Question

A close-coiled helical spring with n coils, mean radius R and diameter 'd' is subjected to an axial load W. What is the compression in spring?

• A. $\frac{64W{R}^{3}n}{C{d}^3}$
• B. $\frac{64W{R}^{3}n}{C{d}^4}$
• C. $\frac{32W{R}^{3}n}{C{d}^3}$
• D. $\frac{32W{R}^{3}n}{C{d}^4}$

Answer 1

The correct option is B $\frac{64W{R}^{3}n}{C{d}^4}$

We know that strain energy stored in the spring is given by

$U=\frac{T^{2}L}{2C{I}_P}$

Here $T=WR,L=2\pi Rn$

$I_P=\frac{\pi }{32}{d}^4$

$\therefore U=\frac{W^2{R}^2\times 2\pi Rn}{2C\times \left(\frac{\pi }{32}{d}^2\right)}$

$\Rightarrow U=\frac{32W^2{R}^{3}n}{C{d}^4}$

Compression in the spring,

$\delta =\frac{\delta U}{\delta W}=\frac{\delta }{\delta W}\left(\frac{32W^2{R}^{3}n}{C{d}^3}\right)$

$=\frac{64W{R}^{3}n}{C{d}^4}$