Poisson's Ratio - UNSOLVED PRACTICE SET

Q7. Define Poisson's ratio. Why is there a negative sign in its definition?

Q8. A wire of original diameter 2 mm is stretched by 0.2%. If Poisson's ratio is 0.3, calculate the decrease in diameter.

Q9. Explain why Poisson's ratio cannot be greater than 0.5 for most common materials. What would happen if it were?

Q10. In your school, a student stretches a rubber band and notices it becomes noticeably thinner. Another student stretches a steel wire of the same original diameter by the same percentage and observes very little change in diameter. What does this tell you about their Poisson's ratios?

Q11. A cube of side L is compressed uniformly. If Poisson's ratio is σ, derive the expression for the change in volume in terms of longitudinal strain.

Q12. Why is Poisson's ratio important in engineering design? Give one practical example.

Q13. Define Poisson's ratio and explain its physical significance. Discuss:

(i) Why it is dimensionless

(ii) Why it is typically positive for most materials

(iii) The theoretical limits (−1 to 0.5) and what they represent physically

(iv) Materials with negative Poisson's ratio (auxetic materials)

(v) The relationship between Poisson's ratio and the three elastic moduli

Q14. A steel rod of length 2 m and diameter 10 mm is subjected to a tensile force of 15.7 kN. Young's modulus of steel is 2 × 10¹¹ Pa and Poisson's ratio is 0.3.

(a) Calculate the longitudinal strain.

(b) Calculate the extension of the rod.

(c) Calculate the lateral strain.

(d) Calculate the decrease in diameter.

(e) Calculate the percentage change in cross-sectional area.

(f) Verify that the volume change is consistent with the value of Poisson's ratio.

Q15. Discuss the concept of volume change during deformation:

(a) Show that for a material under longitudinal stress, the fractional volume change is given by ΔV/V = (1 − 2σ) × (ΔL/L)

(b) What does this imply for σ = 0.5?

(c) What does this imply for σ = 0?

(d) For a material with σ = 0.25, calculate the percentage volume change when the longitudinal strain is 0.01.

(e) Discuss why cork has Poisson's ratio ≈ 0 and rubber has σ ≈ 0.5.

Q16. A wire of original length 3 m and diameter 2 mm is stretched by 3 mm. The diameter decreases to 1.996 mm.

(a) Calculate the longitudinal strain.

(b) Calculate the lateral strain.

(c) Calculate Poisson's ratio.

(d) If Young's modulus is 7 × 10¹⁰ Pa, calculate the stretching force.

(e) Calculate the percentage change in the cross-sectional area.

(f) Verify that the volume of the wire remains approximately constant.

Q17. The following data is given for three materials:

Table

Material Y (GPa) η (GPa) K (GPa)

A 200 80 140

B 70 26 70

C 3 1 1

(a) Calculate Poisson's ratio for each material using Y = 2η(1 + σ).

(b) Verify your answers using Y = 3K(1 − 2σ).

(c) Classify each material as metallic, polymeric, or rubber-like.

(d) For material A, calculate the percentage change in volume when the longitudinal strain is 0.001.

(e) Discuss why the two formulas give slightly different results for real materials.

Q18. In a school project, students design an experiment to measure Poisson's ratio for a rubber tube:

(a) Describe how you would measure the change in length and change in diameter when the tube is stretched.

(b) List the instruments needed and their least counts.

(c) A rubber tube of original length 20 cm and internal diameter 1 cm stretches by 2 cm when pulled. The external diameter decreases from 1.2 cm to 1.18 cm. Calculate Poisson's ratio.

(d) Discuss why measuring Poisson's ratio for rubber is more challenging than for metals.

(e) Suggest two applications where knowing Poisson's ratio is critical.