If the mean coil diameter and wire diameter of a helical compression or tension spring are doubled, then the deflection of the spring under the same applied load will be

Concept:

Deflection of Helical Springs of Circular Wire:

\(\delta = \frac{{8W{D^3}n}}{{Gd^4}}\)

Where D = Mean diameter of the spring coil, d = Diameter of the spring wire, n = Number of active coils, G = Modulus of rigidity for the spring material, W = Axial load on the spring

Calculation:

It is given that D2 = 2D1

d2 = 2d1

\({\delta _1} = \frac{{8\;W\;D_1^3n}}{{G\;{d^4_1}}}\)

\({\delta _2} = \frac{{8\;W\;D_2^3n}}{{G\;{d^4_2}}}\)

\(\delta = \frac{{8W{D^3}n}}{{Gd^4}} \Rightarrow \delta \propto \frac{D^3}{d^4}\) (rest other parameters are same)

\(\frac{{{\delta _2}}}{{{\delta _1}}} = (\frac{{{D_2}}}{{{D_1}}})^3 \times(\frac{d_1}{d_2})^4= \frac{{{2^3}}}{{{2^4}}} = \frac{1}{2}\)

\({\delta _2} = \frac{1}{2}{\delta _1}\)