A helical spring, made of music wire of diameter (d), has a mean coil diameter ((D)) of
Question:
A helical spring, made of music wire of diameter \(d\), has a mean coil diameter \((D)\) of \(14 \mathrm{~mm}\) and \(N\) active coils (turns). It is found to have a frequency of vibration
(f) of \(193 \mathrm{~Hz}\) and a spring rate \(k\) of \(4.6 \mathrm{~N} / \mathrm{mm}\). Determine the wire diameter \(d\) and the number of coils \(N\), assuming the shear modulus \(G\) is \(80 \mathrm{GPa}\) and density \(ho\) is \(8000 \mathrm{~kg} / \mathrm{m}^{3}\). The spring rate \((k)\) and frequency ( \(f\) ) are given by
\[k=\frac{d^{4} G}{8 D^{3} N}, \quad f=\frac{1}{2} \sqrt{\frac{k g}{W}}\]
where \(W\) is the weight of the helical spring and \(g\) is the acceleration due to gravity.
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