Two closely coiled helical springs A and B are equal in all respects but for the number of turns, with A having just half the number of turns of that of B. What is the ratio of deflections in terms of spring A to spring B?

Concept:

For closed coil helical spring,

\(Deflection\;under\;load,\;\delta = \frac{{4W{R^3}n}}{{G{r^4}}}\)

Where, R = radius of coil of spring, n = number of turns, G = modulus of elasticity and r = radius of wire of coils

From the above formula, it is clear that the deflection is directly proportional to the number of turns keeping all other parameters same.

Calculation:

Given,

Number of turns in spring A = ½ time the number of turns in B

\(\frac{{{\delta _A}}}{{{\delta _B}}} = \frac{{{n_A}}}{{{n_B}}}\)

We know, \({n_A} = \frac{1}{2}{n_B}\)

\(\therefore \frac{{{\delta _A}}}{{{\delta _B}}} = \frac{1}{2} \Rightarrow {\delta _A} = \frac{1}{2}{\delta _B}\)