Question

A rope 1 cm in diameter breaks if the tension in it exceeds 500 N. The maximum tension that may be given to a similar rope of diameter 2 cm is
(a) 500 N
(b) 250 N
(c) 1000 N
(d) 2000 N

Answer 1

Correct option: (d) 2000 N

$$\begin{gathered} {\mathrm{F}}_1=500\mathrm{N} \\ \mathrm{Let}\mathrm{the}\mathrm{required}\mathrm{breaking}\mathrm{force}\mathrm{on}\mathrm{the}2\mathrm{cm}\mathrm{wire}\mathrm{be}\mathrm{F}. \\ \mathrm{Breaking}\mathrm{stress}\mathrm{in}1\mathrm{cm}\mathrm{wire}=\frac{{\mathrm{F}}_1}{{\mathrm{A}}_1}=\frac{500}{\mathrm{\pi }{\left({\displaystyle \frac{0.01}{2}}\right)}^2} \\ \mathrm{Breaking}\mathrm{stress}\mathrm{in}2\mathrm{cm}\mathrm{wire}=\frac{{\mathrm{F}}_2}{{\mathrm{A}}_2}=\frac{{\mathrm{F}}_2}{\mathrm{\pi }{\left({\displaystyle \frac{0.02}{2}}\right)}^2} \\ \mathrm{The}\mathrm{breaking}\mathrm{stress}\mathrm{is}\mathrm{the}\mathrm{same}\mathrm{for}\mathrm{a}\mathrm{material}. \\ \Rightarrow \frac{500}{\mathrm{\pi }{\left({\displaystyle \frac{0.01}{2}}\right)}^2}=\frac{{\mathrm{F}}_2}{\mathrm{\pi }{\left({\displaystyle \frac{0.02}{2}}\right)}^2} \\ =>{\mathrm{F}}_2=2000\mathrm{N} \end{gathered}$$