Natural Length from Multiple Loads
For a fixed elastic wire in its linear regime, loaded length obeys l = L0 + T/k; subtracting two observations determines compliance and then the natural length L0.
Why this shows up in the exam
Finding unstretched string length · Calibrating elastic cords · Interpreting load-length graphs
Learn the idea
Two load-length observations eliminate unknown stiffness and reveal the unloaded length. In the linear region, measured length is natural length plus an extension proportional to tension, so two points determine the line's zero-load intercept.
🧠 Memory hook: Natural length is the zero-load intercept.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- l = L0 + T/k — loaded length as a linear function of tension
- (l₂-L0)/(l₁-L0) = T₂/T₁ — ratio relation for two load states
- L0 = (T₂ l₁ - T₁ l₂)/(T₂-T₁) — zero-tension intercept
How to approach it
- 1Write one linear load-length equation per observation
- 2Eliminate the unknown compliance
- 3Check that the natural length is below tensile loaded lengths
Common slip-ups that cost marks
- •Using loaded length as the denominator for strain
- •Assuming extension is proportional to loaded length
- •Applying the method outside the linear range
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A wire 1 m long and cross-sectional area 2 mm^2 extends by 1 mm under a 200 N load. Find Young modulus.
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