Two rods, one of aluminium and the other made of steel, having initial length `l_1 and l_2` are connected together to from a single rod of length `l_1+l_2.` The coefficients of linear expansion for aluminium and steel are `alpha_a and alpha_s` and respectively. If the length of each rod increases by the same amount when their temperature are raised by `t^0C,` then find the ratio `l_1//(l_1+l_2)`

Options

A. `alpha_s//alpha_a`
B. `alpha_a//alpha_s`
C. `alpha_3//(alpha_a+alpha_s)`
D. `alpha_a//(alpha_s+alpha_s)`

Answer

C

Explanation

The length of each rod increases by the same amount
`:. Deltal_(a)=Deltal_(s) implies l_(1)alpha_(a)t=l_(2)alpha_(s)t`
`implies (l_(2))/(l_(1))=(alpha_(a))/(alpha_(s)) implies (l_(2))/(l_(1))+1=(alpha_(a))/(alpha_(s))+1`
`implies (l_(2)+l_(1))/(l_(1))=(alpha_(a)+alpha_(s))/(alpha_(s)) implies (l_(1))/(l_(1)+l_(2))=(alpha_(s))/(alpha_(a)+alpha_(s))`.