Lessons 40–41 · 2 lessons · P. Habibullayev, A. Boydedayev, A. Bahromov, K. Suyarov, J. Usarov, M. Yuldasheva. Physics Grade 9, revised and expanded 3rd edition. G‘afur G‘ulom Publishing House, Tashkent, 2019
40
Problem solving
Textbook: pp. 109–110
GoalSolve problems on deformation, stress and Hooke’s law; understand the strength limit and the safety factor.
New words
absolute elongation Δl · absolyut uzayish Δlstrength limit · mustahkamlik chegarasisafety factor · zaxira koeffitsiyenticross-section area · ko‘ndalang kesim yuzi
Explanation
In a problem first convert the data to SI: mm² to m² (1 mm² = 10⁻⁶ m²), mm to metres. Then find the stress with σ = F/S and link it to Young’s modulus: the relative elongation is ε = σ/E and the absolute elongation Δl = ε·l₀; combined, they give Δl = F·l₀/(E·S). The elongation is directly proportional to the length and the force, and inversely proportional to the cross-section area and Young’s modulus. Structures are designed for a stress far below the strength limit: the ratio of the limit to the working stress is called the safety factor. The greatest length of a rope that does not break under its own weight is l = σ_max/(ρg), independent of the cross-section, because the weight and the strength both scale with S.
Worked examples
Steel wire: l₀ = 3 m, S = 2 mm², F = 400 N, E = 2·10¹¹ Pa. Δl = 400·3/(2·10¹¹·2·10⁻⁶) = 1200/(4·10⁵) = 3·10⁻³ m = 3 mm.
For a steel rope σ_max = 5·10⁸ Pa, ρ = 7800 kg/m³, g = 10 m/s²: l = 5·10⁸/(7800·10) ≈ 6400 m, the length at which it breaks under its own weight.
Class activity
Think: if the wire is twice as thick (cross-section twice as large), how does the elongation change under the same force? And if it is twice as long? Justify your answer with the formula and, with an adult, check it with a light rubber cord.
Practice
1
From Δl = F·l₀/(E·S), how does Δl change if S doubles?
It halves.
2
Aluminium wire (E = 7·10¹⁰ Pa): l₀ = 2 m, S = 1 mm², F = 70 N. Find Δl.
Δl = 70·2/(7·10¹⁰·10⁻⁶) = 2·10⁻³ m = 2 mm
3
A wire’s strength limit is 4·10⁸ Pa and the working stress 8·10⁷ Pa. Find the safety factor.
4·10⁸/(8·10⁷) = 5
4
Why are long suspension bridges built with steel cables rather than copper?
Steel has a high strength and Young’s modulus, so it stretches little and carries large loads.