A composite heavy rope of two materials is suspended vertically f… - Vidyadip

MCQ

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

A
$\frac{2}{3}$
B
$\frac{3}{4}$
✓
$\frac{5}{6}$
D
$\frac{4}{5}$

✓

Answer

Correct option: C.

$\frac{5}{6}$

c
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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