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

$\Large\underline{\underline{\sf \red{Given}:}}$

• Diameter of first rope $\sf{(d_1)}$ =1 cm or 0.01m

• Tension in first rope $\sf{(T_1)}$ = 500N

• Diameter of second rope $\sf{(d_2)}$ = 2cm or 0.02m

$\Large\underline{\underline{\sf \red{To\:Find}:}}$

• Maximum tension in second rope $\sf{(T_2)}$= ?

$\large\underline{\underline{\sf \red{Solution}:}}$

$\large{\boxed{\sf Breaking\:Stress(B)=\dfrac{F}{A} }}$

For first rope :-

$\implies{\sf B_1=\frac{F_1}{A}}$

$\implies{\sf \dfrac{500}{π×\left(\dfrac{0.01}{2}\right)^2} }$

For second rope :-

$\implies{\sf B_2=\dfrac{F_2}{A} }$

$\implies{\sf B_2=\dfrac{F_2}{π×\left(\dfrac{0.02}{2}\right)^2}}$

______________________________________

$\implies{\sf B_1\:\:=\:\:B_2}$

$\implies{\sf \dfrac{500}{π×\left(\dfrac{0.01}{2}\right)^2}=\dfrac{F_2}{π×\left(\dfrac{0.02}{2}\right)^2}}$

$\implies{\sf F_2=2000N}$

$\Large\underline{\underline{\sf \red{Answer}:}}$

•°• Maximum tension that may be given to a similar rope of diameter 2 cm is 2000N

Answer 2

The maximum tension that may be given to a similar rope of diameter 2 cm is $2000 \mathrm{N}$

Explanation:

• When the maximum permissible stress in the wire cross section is reached, the wire breaks.
• The maximum permissible stress of a 1 cm diameter wire is given here by 500 N.
• Since the cross-section area of a wire 2 cm in diameter is four times that of a wire 1 cm in diameter, It will also carry four times its load capacity,

i.e. $4^{*} 500 \mathrm{N}=2000 \mathrm{N}$