Question

A metal rope of density $6000kg/m^3$ has breaking stress $9.8\times {10}^{8}N{m}^{-2}$. This rope is used to measure the depth of the sea. Then the depth of the sea that can be measured with out breaking is _____

• A. $10\times {10}^3$
• B. $20\times {10}^3$
• C. $30\times {10}^3$
• D. $40\times {10}^3$ m

Answer 1

The correct option is B $20\times {10}^3$

Given,

Rope density ${\rho }_r=6000kg/m^3$

Density of water ${\rho }_w=1000kg/m^3$

Breaking stress$\sigma =9.8\times {10}^{8}N{m}^{-2}$

Net force on rope upper end $F_{net}$ = $W$ (weight of rope) – $F_b$ (buoyancy force of rope)

$F_{net}={\rho }_r\left(AL\right)g-{\rho }_w\left(AL\right)g=({\rho }_r-{\rho }_w)ALg$

Stress $\sigma =\frac{F_{net}}{A}=\frac{({\rho }_r-{\rho }_w)ALg}{A}$

$\Rightarrow L=\frac{\sigma }{({\rho }_r-{\rho }_w)g}=\frac{9.8\times {10}^8}{(6000-1000)\times 9.8}=20\times {10}^{3}m$

Depth of sea that can be measured without breaking is $20\times {10}^{3}m$