Question

A rope 1cm in diameter breaks if the tensionin it exceeds 500N.The maximum tension that may be given to a similar rope of diameter 2 cm is

Answer 1

we know that tension in rope is directly proportional to r^2.
so if we double the diameter the radius also doubles and tension is 4times.
so maximum tension will be 2000N.

Answer 2

it is based on Young's modulus concepts .

we know, Young's modulus = stress/strain
stress = F/A , strain = ∆L/L

so, Young's modulus, Y = FL/A∆L
so, F = YA∆L/L
hence, it is clear that tension in rope is directly proportional to area of cross section of rope.

area of cross section of rope , A = πd²/4

so, $F\propto d^2$
we can write , $\frac{F_1}{F_2}=\frac{d_1^2}{d_2^2}$

so, $\frac{500}{F_2}=\frac{1^2}{2^2}$

$F_2=500\times4=2000N$

hence, maximum tension that may be given to a similar role of diameter 2cm is 2000N