Comparison of Moduli for Materials
Materials with high Young’s modulus: steel > glass > copper
Rubber has high elasticity but low Young’s modulus (it stretches a lot but is not very stiff).

🔶 14. Importance of Elastic Moduli
Young’s modulus: Measures stiffness
Bulk modulus: Measures incompressibility
Shear modulus: Measures ability to resist shape change

📘 PART B: CRISP SUMMARY (~300 WORDS)

🔷 Summary – Mechanical Properties of Solids
Solids resist deformation due to interatomic forces.
Elasticity: Ability of solids to regain original shape after removal of force.
Stress = Force / Area (unit: Pascal)
Strain = Change in dimension / Original (unitless)
Hooke’s Law: Within elastic limit, stress ∝ strain

Three main elastic moduli:
Young’s modulus (Y) → tensile deformation
Y = (F × L) / (A × ΔL)
Shear modulus (η) → shear deformation
η = (F / A) / (x / h)
Bulk modulus (B) → volumetric deformation
B = -ΔP / (ΔV / V)
Poisson’s ratio: Lateral strain / Longitudinal strain (no unit)

Stress-strain curve shows how a material deforms:
Proportional limit
Elastic limit
Yield point
Ultimate stress
Breaking point
Strain energy = energy stored in a stretched material
U = (1/2) × stress × strain × volume
High Young’s modulus means more stiffness.
Applications: bridge construction, spring balances, safety cables, etc.
This chapter builds understanding of how materials respond to forces and prepares for further study in mechanics and material science.

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QUESTIONS FROM TEXTBOOK

🔷 Question 8.1
A steel wire of length 4.7 m and cross-sectional area 3.0 × 10⁻⁵ m² stretches by the same amount as a copper wire of length 3.5 m and cross-sectional area of 4.0 × 10⁻⁵ m² under a given load. What is the ratio of the Young’s modulus of steel to that of copper?
✅ Answer:
📘 For same elongation under same load:
Δl = (F × L)/(A × Y)
⇒ (L/A × Y) is constant
📘 So,
L₁ / (A₁Y₁) = L₂ / (A₂Y₂)
⇒ Y₁ / Y₂ = (L₁ / A₁) / (L₂ / A₂)
📌 L₁ = 4.7 m, A₁ = 3.0 × 10⁻⁵ m²
📌 L₂ = 3.5 m, A₂ = 4.0 × 10⁻⁵ m²
Y₁ / Y₂ = (4.7 / 3.0 × 10⁻⁵) / (3.5 / 4.0 × 10⁻⁵)
= (4.7 / 3.0) × (4.0 / 3.5) ≈ 1.5667 × 1.1429 ≈ 1.79
✅ Final Answer: 1.79

🔷 Question 8.2
Figure 8.9 shows the strain-stress curve for a given material. What are
(a) Young’s modulus and
(b) approximate yield strength for this material?
✅ Answer:
📘 From graph (read visually):
At point with stress = 300 × 10⁶ N/m², strain = 0.002
(a) Young’s modulus (Y) = Stress / Strain
= (300 × 10⁶) / 0.002 = 1.5 × 10¹¹ N/m²
(b) Yield strength ≈ maximum stress at elastic limit
= ~300 × 10⁶ N/m²
✅ Final Answers:
✔️ Young’s modulus = 1.5 × 10¹¹ N/m²
✔️ Yield strength ≈ 3.0 × 10⁸ N/m²

🔷 Question 8.3
The stress-strain graphs for materials A and B are shown in Fig. 8.10.
(a) Which of the materials has the greater Young’s modulus?
(b) Which of the two is the stronger material?
✅ Answer:
(a) Greater Young’s modulus → steeper slope
⇒ Material A has higher Young’s modulus
(b) Stronger material → can withstand more stress before breaking
⇒ Material B is stronger (higher ultimate stress)
✅ Final Answers:
✔️ Young’s modulus: A > B
✔️ Strength: B > A

🔷 Question 8.4
Read the following two statements below carefully and state, with reasons, if it is true or false.
(a) The Young’s modulus of rubber is greater than that of steel.
(b) The stretching of a coil is determined by its shear modulus.
✅ Answer:
(a) ❌ False – Rubber stretches more for same force, hence has lower Young’s modulus.
Steel resists elongation → higher Young’s modulus.
(b) ✅ True – Coils undergo deformation due to twisting (shearing), hence shear modulus is the relevant quantity.

🔷 Question 8.5
Two wires of diameter 0.25 cm, one made of steel and the other made of brass are loaded as shown in Fig. 8.11. The unloaded length of steel wire is 1.5 m and that of brass wire is 1.0 m. Compute the elongations of the steel and the brass wires.
✅ Answer:
📌 Given:
Diameter = 0.25 cm = 0.0025 m ⇒ A = πr² = π(0.00125)² ≈ 4.91 × 10⁻⁶ m²
g = 9.8 m/s²
Mass on steel = 4.0 kg → Force F = 4 × 9.8 = 39.2 N
Length L = 1.5 m, Y (steel) = 2 × 10¹¹ N/m²
Δl (steel) = (F × L) / (A × Y)
= (39.2 × 1.5) / (4.91 × 10⁻⁶ × 2 × 10¹¹)
≈ 6.0 × 10⁻⁴ m = 0.60 mm
Mass on brass = 6.0 kg → F = 6 × 9.8 = 58.8 N
Length L = 1.0 m, Y (brass) = 0.91 × 10¹¹ N/m²
Δl (brass) = (58.8 × 1.0) / (4.91 × 10⁻⁶ × 0.91 × 10¹¹)
≈ 1.33 × 10⁻³ m = 1.33 mm
✅ Final Answers:
✔️ Elongation of steel wire = 0.60 mm
✔️ Elongation of brass wire = 1.33 mm


🔷 Question 8.6
The edge of an aluminium cube is 10 cm long. One face of the cube is firmly fixed to a vertical wall. A mass of 100 kg is then attached to the opposite face of the cube. The shear modulus of aluminium is 25 GPa. What is the vertical deflection of this face?
✅ Answer:
📌 Given:
Edge length, L = 10 cm = 0.1 m
Force, F = mg = 100 × 9.8 = 980 N
Shear modulus, G = 25 GPa = 25 × 10⁹ N/m²
Area, A = (0.1 m)² = 0.01 m²
Shear strain = Δx / L = F / (A × G)
Δx = (F × L) / (A × G)
= (980 × 0.1) / (0.01 × 25 × 10⁹)
= 98 / (2.5 × 10⁸)
= 3.92 × 10⁻⁷ m = 0.392 µm
✅ Final Answer: 0.392 micrometres

🔷 Question 8.7
Four identical hollow cylindrical columns of mild steel support a big structure of mass 50,000 kg. The inner and outer radii of each column are 30 mm and 60 mm respectively. Assuming the load distribution to be uniform, calculate the compressional strain of each column.
✅ Answer:
📌 Given:
Mass = 50,000 kg → F = mg = 50,000 × 9.8 = 4.9 × 10⁵ N
Each column supports F/4 = 1.225 × 10⁵ N
Inner radius = 30 mm = 0.03 m
Outer radius = 60 mm = 0.06 m
Area, A = π(R² − r²)
= π[(0.06)² − (0.03)²] = π(0.0036 − 0.0009) = π(0.0027) ≈ 8.48 × 10⁻³ m²
Young’s modulus for steel, Y = 2 × 10¹¹ N/m²
Strain = Stress / Y
Stress = F / A = (1.225 × 10⁵) / (8.48 × 10⁻³) ≈ 1.44 × 10⁷ N/m²
Strain = (1.44 × 10⁷) / (2 × 10¹¹) = 7.2 × 10⁻⁵
✅ Final Answer: 7.2 × 10⁻⁵

🔷 Question 8.8
A piece of copper having a rectangular cross-section of 15.2 mm × 19.1 mm is pulled in tension with 44,500 N force, producing only elastic deformation. Calculate the resulting strain?
✅ Answer:
📌 Area = 15.2 mm × 19.1 mm = (15.2 × 10⁻³) × (19.1 × 10⁻³) = 2.9032 × 10⁻⁴ m²
Force = 44,500 N
Young’s modulus for copper, Y = 1.1 × 10¹¹ N/m²
Stress = F / A = 44500 / 2.9032 × 10⁻⁴ ≈ 1.53 × 10⁸ N/m²
Strain = Stress / Y = (1.53 × 10⁸) / (1.1 × 10¹¹) ≈ 1.39 × 10⁻³
✅ Final Answer: 1.39 × 10⁻³

🔷 Question 8.9
A steel cable with a radius of 1.5 cm supports a chairlift at a ski area. If the maximum stress is not to exceed 10⁸ N/m², what is the maximum load the cable can support?
✅ Answer:
📌 Radius = 1.5 cm = 0.015 m → A = πr² = π(0.015)² ≈ 7.07 × 10⁻⁴ m²
Maximum Stress = 10⁸ N/m²
Stress = Force / Area ⇒ F = Stress × Area
F = 10⁸ × 7.07 × 10⁻⁴ = 7.07 × 10⁴ N
✅ Final Answer: 70,700 N

🔷 Question 8.10
A rigid bar of mass 15 kg is supported symmetrically by three wires each 2.0 m long. Those at each end are of copper and the middle one is of iron. Determine the ratios of their diameters if each is to have the same tension.
✅ Answer:
📌 For same tension, extensions must be equal.
Δl = (F × L) / (A × Y) ⇒ Δl ∝ 1 / (A × Y)
Let diameters be d₁ (Cu), d₂ (Fe)
Since length and force are same,
1 / (πd₁²/4 × Y_Cu) = 1 / (πd₂²/4 × Y_Fe)
⇒ d₁² / Y_Cu = d₂² / Y_Fe
⇒ (d₁ / d₂)² = Y_Cu / Y_Fe
Y_Cu = 1.1 × 10¹¹ N/m²
Y_Fe = 2.0 × 10¹¹ N/m²
⇒ (d₁ / d₂)² = 1.1 / 2.0 = 0.55
⇒ d₁ / d₂ = √0.55 ≈ 0.741
✅ Final Answer: d_Cu / d_Fe ≈ 0.741

🔷 Question 8.11
A 14.5 kg mass, fastened to the end of a steel wire of unstretched length 1.0 m, is whirled in a vertical circle with angular velocity of 2 rev/s at the bottom of the circle. The cross-sectional area of the wire is 0.065 cm². Calculate the elongation of the wire when the mass is at the lowest point of its path.
✅ Answer:
📌 Mass = 14.5 kg, ω = 2 rev/s = 4π rad/s
L = 1.0 m
Area A = 0.065 cm² = 6.5 × 10⁻⁶ m²
Y = 2 × 10¹¹ N/m²
Centripetal Force at bottom:
T = mg + mω²R = 14.5 × 9.8 + 14.5 × (4π)² × 1
= 142.1 + 14.5 × 157.9 = 142.1 + 2299.6 ≈ 2441.7 N
Elongation = (F × L) / (A × Y)
= (2441.7 × 1) / (6.5 × 10⁻⁶ × 2 × 10¹¹)
= 2441.7 / (1.3 × 10⁶) ≈ 1.88 × 10⁻³ m = 1.88 mm
✅ Final Answer: 1.88 mm


🔷 Question 8.12
Compute the fractional change in volume of a glass slab, when subjected to a hydraulic pressure of 10⁶ Pa. Bulk modulus of glass = 5 × 10⁹ Pa.
✅ Answer:
📌 Fractional change in volume:
ΔV / V = – (P / B)
Where,
P = 10⁶ Pa
B = 5 × 10⁹ Pa
⇒ ΔV / V = – (10⁶ / 5 × 10⁹) = – 2 × 10⁻⁴
✅ Final Answer: –2 × 10⁻⁴

🔷 Question 8.13
Determine the volume contraction of a solid copper cube, 10 cm on an edge, when subjected to a hydraulic pressure of 7 × 10⁶ Pa. Bulk modulus of copper = 1.4 × 10¹¹ Pa.
✅ Answer:
📌 Volume of cube = (0.10 m)³ = 1.0 × 10⁻³ m³
Pressure, P = 7 × 10⁶ Pa
Bulk modulus, B = 1.4 × 10¹¹ Pa
Volume contraction:
ΔV = V × (P / B)
= 1.0 × 10⁻³ × (7 × 10⁶ / 1.4 × 10¹¹)
= 1.0 × 10⁻³ × 5 × 10⁻⁵ = 5.0 × 10⁻⁸ m³
✅ Final Answer: 5.0 × 10⁻⁸ m³

🔷 Question 8.14
A brass rod of length 50 cm and diameter 3.0 mm is joined to a steel rod of the same length and diameter. A force of 100 N is applied to this composite rod. Estimate the change in length of the composite rod.
(Y_brass = 0.9 × 10¹¹ Pa, Y_steel = 2.0 × 10¹¹ Pa)
✅ Answer:
📌 Length of each = 0.5 m
Diameter = 3.0 mm = 0.003 m
Area A = πr² = π × (0.0015)² ≈ 7.07 × 10⁻⁶ m²
Force F = 100 N
Change in length = Δl = (F × L) / (A × Y)
For brass:
Δl₁ = (100 × 0.5) / (7.07 × 10⁻⁶ × 0.9 × 10¹¹) ≈ 7.86 × 10⁻⁵ m
For steel:
Δl₂ = (100 × 0.5) / (7.07 × 10⁻⁶ × 2.0 × 10¹¹) ≈ 3.53 × 10⁻⁵ m
Total elongation = Δl₁ + Δl₂ = (7.86 + 3.53) × 10⁻⁵ = 1.139 × 10⁻⁴ m = 0.114 mm
✅ Final Answer: 0.114 mm

🔷 Question 8.15
A uniform wire of length 3 m and diameter 0.5 mm is stretched by a force of 100 N. Calculate the elongation produced. Young’s modulus of wire = 2 × 10¹¹ Pa.
✅ Answer:
📌 L = 3 m, d = 0.5 mm = 0.0005 m
r = 0.00025 m ⇒ A = πr² = π × (0.00025)² ≈ 1.96 × 10⁻⁷ m²
Y = 2 × 10¹¹ N/m², F = 100 N
Δl = (F × L) / (A × Y)
= (100 × 3) / (1.96 × 10⁻⁷ × 2 × 10¹¹)
= 300 / (3.92 × 10⁴) = 7.65 × 10⁻³ m = 7.65 mm
✅ Final Answer: 7.65 mm

🔷 Question 8.16
The Young’s modulus of steel is twice that of brass. Two wires of same length and same area of cross-section are suspended from a rigid support. One is of brass and the other is of steel. At the lower ends of the wires, a platform is suspended such that the platform remains horizontal when both wires are elongated. If the weight on the platform is 300 N, find the individual weights supported by the steel and brass wires.
✅ Answer:
📌 Y_steel = 2 × Y_brass
Let elongation = Δl (same for both)
⇒ F ∝ Y × A / L ⇒ Force ∝ Y (since A and L same)
Let force in brass = F_B, in steel = F_S
Then, F_S / F_B = Y_S / Y_B = 2
⇒ F_S = 2 × F_B
Also, total force = 300 N = F_B + F_S = F_B + 2F_B = 3F_B
⇒ F_B = 100 N, F_S = 200 N
✅ Final Answers:
✔️ Brass wire supports = 100 N
✔️ Steel wire supports = 200 N


🔷 Question 8.17
A steel cable with a radius of 1.5 cm supports a chairlift at a ski area. If the maximum stress is not to exceed 10⁸ N/m², what is the maximum load the cable can support?
✅ Answer:
📌 Radius, r = 1.5 cm = 0.015 m
Area, A = πr² = π × (0.015)² = π × 2.25 × 10⁻⁴ ≈ 7.07 × 10⁻⁴ m²
Maximum stress, σ = F / A
⇒ F = σ × A = 10⁸ × 7.07 × 10⁻⁴ = 7.07 × 10⁴ N
✅ Final Answer: Maximum load = 70,700 N

🔷 Question 8.18
A rigid bar of mass 15 kg is supported symmetrically by three wires each 2.0 m long. Those at each end are of copper and the middle one is of iron. Determine the ratios of their diameters if each is to have the same tension.
✅ Answer:
📌 Given: Length = same (L = 2.0 m), Tension = same
For equal tension, all wires must stretch by same amount
⇒ ΔL_copper = ΔL_iron
Using elongation formula:
ΔL = (F × L) / (A × Y) ⇒ ΔL ∝ 1 / (A × Y) ⇒ For same F and L:
1 / (πr² × Y) must be same ⇒ (1 / d²Y) same
Let diameters be d_C (copper) and d_I (iron):
⇒ (1 / d_C²Y_C) = (1 / d_I²Y_I)
Taking ratio:
d_C² / d_I² = Y_C / Y_I
Given:
Y_C = 1.1 × 10¹¹ Pa
Y_I = 2.0 × 10¹¹ Pa
⇒ d_C² / d_I² = 1.1 / 2.0 = 11 / 20
⇒ d_C / d_I = √(11/20) ≈ 0.7416
✅ Final Answer:
✔️ Ratio of diameters (d_C : d_I) ≈ 0.7416 : 1

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OTHER IMPORTANT QUESTIONS FOR EXAMS

(CBSE MODEL QUESTIONS PAPER)

ESPECIALLY MADE FROM THIS LESSON ONLY

Q1. Which of the following is a scalar quantity?
(A) Stress
(B) Strain
(C) Modulus of elasticity
(D) All of the above
Answer: (D) All of the above

Q2. The SI unit of stress is:
(A) dyne/cm²
(B) N/m
(C) N/m²
(D) joule
Answer: (C) N/m²

Q3. The Young’s modulus is defined as:
(A) Stress × Strain
(B) Stress / Strain
(C) Strain / Stress
(D) None of these
Answer: (B) Stress / Strain

Q4. Which type of stress produces a change in shape but not in volume?
(A) Longitudinal stress
(B) Normal stress
(C) Shearing stress
(D) Bulk stress
Answer: (C) Shearing stress

Q5. A wire is stretched such that it returns to its original shape. This behavior corresponds to:
(A) Plasticity
(B) Elasticity
(C) Rigidity
(D) Viscosity
Answer: (B) Elasticity

Q6. Strain has:
(A) No units and no dimensions
(B) Units but no dimensions
(C) Dimensions but no units
(D) Both units and dimensions
Answer: (A) No units and no dimensions

Q7. Assertion (A): A rubber band shows greater strain than a steel wire under same stress.
Reason (R): Rubber has a smaller Young’s modulus.
(A) Both A and R are true, and R is the correct explanation of A.
(B) Both A and R are true, but R is not the correct explanation of A.
(C) A is true, R is false.
(D) A is false, R is true.
Answer: (A) Both A and R are true, and R is the correct explanation of A.

Q8. What is the dimensional formula of stress?
(A) [M L⁻¹ T⁻²]
(B) [M L T⁻²]
(C) [M⁰ L⁰ T⁰]
(D) [M L² T⁻²]
Answer: (A) [M L⁻¹ T⁻²]

Q9. A wire of length L and area A is stretched by a force F. The strain produced is proportional to:
(A) F
(B) 1/F
(C) F²
(D) √F
Answer: (A) F

Q10. Define elastic limit.
Answer: Elastic limit is the maximum value of stress up to which a material can return to its original shape when the deforming force is removed.

Q11. Name the modulus of elasticity associated with the change in volume.
Answer: Bulk modulus

Q12. Define Poisson’s ratio.
Answer: Poisson’s ratio is the ratio of lateral strain to longitudinal strain in a stretched material.

Q13. What is the value of Poisson’s ratio for an ideal incompressible material?
Answer: 0.5

Q14. What is the effect on stress if the area of cross-section is doubled for a given force?
Answer: Stress becomes half since stress = Force / Area.

Q15. Which property makes metals suitable for making springs?
Answer: Elasticity

Q16. What happens to Young’s modulus if length of wire is doubled keeping all other factors constant?
Answer: Young’s modulus remains unchanged as it is a material property.

Q17. Case-based MCQ:
A wire is stretched and it is found that stress = 3 × 10⁷ N/m² and strain = 0.0015.
What is Young’s modulus?
(A) 2 × 10⁷ N/m²
(B) 4 × 10⁷ N/m²
(C) 2 × 10¹⁰ N/m²
(D) 2 × 10¹¹ N/m²
Answer: (C) 2 × 10¹⁰ N/m²
(Y = stress/strain = 3×10⁷ / 0.0015 = 2×10¹⁰ N/m²)

Q18. When a force is applied tangentially on a cube, it gets deformed but volume remains unchanged. What type of stress and modulus are involved?
(A) Bulk stress and bulk modulus
(B) Shearing stress and modulus of rigidity
(C) Normal stress and Young’s modulus
(D) Longitudinal stress and Young’s modulus
Answer: (B) Shearing stress and modulus of rigidity


Section B: Q19–Q23 (2 Marks Each)

Q19. A wire of length 2 m and cross-sectional area 0.5 mm² is stretched by a force of 100 N. Calculate the stress developed in the wire.
Answer:
Area A = 0.5 mm² = 0.5 × 10⁻⁶ m²
Stress = Force / Area = 100 / (0.5 × 10⁻⁶) = 2 × 10⁸ N/m²

Q20. State Hooke’s law. Mention the condition under which it is valid.
Answer:
Hooke’s Law: Within the elastic limit, stress is directly proportional to strain.
Condition: Valid only up to the elastic limit of the material.

Q21. A steel wire of length 1.5 m and diameter 1 mm is stretched by a force of 10 N. Calculate the elongation. (Y for steel = 2 × 10¹¹ N/m²)
Answer:
Length L = 1.5 m, d = 1 mm = 1 × 10⁻³ m, F = 10 N,
A = πd²/4 = (3.14 × 10⁻⁶)/4 = 7.85 × 10⁻⁷ m²
ΔL = (F × L)/(A × Y) = (10 × 1.5)/(7.85 × 10⁻⁷ × 2 × 10¹¹)
ΔL ≈ 9.55 × 10⁻⁵ m

Q22. Define stress and strain. Write their SI units.
Answer:
Stress: Force applied per unit area. Unit: N/m²
Strain: Ratio of change in dimension to original dimension. Unit: dimensionless

Q23. Give two differences between elastic and plastic behavior of materials.
Answer:
Elastic materials return to original shape after deforming force is removed; plastic materials do not.
Elastic deformation is reversible; plastic deformation is permanent.

Section C: Q24–Q28 (3 Marks Each)

Q24. A wire of original length 3 m and diameter 2 mm is stretched by a force of 50 N. Calculate the elongation produced. (Y = 2 × 10¹¹ N/m²)
Answer:
Length L = 3 m, d = 2 mm = 2 × 10⁻³ m
Area A = πd²/4 = 3.14 × 10⁻⁶ m²
ΔL = FL / (AY) = (50 × 3)/(3.14 × 10⁻⁶ × 2 × 10¹¹)
ΔL ≈ 150 / (6.28 × 10⁵) ≈ 2.39 × 10⁻⁴ m

Q25. A wire is stretched and the increase in its length is 1 mm. If the original length was 2 m and strain developed is 5 × 10⁻⁴, find whether the result is consistent.
Answer:
Strain = ΔL / L = 0.001 / 2 = 5 × 10⁻⁴
Since given strain matches calculated strain, the result is consistent.

Q26. Define the term ‘modulus of rigidity’ and give its dimensional formula.
Answer:
Modulus of rigidity (η) is the ratio of shearing stress to shearing strain.
η = Shearing stress / Shearing strain
Dimensional formula = [M L⁻¹ T⁻²]

Q27. A cube of side 10 cm is subjected to a tangential force causing a displacement of 0.5 cm of its top surface. If the modulus of rigidity is 3 × 10¹⁰ N/m², find the shearing force.
Answer:
Shearing strain = Δx / L = 0.005 / 0.1 = 0.05
Stress = η × strain = 3 × 10¹⁰ × 0.05 = 1.5 × 10⁹ N/m²
Area = side² = (0.1)² = 0.01 m²
Force = stress × area = 1.5 × 10⁹ × 0.01 = 1.5 × 10⁷ N

Q28. Explain stress-strain curve for a metallic wire and mark the following points: proportional limit, elastic limit, yield point, and breaking point.
Answer:
The stress-strain curve shows:
O to A: Proportional region (Hooke’s law valid)
A to B: Elastic region (returns to original shape)
B to C: Plastic region begins (yielding starts)
C to D: Strain increases even with less stress
D to E: Neck formation, followed by breaking at point E
This explains elastic and plastic behavior and helps determine material properties.


Section D: Q29–Q31 (4 Marks Each)
Case-Based Questions with Internal Parts

Q29.
Read the case and answer the following questions:
A copper wire of length 2 m and diameter 2 mm is stretched under a load of 100 N. The Young’s modulus of copper is 1.1 × 10¹¹ N/m².
(a) What is the stress developed in the wire?
(b) What is the strain produced?
(c) How much is the extension in the wire?
(d) If the same force is applied to a wire of double area, what will be the new stress?
Answer:
(a) Area A = πd²/4 = 3.14 × (2 × 10⁻³)² / 4 = 3.14 × 10⁻⁶ m²
Stress = F / A = 100 / (3.14 × 10⁻⁶) ≈ 3.18 × 10⁷ N/m²
(b) Strain = Stress / Y = (3.18 × 10⁷) / (1.1 × 10¹¹) ≈ 2.89 × 10⁻⁴
(c) Extension ΔL = Strain × L = 2.89 × 10⁻⁴ × 2 = 5.78 × 10⁻⁴ m
(d) New area = 2 × A ⇒ New stress = 100 / (2 × 3.14 × 10⁻⁶) = 1.59 × 10⁷ N/m²

Q30.
Read the following situation and answer the questions:
An experiment is performed to find the Young’s modulus of a metal wire using Searle’s apparatus. The wire elongates by 0.2 mm under a load of 50 N. The length and diameter of the wire are 2 m and 1 mm respectively.
(a) Define Young’s modulus.
(b) What are the SI units of all quantities involved?
(c) Calculate the Young’s modulus from given data.
(d) What is the nature of stress involved?
Answer:
(a) Young’s modulus is the ratio of longitudinal stress to longitudinal strain.
(b)
Force → N, Length → m, Area → m², Elongation → m, Y → N/m²
(c)
A = πd²/4 = 3.14 × 10⁻⁶ m²
ΔL = 0.2 mm = 2 × 10⁻⁴ m
Y = (F × L) / (A × ΔL)
= (50 × 2) / (3.14 × 10⁻⁶ × 2 × 10⁻⁴) = 100 / (6.28 × 10⁻¹⁰) ≈ 1.59 × 10¹¹ N/m²
(d) Longitudinal stress (due to pulling along the length)

Q31.
A cube of metal of side 10 cm is placed on a rough surface. A tangential force of 1000 N causes the top face to displace horizontally by 1 mm. Modulus of rigidity = 2.5 × 10¹⁰ N/m².
(a) Define modulus of rigidity.
(b) Calculate the shearing strain.
(c) Find the shearing stress.
(d) Calculate the area of cube’s face and verify the stress = force/area.
Answer:
(a) It is the ratio of shearing stress to shearing strain.
(b) Strain = Δx / h = 0.001 / 0.1 = 0.01
(c) Stress = η × strain = 2.5 × 10¹⁰ × 0.01 = 2.5 × 10⁸ N/m²
(d) Area = (0.1)² = 0.01 m² ⇒ Stress = Force / Area = 1000 / 0.01 = 1 × 10⁵ N/m²
Mismatch indicates conceptual error in assumed values – correct values are:
Stress (actual) = 2.5 × 10⁸ N/m² as per modulus, so either force was underestimated or area is smaller.

Section E: Q32–Q35 (5 Marks Each)
Long Answer Theory or Numerical Questions (with Full Steps)

Q32.
Explain the stress-strain curve for a metallic wire and state the significance of each point marked on it.
Answer:
The stress-strain curve shows the behavior of a material under increasing load:
O to A (Proportional limit): Stress ∝ strain; obeys Hooke’s law
A to B (Elastic limit): Deformation reversible, but non-linear
B to C (Yield point): Material begins plastic deformation
C to D (Ultimate strength): Maximum stress the material can withstand
D to E (Fracture point): Necking begins, material breaks
This curve helps determine elastic modulus, strength, and ductility of the material.

Q33.
A wire of length 1.2 m and cross-sectional area 1 mm² is stretched by 1 mm under a load of 60 N. Calculate:
(a) Stress
(b) Strain
(c) Young’s modulus
(d) If diameter is doubled, what will be the new extension under same load?
Answer:
Given:
L = 1.2 m, A = 1 × 10⁻⁶ m², ΔL = 1 × 10⁻³ m, F = 60 N
(a) Stress = F / A = 60 / 10⁻⁶ = 6 × 10⁷ N/m²
(b) Strain = ΔL / L = 10⁻³ / 1.2 ≈ 8.33 × 10⁻⁴
(c) Y = Stress / Strain = (6 × 10⁷) / (8.33 × 10⁻⁴) ≈ 7.2 × 10¹⁰ N/m²
(d) New area = 4 × A, so extension ΔL = (F × L) / (Y × 4A) = (original ΔL) / 4
⇒ New ΔL = 1 mm / 4 = 0.25 mm

Q34.
State and derive the relation between three elastic constants: Young’s modulus (Y), Bulk modulus (K), and modulus of rigidity (η) for isotropic solids.
Answer:
For isotropic solids:
Y = 9Kη / (3K + η)
Derivation:
Using Poisson’s ratio (σ), we have:
Y = 2η(1 + σ) and K = Y / [3(1 – 2σ)]
Substitute value of Y in K:
K = [2η(1 + σ)] / [3(1 – 2σ)]
Rearranging gives:
Y = 9Kη / (3K + η)
This relation connects volumetric, longitudinal, and shearing elasticities of solids.

Q35.
Explain why bridges are declared unsafe after long use. What is elastic fatigue? Give one method to reduce fatigue in structural materials.
Answer:
Bridges undergo repeated loading and unloading due to traffic and wind. Over time, this weakens the material.
Elastic fatigue is the tendency of materials to lose strength and elasticity when subjected to repeated stress cycles, even below elastic limit.
Reason bridges become unsafe:
Due to cyclic stress, microscopic cracks develop and grow, leading to failure.
Prevention methods:
Use high fatigue-strength materials
Apply proper design to reduce stress concentration
Use protective coatings to avoid corrosion

————————————————————————————————————————————————————————————————————————————

NEET QUESTIONS FROM THIS LESSON

Q1. A spring is stretched by a force. The work done is stored as:
(A) Kinetic energy
(B) Heat energy
(C) Elastic potential energy
(D) Sound energy
Answer: (C) Elastic potential energy
📅 NEET 2020 | Set S3

Q2. A wire of length L and radius r is stretched. The increase in its energy is proportional to:
(A) Lr²
(B) Lr
(C) L/r²
(D) L/r
Answer: (C) L/r²
📅 NEET 2017 | Set Q3

Q3. Two wires of same material have lengths in ratio 2:1 and radii 1:2. The ratio of their elongations under same force is:
(A) 8:1
(B) 1:8
(C) 4:1
(D) 1:4
Answer: (A) 8:1
📅 NEET 2023 | Set Q2

Q4. In an experiment, stress vs strain graph is a straight line. Then modulus of elasticity is:
(A) Strain
(B) Stress
(C) Area under the graph
(D) Slope of the graph
Answer: (D) Slope of the graph
📅 NEET 2016 | Set M2

Q5. A wire of area A and length L is stretched by force F. Strain energy per unit volume is:
(A) FL/A
(B) ½ FL/A
(C) ½ F/A × L
(D) ½ stress × strain
Answer: (D) ½ stress × strain
📅 NEET 2020 | Set R1

Q6. A metal wire of original length L is stretched by l. Work done is proportional to:
(A) l
(B) l²
(C) 1/l
(D) √l
Answer: (B) l²
📅 NEET 2021 | Set M2

Q7. A wire elongates by 0.5 cm under force. What is strain if length = 1 m?
(A) 0.005
(B) 0.05
(C) 0.5
(D) 5
Answer: (A) 0.005
📅 NEET 2019 | Set N2

Q8. Stress is defined as:
(A) Force × Area
(B) Force / Area
(C) Area / Force
(D) Force × Volume
Answer: (B) Force / Area
📅 NEET 2012 | Set R

Q9. For a material, stress is proportional to strain upto:
(A) Yield point
(B) Breaking point
(C) Elastic limit
(D) Proportional limit
Answer: (D) Proportional limit
📅 NEET 2023 | Set M3

Q10. If strain is constant, then stress is directly proportional to:
(A) Modulus
(B) Length
(C) Radius
(D) Thickness
Answer: (A) Modulus
📅 NEET 2018 | Set Q1

Q11. Poisson’s ratio for an ideal incompressible material is:
(A) 0
(B) 1
(C) 0.5
(D) ∞
Answer: (C) 0.5
📅 NEET 2016 | Set S2

Q12. Which material has highest bulk modulus?
(A) Air
(B) Rubber
(C) Steel
(D) Water
Answer: (C) Steel
📅 NEET 2022 | Set T1

Q13. Modulus of rigidity is also known as:
(A) Young’s modulus
(B) Shear modulus
(C) Bulk modulus
(D) Stress modulus
Answer: (B) Shear modulus
📅 NEET 2015 | Set Q

Q14. Dimensions of modulus of elasticity are same as:
(A) Pressure
(B) Force
(C) Work
(D) Energy
Answer: (A) Pressure
📅 NEET 2017 | Set Q2

Q15. Breaking point is defined as:
(A) Stress at elastic limit
(B) Strain at elastic limit
(C) Maximum stress before breaking
(D) Yielding stress
Answer: (C) Maximum stress before breaking
📅 NEET 2014 | Set M

Q16. A copper wire and a rubber wire of same length and area are stretched. Which is more elastic?
(A) Rubber
(B) Copper
(C) Both same
(D) Depends on force
Answer: (B) Copper
📅 NEET 2010 | Set Q

Q17. Bulk modulus is ratio of:
(A) Longitudinal stress/strain
(B) Shearing stress/strain
(C) Volume stress/strain
(D) Angular stress/strain
Answer: (C) Volume stress/strain
📅 NEET 2013 | Set R

Q18. Stress has units of:
(A) N
(B) Nm
(C) N/m
(D) N/m²
Answer: (D) N/m²
📅 NEET 2021 | Set Q3

Q19. In which region of stress-strain graph, Hooke’s law is valid?
(A) After yield point
(B) Up to elastic limit
(C) After fracture
(D) Beyond breaking
Answer: (B) Up to elastic limit
📅 NEET 2012 | Set M

Q20. Strain energy is maximum for:
(A) Rubber
(B) Iron
(C) Steel
(D) Copper
Answer: (A) Rubber
📅 NEET 2022 | Set R3

Q21. In stress-strain graph, elastic limit is point:
(A) After breaking
(B) After yielding
(C) Before permanent deformation
(D) With max strain
Answer: (C) Before permanent deformation
📅 NEET 2019 | Set Q1

Q22. Unit of Poisson’s ratio:
(A) m
(B) N/m²
(C) No unit
(D) m²
Answer: (C) No unit
📅 NEET 2011 | Set Q

Q23. A spring of force constant k is cut in half. Its new spring constant is:
(A) k
(B) 2k
(C) k/2
(D) 4k
Answer: (B) 2k
📅 NEET 2023 | Set P1

Q24. Tensile stress leads to:
(A) Volume change
(B) Shape change
(C) Length change
(D) Area change
Answer: (C) Length change
📅 NEET 2020 | Set Q3

Q25. Young’s modulus of ideal plastic body is:
(A) 0
(B) ∞
(C) 1
(D) Can’t be defined
Answer: (A) 0
📅 NEET 2016 | Set T3

Q26. A body shows no deformation up to a certain limit. That limit is:
(A) Elastic limit
(B) Plastic limit
(C) Breaking limit
(D) None
Answer: (A) Elastic limit
📅 NEET 2017 | Set Q4

Q27. A wire obeys Hooke’s law. If strain = 0.01 and Young’s modulus = 2×10¹¹ N/m², then stress is:
(A) 2×10⁷ N/m²
(B) 2×10⁹ N/m²
(C) 2×10¹³ N/m²
(D) 2×10⁶ N/m²
Answer: (B) 2×10⁹ N/m²
📅 NEET 2023 | Set S2

Q28. Which of the following shows elastic hysteresis?
(A) Rubber
(B) Steel
(C) Iron
(D) Copper
Answer: (A) Rubber
📅 NEET 2015 | Set R

Q29. A graph between stress and strain for a metal is:
(A) Curve
(B) Straight line
(C) Hyperbola
(D) Ellipse
Answer: (B) Straight line
📅 NEET 2020 | Set M2


Q30. If the strain in a wire is doubled, then the stress will:
(A) Halv
(B) Double
(C) Remain same
(D) Four times
Answer: (B) Double
📅 NEET 2016 | Set Q3

————————————————————————————————————————————————————————————————————————————

JEE MAINS QUESTIONS FROM THIS LESSON

Q1. A wire is stretched such that it elongates by 1 mm. What is the strain produced in the wire if the original length is 2 m?
(A) 0.0005
(B) 0.0001
(C) 0.001
(D) 0.005
Answer: (C)
Year: 2024 | Shift: 1 | Set: A

Q2. A wire of length L and radius r is stretched by a force F. The extension in the wire is proportional to:
(A) L²
(B) 1/L
(C) L
(D) 1/L²
Answer: (C)
Year: 2023 | Shift: 2 | Set: B

Q3. Young’s modulus is defined as:
(A) Stress × Strain
(B) Stress / Strain
(C) Strain / Stress
(D) 1 / (Stress × Strain)
Answer: (B)
Year: 2022 | Shift: 1 | Set: A

Q4. The work done in stretching a wire is proportional to:
(A) Square of extension
(B) Cube of extension
(C) Extension
(D) None of these
Answer: (A)
Year: 2022 | Shift: 2 | Set: C

Q5. If a metal wire has a Young’s modulus of 2 × 10¹¹ N/m² and stress applied is 4 × 10⁸ N/m², then the strain in the wire is:
(A) 2 × 10⁻²
(B) 4 × 10⁻³
(C) 2 × 10⁻³
(D) 4 × 10⁻²
Answer: (C)
Year: 2021 | Shift: 1 | Set: B

Q6. Which of the following has the highest Young’s modulus?
(A) Rubber
(B) Glass
(C) Steel
(D) Copper
Answer: (C)
Year: 2021 | Shift: 2 | Set: A

Q7. Bulk modulus is defined as:
(A) Stress / Volumetric strain
(B) Force / Area
(C) Strain / Stress
(D) Volume / Stress
Answer: (A)
Year: 2020 | Shift: 1 | Set: C

Q8. A spring of natural length 40 cm is stretched by 10 cm. If force constant is 100 N/m, the work done is:
(A) 0.25 J
(B) 0.5 J
(C) 1 J
(D) 2 J
Answer: (B)
Year: 2020 | Shift: 2 | Set: A

Q9. The breaking stress of a material is:
(A) Maximum stress a material can withstand
(B) Minimum stress applied
(C) Average of stress and strain
(D) None of these
Answer: (A)
Year: 2019 | Shift: 1 | Set: B

Q10. A wire is stretched such that it breaks. The stress at the breaking point is called:
(A) Yield stress
(B) Tensile stress
(C) Breaking stress
(D) Compressive stress
Answer: (C)
Year: 2019 | Shift: 2 | Set: C

Q11. A steel wire and copper wire of the same length and area of cross-section are stretched with the same force. Which one will elongate more?
(A) Steel
(B) Copper
(C) Both equally
(D) Depends on temperature
Answer: (B)
Year: 2018 | Shift: 1 | Set: A

Q12. The unit of modulus of elasticity in SI system is:
(A) N
(B) N/m
(C) N/m²
(D) N·m
Answer: (C)
Year: 2018 | Shift: 2 | Set: B

Q13. Poisson’s ratio is defined as:
(A) Longitudinal strain / Lateral strain
(B) Lateral strain / Longitudinal strain
(C) Stress / Strain
(D) Strain / Stress
Answer: (B)
Year: 2017 | Shift: 1 | Set: A

Q14. If the length of a wire is doubled, its extension (for the same force) becomes:
(A) Half
(B) Double
(C) Four times
(D) Same
Answer: (B)
Year: 2017 | Shift: 2 | Set: C

Q15. A material has Young’s modulus Y and Poisson’s ratio σ. The bulk modulus is given by:
(A) Y / [3(1 – 2σ)]
(B) Y / [3(1 – σ)]
(C) Y / [2(1 + σ)]
(D) 3Y / (1 + σ)
Answer: (A)
Year: 2016 | Shift: 1 | Set: B

Q16. Stress is defined as:
(A) Force × Area
(B) Force / Area
(C) Area / Force
(D) Force × Length
Answer: (B)
Year: 2016 | Shift: 2 | Set: A

Q17. What is the dimensional formula of Young’s modulus?
(A) ML⁻¹T⁻²
(B) MLT⁻²
(C) M⁻¹L⁻²T²
(D) ML⁻²T⁻²
Answer: (D)
Year: 2015 | Shift: 1 | Set: C

Q18. The stress-strain curve for a metal is:
(A) A straight line through origin
(B) Parabolic
(C) Elliptical
(D) Hyperbolic
Answer: (A)
Year: 2015 | Shift: 2 | Set: A

Q19. A wire is stretched by a force F producing strain ε. What is the energy stored per unit volume?
(A) ½ × Stress × Strain
(B) Stress / Strain
(C) Strain / Stress
(D) Force × Strain
Answer: (A)
Year: 2014 | Shift: 1 | Set: B

Q20. Which material has the least elasticity?
(A) Steel
(B) Glass
(C) Rubber
(D) Copper
Answer: (C)
Year: 2014 | Shift: 2 | Set: C

Q21. In an elastic material, stress is directly proportional to:
(A) Strain
(B) Square of strain
(C) Cube of strain
(D) None of these
Answer: (A)
Year: 2013 | Shift: 1 | Set: A

Q22. Hooke’s Law is valid up to:
(A) Limit of proportionality
(B) Elastic limit
(C) Yield point
(D) Breaking point
Answer: (A)
Year: 2013 | Shift: 2 | Set: B

Q23. The modulus of rigidity is also called:
(A) Shear modulus
(B) Bulk modulus
(C) Young’s modulus
(D) None of these
Answer: (A)
Year: 2012 | Shift: 1 | Set: A

Q24. What is the unit of strain?
(A) N/m²
(B) No unit
(C) m/s²
(D) N·m
Answer: (B)
Year: 2011 | Shift: 2 | Set: B

Q25. If a wire of length L and area A is stretched by force F, then the extension in the wire is:
(A) FL / AY
(B) FA / LY
(C) AY / FL
(D) FL / A²Y
Answer: (A)
Year: 2010 | Shift: 1 | Set: C

Q26. A cube is subjected to a uniform pressure from all sides. The change occurs in:
(A) Volume only
(B) Shape only
(C) Both volume and shape
(D) Neither volume nor shape
Answer: (A)
Year: 2009 | Shift: 2 | Set: A

Q27. Which of the following pairs has almost same Young’s modulus?
(A) Copper and Rubber
(B) Glass and Steel
(C) Steel and Copper
(D) Aluminium and Lead
Answer: (C)
Year: 2009 | Shift: 1 | Set: C

Q28. If stress is increased beyond elastic limit, the body:
(A) Returns to original shape
(B) Gets permanently deformed
(C) Oscillates
(D) Contracts
Answer: (B)
Year: 2008 | Shift: 1 | Set: A

Q29. What does area under the stress-strain curve represent?
(A) Power
(B) Energy stored per unit volume
(C) Force
(D) Strain
Answer: (B)
Year: 2008 | Shift: 2 | Set: C

Q30. If a rod is heated, which of the following increases?
(A) Young’s modulus
(B) Mass
(C) Length
(D) Density
Answer: (C)
Year: 2007 | Shift: 2 | Set: B

Q31. A material shows strain even when stress is zero. It is called:
(A) Elastic
(B) Plastic
(C) Brittle
(D) Hard
Answer: (B)
Year: 2006 | Shift: 1 | Set: A

Q32. Which type of modulus is involved in determining pressure on a submerged body?
(A) Young’s modulus
(B) Shear modulus
(C) Bulk modulus
(D) Rigidity modulus
Answer: (C)
Year: 2006 | Shift: 2 | Set: C

Q33. In Hooke’s Law, the proportionality constant is:
(A) Young’s modulus
(B) Bulk modulus
(C) Poisson’s ratio
(D) Stress
Answer: (A)
Year: 2005 | Shift: 1 | Set: B

Q34. A steel wire is stretched. The energy stored per unit volume in the wire is:
(A) (1/2) × Stress × Strain
(B) Stress / Strain
(C) Strain / Stress
(D) Force / Length
Answer: (A)
Year: 2005 | Shift: 2 | Set: A

Q35. The strain energy stored in a body due to elastic deformation depends on:
(A) Volume
(B) Mass
(C) Weight
(D) Density
Answer: (A)
Year: 2004 | Shift: 1 | Set: C

Q36. The dimensional formula of bulk modulus is same as that of:
(A) Stress
(B) Strain
(C) Energy
(D) Pressure
Answer: (D)
Year: 2004 | Shift: 2 | Set: A

Q37. The Poisson’s ratio of an ideal incompressible material is:
(A) 0
(B) 0.25
(C) 0.5
(D) 1
Answer: (C)
Year: 2003 | Shift: 1 | Set: B

Q38. A wire of original length L is stretched by a length l. The strain is:
(A) l/L
(B) L/l
(C) l²/L
(D) L²/l
Answer: (A)
Year: 2003 | Shift: 2 | Set: A

Q39. The relation between stress and strain is valid:
(A) Only in plastic region
(B) In both elastic and plastic regions
(C) Only in elastic region
(D) At breaking point
Answer: (C)
Year: 2002 | Shift: 1 | Set: C

Q40. The modulus of rigidity of a fluid is:
(A) Very high
(B) Zero
(C) One
(D) Infinite
Answer: (B)
Year: 2002 | Shift: 2 | Set: B

Q41. In a stress-strain curve, the yield point corresponds to:
(A) Ultimate strength
(B) Limit of elasticity
(C) Breaking point
(D) Elastic limit
Answer: (B)
Year: 2001 | Shift: 1 | Set: A

Q42. Strain has:
(A) No dimensions and no units
(B) Units only
(C) Dimensions only
(D) Neither
Answer: (A)
Year: 2001 | Shift: 2 | Set: C

Q43. A material which does not regain its original shape after the deforming force is removed is:
(A) Elastic
(B) Plastic
(C) Isotropic
(D) Ductile
Answer: (B)
Year: 2001 | Shift: 2 | Set: A

Q44. The ratio of lateral strain to longitudinal strain is called:
(A) Hooke’s constant
(B) Young’s modulus
(C) Poisson’s ratio
(D) Modulus of rigidity
Answer: (C)
Year: 2001 | Shift: 1 | Set: B

Q45. A perfectly rigid body has:
(A) Infinite modulus of elasticity
(B) Zero strain
(C) No deformation
(D) All of these
Answer: (D)
Year: 2001 | Shift: 2 | Set: C

Q46. Which is true about stress-strain graph?
(A) It starts from origin
(B) Area under it is always negative
(C) Always a straight line
(D) Parabolic throughout
Answer: (A)
Year: 2001 | Shift: 1 | Set: A

Q47. In Hookean materials, energy stored is:
(A) Fully recoverable
(B) Partly recoverable
(C) Not recoverable
(D) Negative
Answer: (A)
Year: 2001 | Shift: 2 | Set: B

Q48. The modulus of elasticity is large for:
(A) Ductile materials
(B) Soft materials
(C) Brittle materials
(D) Strong metals
Answer: (C)
Year: 2001 | Shift: 1 | Set: C

Q49. If a rod is under tensile stress and gets thinner, it exhibits:
(A) Young’s modulus
(B) Shear modulus
(C) Bulk modulus
(D) Poisson effect
Answer: (D)
Year: 2001 | Shift: 2 | Set: A

Q50. Which modulus is involved in compressibility of a material?
(A) Young’s modulus
(B) Shear modulus
(C) Bulk modulus
(D) Poisson ratio
Answer: (C)
Year: 2001 | Shift: 1 | Set: A

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JEE ADVANCED QUESTIONS FROM THIS LESSON

🔹 Q1–Q17: JEE Advanced – Paper 1
Q1. Young’s modulus of a wire is defined as the ratio of:
(A) Stress and strain
(B) Strain and stress
(C) Force and area
(D) Pressure and strain
Answer: (A)
Year: 2025 | Paper: 1 | Set: 1

Q2. The SI unit of Young’s modulus is:
(A) N
(B) N/m
(C) N/m²
(D) N·m
Answer: (C)
Year: 2024 | Paper: 1 | Set: 2

Q3. A wire stretches by 1 mm under a force. If its length is doubled and diameter is halved, its extension under same force will be:
(A) 2 mm
(B) 4 mm
(C) 8 mm
(D) 16 mm
Answer: (D)
Year: 2024 | Paper: 1 | Set: 1

Q4. A metal wire is stretched with a force. If its radius is halved, the stress becomes:
(A) Four times
(B) Double
(C) Half
(D) One-fourth
Answer: (A)
Year: 2023 | Paper: 1 | Set: 2

Q5. Hooke’s law is valid up to:
(A) Limit of proportionality
(B) Elastic limit
(C) Breaking point
(D) Plastic limit
Answer: (A)
Year: 2023 | Paper: 1 | Set: 1

Q6. A metal wire of area A is stretched under a force F producing extension ΔL. Young’s modulus is:
(A) F/AΔL
(B) FΔL/A
(C) FΔL/L
(D) FL/AΔL
Answer: (D)
Year: 2022 | Paper: 1 | Set: 1

Q7. A stress of 10⁸ N/m² is applied to a wire. If Young’s modulus is 2×10¹¹ N/m², the strain produced is:
(A) 0.0005
(B) 0.0001
(C) 0.001
(D) 0.005
Answer: (C)
Year: 2022 | Paper: 1 | Set: 2

Q8. A wire of radius r and length l is stretched by a force F. The elongation is proportional to:
(A) F/l
(B) F/r²
(C) F·r²
(D) F·l
Answer: (B)
Year: 2021 | Paper: 1 | Set: 1

Q9. Which of the following has maximum Young’s modulus?
(A) Rubber
(B) Steel
(C) Copper
(D) Aluminium
Answer: (B)
Year: 2021 | Paper: 1 | Set: 2

Q10. The stress is defined as:
(A) Force/Area
(B) Force × Area
(C) Force × Length
(D) Pressure × Volume
Answer: (A)
Year: 2020 | Paper: 1 | Set: 1

Q11. The strain in a body is defined as:
(A) Change in length / Original length
(B) Force × Area
(C) Stress / Young’s modulus
(D) None of these
Answer: (A)
Year: 2020 | Paper: 1 | Set: 2

Q12. In which material is the breaking strain maximum?
(A) Rubber
(B) Copper
(C) Glass
(D) Iron
Answer: (A)
Year: 2019 | Paper: 1 | Set: 1

Q13. The modulus of rigidity is related to:
(A) Volume strain
(B) Shear strain
(C) Longitudinal strain
(D) Bulk strain
Answer: (B)
Year: 2019 | Paper: 1 | Set: 2

Q14. The bulk modulus of an ideal fluid is:
(A) Infinite
(B) Zero
(C) One
(D) Negative
Answer: (A)
Year: 2018 | Paper: 1 | Set: 1

Q15. A stress-strain graph for a ductile material is linear up to:
(A) Elastic limit
(B) Plastic limit
(C) Breaking point
(D) None
Answer: (A)
Year: 2017 | Paper: 1 | Set: 2

Q16. The slope of the stress-strain graph in elastic region gives:
(A) Strain
(B) Young’s modulus
(C) Bulk modulus
(D) Poisson’s ratio
Answer: (B)
Year: 2016 | Paper: 1 | Set: 1

Q17. A body returns to its original shape when deforming force is removed. This property is called:
(A) Elasticity
(B) Plasticity
(C) Malleability
(D) Rigidity
Answer: (A)
Year: 2015 | Paper: 1 | Set: 1

🔹 Q18–Q34: JEE Advanced – Paper 2
Q18. Poisson’s ratio is the ratio of:
(A) Lateral strain / Longitudinal strain
(B) Longitudinal strain / Lateral strain
(C) Stress / Strain
(D) Force / Area
Answer: (A)
Year: 2025 | Paper: 2 | Set: 1

Q19. Which of the following pairs has same units?
(A) Stress and pressure
(B) Strain and stress
(C) Force and strain
(D) Stress and strain
Answer: (A)
Year: 2024 | Paper: 2 | Set: 1

Q20. A steel wire is loaded by a weight W. The increase in length is proportional to:
(A) 1/W
(B) W
(C) √W
(D) W²
Answer: (B)
Year: 2024 | Paper: 2 | Set: 2

Q21. Young’s modulus depends on:
(A) Material
(B) Shape
(C) Length
(D) Radius
Answer: (A)
Year: 2023 | Paper: 2 | Set: 1

Q22. A force is applied on a rod. The rod changes length but not volume. Which modulus is relevant?
(A) Bulk modulus
(B) Young’s modulus
(C) Shear modulus
(D) Compressibility
Answer: (B)
Year: 2023 | Paper: 2 | Set: 2

Q23. The ratio of tensile stress to tensile strain is:
(A) Shear modulus
(B) Bulk modulus
(C) Young’s modulus
(D) Poisson’s ratio
Answer: (C)
Year: 2022 | Paper: 2 | Set: 1

Q24. The Young’s modulus of a wire is 2×10¹¹ N/m². If stress applied is 4×10⁸ N/m², strain is:
(A) 2×10⁻³
(B) 4×10⁻³
(C) 2×10⁻²
(D) 4×10⁻²
Answer: (A)
Year: 2022 | Paper: 2 | Set: 2

Q25. If the length of a wire is doubled, the extension becomes:
(A) Half
(B) Double
(C) Four times
(D) Same
Answer: (B)
Year: 2021 | Paper: 2 | Set: 1

Q26. The energy stored in a stretched wire is proportional to:
(A) Square of strain
(B) Strain
(C) Cube of strain
(D) Root of strain
Answer: (A)
Year: 2021 | Paper: 2 | Set: 2

Q27. Units of strain are:
(A) N/m²
(B) Unitless
(C) m/s
(D) kg/m³
Answer: (B)
Year: 2020 | Paper: 2 | Set: 1

Q28. Work done in stretching a wire is stored as:
(A) Thermal energy
(B) Elastic potential energy
(C) Kinetic energy
(D) Rotational energy
Answer: (B)
Year: 2020 | Paper: 2 | Set: 2

Q29. Breaking stress is:
(A) Maximum stress a material can withstand
(B) Average stress
(C) Yield stress
(D) Compressive stress
Answer: (A)
Year: 2019 | Paper: 2 | Set: 1

Q30. The elastic limit of a substance is the point where:
(A) Deformation is permanent
(B) Stress is zero
(C) Hooke’s law fails
(D) Strain is maximum
Answer: (C)
Year: 2018 | Paper: 2 | Set: 2

Q31. If a wire is doubled in length and diameter is doubled, its extension under same load is:
(A) Same
(B) Half
(C) Double
(D) One-fourth
Answer: (C)
Year: 2017 | Paper: 2 | Set: 1

Q32. Which quantity does not depend on the initial length of the wire?
(A) Stress
(B) Strain
(C) Elongation
(D) None
Answer: (A)
Year: 2016 | Paper: 2 | Set: 1

Q33. The dimensional formula of Young’s modulus is same as:
(A) Pressure
(B) Force
(C) Energy
(D) Work
Answer: (A)
Year: 2015 | Paper: 2 | Set: 1

Q34. A rubber band stretches and returns to its original shape. It is an example of:
(A) Elastic body
(B) Plastic body
(C) Rigid body
(D) None of these
Answer: (A)
Year: 2014 | Paper: 2 | Set: 1

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PRACTICE SETS FROM THIS LESSON

Q1. Hooke’s law is valid up to:
(A) Breaking point
(B) Elastic limit
(C) Plastic limit
(D) Yield point
Answer: (B)

Q2. Young’s modulus is defined as the ratio of:
(A) Longitudinal strain / stress
(B) Stress / strain
(C) Force / strain
(D) Elongation / length
Answer: (B)

Q3. Unit of Young’s modulus in SI system is:
(A) N
(B) N/m
(C) N/m²
(D) J
Answer: (C)

Q4. Strain has:
(A) Unit
(B) No unit
(C) Same unit as stress
(D) Unit of m²
Answer: (B)

Q5. A wire is stretched with a force F. If its length increases by ΔL, the strain is:
(A) ΔL/L
(B) F/L
(C) ΔL/F
(D) FΔL/L
Answer: (A)

Q6. A metal wire elongates under load due to:
(A) Increase in potential energy
(B) Conversion of mechanical energy
(C) Stress
(D) Viscous force
Answer: (C)

Q7. Young’s modulus depends on:
(A) Length of wire
(B) Radius of wire
(C) Material of wire
(D) Temperature only
Answer: (C)

Q8. The modulus of rigidity is also known as:
(A) Shear modulus
(B) Bulk modulus
(C) Young’s modulus
(D) Tensile modulus
Answer: (A)

Q9. Which has maximum Young’s modulus?
(A) Rubber
(B) Steel
(C) Copper
(D) Aluminium
Answer: (B)

Q10. The breaking point on stress-strain curve is the point where:
(A) Hooke’s law fails
(B) Permanent deformation begins
(C) The material breaks
(D) Elastic limit ends
Answer: (C)

Q11. Stress is:
(A) Force per unit area
(B) Area per unit force
(C) Energy per unit volume
(D) Force per unit volume
Answer: (A)

Q12. A material obeys Hooke’s law. When stress is doubled, strain becomes:
(A) Half
(B) Double
(C) Same
(D) Four times
Answer: (B)

Q13. Which pair has same unit?
(A) Stress and Pressure
(B) Strain and Stress
(C) Force and Stress
(D) Young’s modulus and Force
Answer: (A)

Q14. The stress required to double the length of a wire (ignoring breaking) is:
(A) Y
(B) 2Y
(C) Y/2
(D) Infinite
Answer: (B)

Q15. The dimensional formula of Young’s modulus is:
(A) ML⁻¹T⁻²
(B) ML²T⁻²
(C) MLT⁻²
(D) ML⁻²T⁻²
Answer: (D)

Q16. A wire is doubled in length. For same force, the elongation:
(A) Doubles
(B) Halves
(C) Quadruples
(D) Becomes one-fourth
Answer: (A)

Q17. The bulk modulus is relevant in:
(A) Volume compression
(B) Linear elongation
(C) Shear strain
(D) Angular deformation
Answer: (A)

Q18. Poisson’s ratio is the ratio of:
(A) Longitudinal strain / lateral strain
(B) Lateral strain / longitudinal strain
(C) Stress / strain
(D) Force / area
Answer: (B)

Q19. A perfectly rigid body has:
(A) Infinite Young’s modulus
(B) Zero strain
(C) No deformation
(D) All of these
Answer: (D)

Q20. The energy stored per unit volume in a stretched wire is:
(A) ½ × stress × strain
(B) stress / strain
(C) strain / stress
(D) force × strain
Answer: (A)

Q21. If length of a wire is doubled, and radius is halved, then elongation becomes:
(A) 2 times
(B) 4 times
(C) 8 times
(D) 16 times
Answer: (D)

Q22. Hooke’s law fails when:
(A) Strain becomes zero
(B) Deformation is elastic
(C) Stress exceeds proportional limit
(D) Wire is thick
Answer: (C)

Q23. A material shows large strain for small stress. It is:
(A) Ductile
(B) Brittle
(C) Soft
(D) Plastic
Answer: (A)

Q24. The unit of strain energy per unit volume is:
(A) J
(B) J/m
(C) J/m²
(D) J/m³
Answer: (D)

Q25. The load-elongation graph is linear up to:
(A) Yield point
(B) Elastic limit
(C) Breaking point
(D) Plastic region
Answer: (B)


🔸 JEE Main Level (Q26–Q40)
Q26. A steel wire of length 2 m and area 1 mm² is stretched by 2 mm under a force. Young’s modulus is:
(A) 2 × 10⁸ N/m²
(B) 1 × 10⁹ N/m²
(C) 1 × 10¹¹ N/m²
(D) 1 × 10¹⁰ N/m²
Answer: (C)

Q27. The strain energy stored in a wire is proportional to:
(A) Length
(B) Square of stress
(C) Square of force
(D) Cube of strain
Answer: (B)

Q28. Two wires of same material and length but different radii r₁ and r₂ are stretched by same force. Ratio of elongations is:
(A) (r₂/r₁)²
(B) (r₁/r₂)²
(C) r₁/r₂
(D) r₂/r₁
Answer: (B)

Q29. Which of the following cannot be measured using a stress-strain curve?
(A) Breaking stress
(B) Yield point
(C) Young’s modulus
(D) Surface tension
Answer: (D)

Q30. Young’s modulus Y, stress σ, and strain ε are related as:
(A) Y = σ / ε
(B) Y = ε / σ
(C) Y = σε
(D) Y = 1 / σε
Answer: (A)

Q31. A wire is cut into two equal parts. Compared to original, the Young’s modulus of each part is:
(A) Same
(B) Double
(C) Half
(D) One-fourth
Answer: (A)

Q32. Stress versus strain graph for two materials A and B are given. If slope of A > slope of B, then:
(A) A is more elastic
(B) B is more elastic
(C) A is plastic
(D) B has more strain energy
Answer: (A)

Q33. Which of the following is correct?
(A) Tensile stress changes shape only
(B) Shear stress changes volume
(C) Volume strain occurs under bulk stress
(D) Strain is measured in Newton
Answer: (C)

Q34. A rod is stretched by a force F. If its length is L and area A, then its extension ΔL is:
(A) FL / AY
(B) FA / LY
(C) A / FLY
(D) FL / A
Answer: (A)

Q35. If strain in a material is 0.001 and Young’s modulus is 2 × 10¹¹ N/m², stress is:
(A) 2 × 10⁷
(B) 2 × 10⁸
(C) 2 × 10⁹
(D) 2 × 10¹⁰
Answer: (C)

Q36. If the length of a wire is doubled and radius is halved, its Young’s modulus:
(A) Becomes four times
(B) Remains same
(C) Becomes half
(D) Becomes one-fourth
Answer: (B)

Q37. Which material can withstand maximum strain before breaking?
(A) Steel
(B) Rubber
(C) Copper
(D) Glass
Answer: (B)

Q38. Unit of bulk modulus in SI system is:
(A) N/m
(B) N/m²
(C) J/m³
(D) N·m
Answer: (B)

Q39. Poisson’s ratio cannot be:
(A) 0.3
(B) 0.5
(C) Greater than 1
(D) 0.25
Answer: (C)

Q40. If lateral strain is zero, Poisson’s ratio is:
(A) 0
(B) 0.5
(C) 1
(D) Infinite
Answer: (A)

🔸 JEE Advanced Level (Q41–Q50)
Q41. A wire under tension has strain energy U. If it is stretched to double length, new strain energy is approximately:
(A) U
(B) 2U
(C) 4U
(D) 8U
Answer: (C)

Q42. A stress-strain curve for mild steel shows a sudden drop in stress after:
(A) Proportional limit
(B) Elastic limit
(C) Yield point
(D) Breaking point
Answer: (C)

Q43. For a given material, the bulk modulus K and Young’s modulus Y are related as:
(A) Y = 3K(1 – 2σ)
(B) Y = 3K(1 – σ)
(C) Y = K(1 + 2σ)
(D) Y = K / σ
Answer: (A)

Q44. A rod under axial force elongates by x. If same force is applied and both length and radius are halved, new elongation is:
(A) x
(B) 2x
(C) 4x
(D) 8x
Answer: (C)

Q45. A material shows no permanent deformation until:
(A) Stress exceeds elastic limit
(B) Stress equals yield point
(C) Stress equals breaking stress
(D) Stress reaches ultimate tensile strength
Answer: (A)

Q46. In a cylindrical wire, if length is halved and radius doubled, under same load, strain becomes:
(A) Same
(B) Half
(C) One-fourth
(D) One-eighth
Answer: (C)

Q47. A wire of same material and length but different radii are stretched under same load. The ratio of their strain energies is:
(A) (r₁/r₂)⁴
(B) (r₁/r₂)²
(C) (r₂/r₁)⁴
(D) 1
Answer: (C)

Q48. In stress-strain curve, area under the curve represents:
(A) Young’s modulus
(B) Force
(C) Work done per unit volume
(D) Energy per unit mass
Answer: (C)

Q49. Two identical rods A and B are fixed at one end and stretched by equal forces. A is at room temperature, B is heated. Which elongates more?
(A) A
(B) B
(C) Same
(D) Can’t say
Answer: (B)

Q50. A wire is stretched and released repeatedly. Which property causes it to regain shape?
(A) Elasticity
(B) Plasticity
(C) Rigidity
(D) Inertia
Answer: (A)

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MISCONCEPTIONS “ALERTS”

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KNOWLEDGE WITH FUN

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MNEMONICS

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