Young's Modulus, Bulk Modulus, Shear Modulus - UNSOLVED PRACTICE SET

Q7. Define Young's modulus, bulk modulus, and shear modulus. Write the formula for each.

Q8. A wire of length 2 m and cross-sectional area 10⁻⁶ m² is stretched by 2 mm under a load of 20 N. Calculate Young's modulus of the material.

Q9. Explain why Young's modulus and shear modulus are defined only for solids, while bulk modulus is defined for solids, liquids, and gases.

Q10. In your school, a student compresses a rubber eraser and a steel block with the same force. The eraser shows much more compression. What does this tell you about their bulk moduli?

Q11. A cube of side 0.1 m is subjected to a shearing force of 10⁴ N on its upper face, causing a displacement of 0.01 mm. Calculate the shear modulus.

Q12. The bulk modulus of water is 2.2 GPa. Calculate the pressure required to compress 1 litre of water by 0.1%.

Q13. Explain the three elastic moduli — Young's modulus (Y), bulk modulus (K), and shear modulus (η) — with definitions, formulas, and physical significance. For each modulus, discuss:

(i) The type of deformation it describes

(ii) The type of stress and strain involved

(iii) Why it is relevant only for solids (in the case of Y and η)

(iv) Typical values for common materials

(v) Practical applications

Q14. Derive the relationship between the three elastic moduli and Poisson's ratio:

(a) Show that Y = 2η(1 + σ)

(b) Show that Y = 3K(1 − 2σ)

(c) Hence derive: Y = 9Kη/(3K + η)

(d) For steel (Y = 2 × 10¹¹ Pa, η = 8.4 × 10¹⁰ Pa), calculate Poisson's ratio and bulk modulus.

(e) Verify that these values are consistent with the above relations.

Q15. A cylindrical steel rod of length 2 m and diameter 2 cm is subjected to different types of deformation:

(a) When pulled with a tensile force of 50 kN, it extends by 0.8 mm. Calculate Young's modulus.

(b) When subjected to a uniform pressure of 10⁸ Pa, its volume decreases by 0.01%. Calculate the bulk modulus.

(c) When its top face is displaced by 0.02 mm by a tangential force of 5 kN, calculate the shear modulus.

(d) Using the calculated values, estimate Poisson's ratio.

(e) Discuss whether the calculated values are consistent with typical values for steel.

Q16. The following table gives the elastic moduli of some materials:

Table

Material Y (GPa) K (GPa) η (GPa)

Steel 200 160 84

Aluminium 70 70 26

Glass 65 35 25

Rubber 0.01 1.0 0.001

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

(b) Identify which material is most resistant to stretching, compression, and shearing.

(c) Which material would be best for making a diving suit? Explain.

(d) Which material would be best for the frame of a skyscraper? Explain.

(e) Discuss why no material has the highest value for all three moduli.

Q17. A copper wire (Y = 1.1 × 10¹¹ Pa) and a steel wire (Y = 2.0 × 10¹¹ Pa) are connected end to end. Each has length 1 m and diameter 1 mm. A load of 20 N is applied.

(a) Calculate the extension of each wire.

(b) Calculate the total extension.

(c) If the wires were connected in parallel (side by side) between two rigid supports and the same total load applied, calculate the load shared by each wire.

(d) Calculate the extension in the parallel arrangement.

(e) Compare the series and parallel arrangements for use in suspension bridges.

Q18. In a school physics lab, students measure the Young's modulus of a wire using Searle's apparatus:

(a) Draw a labelled diagram of the experimental setup.

(b) List the measurements needed and how Young's modulus is calculated.

(c) A student records: length = 2.5 m, diameter = 0.4 mm, load = 5 kg, extension = 1.2 mm. Calculate Young's modulus.

(d) Discuss two major sources of error in this experiment.

(e) Suggest two improvements to increase accuracy.

(f) Why is a reference wire used in the apparatus?