Question

The close-coiled helical springs 'A' and 'B' are of same material, same coil diameter, same wire diameter and subjected to same load. If the number of turns of spring 'A' is half that of spring 'B', the ratio of deflection of spring 'A' to spring 'B' is

• A. \(\frac{1}{2}\)
• B. 1
• C. 2
• D. 4

Hint:

Deflection ($\delta$) and stiffness ($k$) behaviors in helical springs: - Deflection is directly proportional to the number of turns: \(\delta \propto n\) - Axial spring stiffness is inversely proportional to the number of active turns: \(k = \frac{W}{\delta} = \frac{G d^4}{8 D^3 n} \implies k \propto \frac{1}{n}\) Cutting a spring in half reduces the turns by half, which doubles its stiffness and halves its deflection under the same load.

Answer 1

The Correct Option is A

Concept: The axial elongation or deflection (\(\delta\)) of a closely coiled helical spring subjected to an axial force load \(W\) is derived using Castigliano's theorem or strain energy methods in torsion. The canonical formula for deflection is: \[ \delta = \frac{8 W D^3 n}{G d^4} \] Where the independent parameters are defined as:
• \(W\) = Applied axial force load on the spring.
• \(D\) = Mean diameter of the spring coil.
• \(n\) = Number of active coils or turns in the spring helix.
• \(G\) = Shear modulus of elasticity (modulus of rigidity) of the material.
• \(d\) = Diameter of the spring wire.

Step 1: Extracting proportional relationships from the given conditions.
The problem specifies that the two springs, designated as 'A' and 'B', share several identical properties:
• Same material \(\Rightarrow G_A = G_B = G\)
• Same coil diameter \(\Rightarrow D_A = D_B = D\)
• Same wire diameter \(\Rightarrow d_A = d_B = d\)
• Subjected to same load \(\Rightarrow W_A = W_B = W\) By grouping all constant parameters into a single proportionality constant \(K\): \[ K = \frac{8 W D^3}{G d^4} \] We can express the deflection of any spring in this problem as a direct linear function of its number of active turns: \[ \delta = K \cdot n \quad \Rightarrow \quad \delta \propto n \]

Step 2: Setting up the deflection ratio.
Using our proportionality relationship, the ratio of the deflection of spring 'A' (\(\delta_A\)) to that of spring 'B' (\(\delta_B\)) simplifies to the ratio of their respective numbers of turns: \[ \frac{\delta_A}{\delta_B} = \frac{n_A}{n_B} \]

Step 3: Calculating the numerical ratio using the given turn condition.
The problem states that the number of turns of spring 'A' is exactly half that of spring 'B': \[ n_A = \frac{1}{2} n_B \quad \Rightarrow \quad \frac{n_A}{n_B} = \frac{1}{2} \] Substituting this value into our ratio equation: \[ \frac{\delta_A}{\delta_B} = \frac{1}{2} \] Thus, the ratio of the deflection of spring 'A' to spring 'B' is \(\frac{1}{2}\), matching Option (1).