Question

If two closely coiled helical springs A & B with the mean diameter of spring A is half of that of spring B and having an equal number of active coils and same wire diameter are subjected to a same axial load of W, then the ratio of deflection in spring A to B. 
1. \(\frac{1}{8}\)
2. \(\frac{1}{4}\)
3. 2
4. 8

Answer 1

Correct Answer - Option 1 : \(\frac{1}{8}\)

Concept:

For closed coil helical spring,

\(Deflection\;under\;load,\;δ = \frac{{8W{D^3}n}}{{G{d^4}}}\)

Where, W = Load, D = Mean diameter of coil spring, n = Number of turns, G = Modulus of elasticity and d = Wire diameter of coil

From the above formula, it is clear that keeping all other parameters same,

Deflection, δ ∝ D³

Calculation:

Given:

\(\frac{D_A}{D_B} = \frac{1}{2}\),

∵ Deflection, δ ∝ D3

∴ \(\frac{{{δ _A}}}{{{δ _B}}} = (\frac{{{D_A}}}{{{D_B}}})^3\)

\(\frac{{{δ _A}}}{{{δ _B}}} = (\frac{1}{2})^3\)

\(\frac{{{δ _A}}}{{{δ _B}}} = \frac{1}{8}\)

Hence the ratio of deflection in spring A to spring B will be \(\frac{1}{8}\)