A composite heavy rope of two materials is suspended vertically from a high ceiling. The ratios of different quantities for upper to lower rope are length $\frac{{{L_u}}}{{{L_l}}} = \frac{1}{2}$ , cross sectional area $\frac{{{A_u}}}{{{A_l}}} = \frac{2}{1}$ ,density $\frac{{{d_u}}}{{{d_l}}} = \frac{2}{3}$ .What is the ratio of maximum stress in the two ropes
Diffcult
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Stress $=\frac{\mathrm{mg}}{\mathrm{A}}$
$\frac{(\text { stress })_{\mathrm{u}}}{(\text { stress })_{\ell}}=\frac{\left(\mathrm{m}_{\mathrm{u}}+\mathrm{m}_{\ell}\right) \mathrm{g} / \mathrm{A}_{\mathrm{u}}}{\left(\mathrm{m}_{\ell}\right) \mathrm{g} / \mathrm{A}_{\ell}}$
$=\frac{\left(\mathrm{d}_{\mathfrak{u}} \mathrm{A}_{\mathfrak{u}} \mathrm{L}_{\mathfrak{u}}+\mathrm{d}_{\ell} \mathrm{A}_{\ell} \mathrm{L}_{\ell}\right) / \mathrm{A}_{\mathfrak{u}}}{\left(\mathrm{d}_{\ell} \mathrm{A}_{\ell} \mathrm{L}_{\ell}\right) / \mathrm{A}_{\ell}}$
$=\left(\frac{\mathrm{d}_{\mathrm{u}} \mathrm{A}_{\mathrm{u}} \mathrm{L}_{\mathrm{u}}}{\mathrm{d}_{\ell} \mathrm{A}_{\ell} \mathrm{L}_{\ell}}+1\right) \frac{\mathrm{A}_{\ell}}{\mathrm{A}_{\mathrm{u}}}=\frac{5}{6}$
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