Answer to Question #104399 in Algebra for Jake

Question

A cable stretches by an amount d when it supports a crate of mass M. The cable is replaced by another cable of the same material having the same length and twice the diameter. If the same crate is supported by the thicker cable, by how much will the cable stretch?

Answer 1

$Stress /Strain=Y(constant)$, this depends on nature of the material.

As both the cables of the same material, their Young's Moduli can be equated.

Initially, stress was $\sigma_1=4Mg/\pi d_i^2$ .

Initial strain was $d/l_o$ .

$Y_1=4Mgl_0/ \pi d_i^2d$ .

Finally stress is $\sigma _2=4Mg/ \pi(2d_i)^2$ .

Thus, final strain is $\Delta x/l_0$ (where $\Delta x$ is the new elongation),

$Y_2=Mgl_0/ \pi d_i^2\Delta x$ ,

$Y_1=Y_2 \implies \Delta x=d/4$ .

Thus, the thicker cable stretches by an amount of d/4.