A lift is tied with thick iron ropes having mass 'M'. The maximum acceleration of the lift is 'a'…
A lift is tied with thick iron ropes having mass 'M'. The maximum acceleration of the lift is 'a' $\mathrm{m} / \mathrm{s}^{2}$ and maximum safe stress is 's' $\mathrm{N} / \mathrm{m}^{2}$. The minimum diameter of the rope is ( $\mathrm{g}=$ acceleration due to gravity $)$
The maximum stress produced in a rope is given by
\(\sigma_{\max }=\frac{\text { Force }}{\text { Area }}=\frac{M g}{\pi r^2}\)
As the lift is accelerating with acceleration \(a\), then
$\begin{aligned}
& \sigma_{\max}=\frac{M(g \pm a)}{\pi r^{2}} \\
& \Rightarrow \quad r^{2}=\frac{M(g \pm a)}{\pi S} \quad\left[\text{Given, } \sigma_{\max}=S\right] \\
& \frac{d^{2}}{4}=\frac{M(g \pm a)}{\pi S} \quad\left[\because r=\frac{d}{2}\right] \\
& \Rightarrow \quad d=\sqrt{\frac{4 M(g \pm a)}{\pi S}}
\end{aligned}$
As acceleration is maximum i.e., \(g^{\prime}=g+a\), so
\(d=\sqrt{\frac{4 M(g+a)}{\pi S}}\)