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Mechanical Properties of Solids
Physics · Class 11NCERT pp. 167–1762 sets × 25 questions
Key facts and formulas for Mechanical Properties of Solids, three worked sample questions, and two full practice sets. Every question has a hint, a reason for every option and a step-by-step solution, with a calculator or animation where one helps. English and Hinglish.
Formula sheet for this chapter
| Idea | Rule or fact |
|---|
| Stress and strain | stress = F / A (N m⁻²); strain = change / original, no unit |
| Young's modulus | Y = (F / A) / (ΔL / L), so ΔL = F L / (A Y); extension ∝ L / d² |
| Shear modulus | G = (F / A) / θ, shear strain θ is an angle in radians |
| Bulk modulus | B = −p / (ΔV / V); solids > liquids > gases; isothermal gas B = p |
| Poisson's ratio | σ = lateral strain / longitudinal strain; unitless; maximum 0.5 |
| Elastic energy | W = ½ F l = ½ k l²; u = ½ σ ε = σ² / 2Y per unit volume; k = Y A / L |
| Constants | Y = 3B(1 − 2σ); Y = 2G(1 + σ); G ≈ Y/3 |
| Stress-strain curve | O→A Hooke's law, B yield point, D ultimate strength, E fracture; beyond B permanent set |
| Other results | Thermal stress = Y α ΔT; wave speed v = √(Y/ρ) or √(B/ρ); fluid G = 0 |
| Material constants | Y, G, B and σ depend only on the material, not on size or shape |
Three sample questions
Q1. Which quantity has the same dimensions as Young's modulus?
- force
- energy
- pressure
- strain
Hint and solution
Hint: Young's modulus is stress divided by strain. Strain has no dimensions, so which quantity matches stress?
- Y = stress / strain — Young's modulus is defined as stress divided by strain.
- strain = ΔL / L → no dimensions — Strain is a ratio of two lengths, so it has no dimensions. Dividing by it changes nothing.
- [Y] = [F / A] = [M L⁻¹ T⁻²] — So Y has the dimensions of stress, which are force per area, the same as pressure. Option C.
Answer: C. Young's modulus has the unit of stress, N m⁻², which is also the unit of pressure: [M L⁻¹ T⁻²].
Q2. Assertion A: Strain has no unit. Reason R: Strain is the ratio of two quantities of the same kind.
- Both A and R are true and R explains A
- Both A and R are true but R does not explain A
- A is true but R is false
- A is false but R is true
Hint and solution
Hint: Ask whether R gives the actual reason for A. What happens to units when you divide a length by a length?
- strain = ΔL / L, Δx / L, ΔV / V — Longitudinal, shear and volume strains are each a change divided by an original value of the same kind.
- m / m = 1 → no unit — Same units cancel in the ratio, so strain has no unit. Statement A is true.
- R gives the reason for A — Statement R is true and it is the cause of A. So the answer is: both true and R explains A. Option A.
Answer: A. ΔL/L, Δx/L and ΔV/V are all ratios of like quantities, so they are dimensionless. R explains A.
Q3. Which statements about the stress-strain curve of a ductile metal are correct? (i) Hooke's law holds from O to A. (ii) B is the elastic limit (yield point). (iii) Unloading beyond B leaves a permanent set. (iv) The wire breaks at D.
- (i) and (iv) only
- (ii), (iii) and (iv) only
- (iii) and (iv) only
- (i), (ii) and (iii) only
Hint and solution
Hint: Recall the labelled points: A is the proportional limit, B the yield point, D the highest stress. Where does the wire actually break?
- O→A linear, B yield, D peak, E break — Mark the standard points: proportional to A, elastic limit at B, maximum stress at D, fracture at E.
- (i), (ii), (iii) are standard facts — Hooke's law holds from O to A, B is the yield point, and beyond B some permanent set remains. All three are true.
- (iv) false: D is not the break point — D is the largest stress, but the wire breaks at E, so (iv) fails. Statements (i), (ii) and (iii) are correct. Option D.
Answer: D. D is the ultimate tensile strength, the largest stress. The wire breaks at E, so (iv) is false.
How to use this chapter
Read the NCERT pages first, then attempt Set 1 without the hints. Note every wrong answer in a mistake list, read the reason for each option, and retry Set 2 two days later. A topic is solid when you can say why each wrong option is wrong.