Equivalent Stiffness and Elastic Elements
The effective stiffness is defined by the quadratic elastic energy U = k_eff q^2/2 or generalized force Q = -k_eff q; series compliances add and parallel stiffnesses add.
Why this shows up in the exam
Series and parallel spring networks · Elastic-wire oscillations · Springs combined with deformable solids or gases
Learn the idea
Replace springs, wires, or deformable bodies by the stiffness seen in the chosen motion coordinate. A complicated elastic assembly behaves like one spring only after every element's extension is related to the same generalized displacement.
🧠 Memory hook: Same stretch adds stiffness; same force adds compliance.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- k_parallel = sum_i k_i — springs sharing the same extension
- 1/k_series = sum_i 1/k_i — springs carrying the same force in series
- k_wire = YA/L — axial stiffness of a uniform wire
- omega = sqrt(k_eff/m_eff) — frequency after reducing the elastic system
How to approach it
- 1Choose one displacement coordinate
- 2Express every extension in that coordinate
- 3Use force or energy to obtain k_eff
Common slip-ups that cost marks
- •Classifying springs from appearance rather than constraints
- •Using YA times L instead of YA over L
- •Ignoring a gas or buoyancy contribution to stiffness
🌟 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 mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
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