A compound spring has two close$-$coiled helical springs connected in series. One

spring is made of steel and the other of bronze, both with a mean coil diameter of $35$

mm and a wire diameter of $4$ mm $.$

$($a$)$ If the deflection of both springs is to be the same, calculate the ratio between the

number of coils in each spring.

$($b$)$ Calculate:

$($i$)$ The greatest load that can be carried by this composite spring if the

maximum allowable stresses are $240$ Mpa and $180$ Mpa in the steel and

bronze respectively, and

$($ii$)$ The corresponding extension if the steel spring has coils

and $($:$G_{st}=40$GPa