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_s)/((alpha_a + alpha_s))`
D. `(alpha_a)/((alpha_ + alpha_s))`

Answer

C

Explanation

As the length of each rod increases by the same amount
`Delta l_a = Delta l_2`
`rArr l_1 alpha_a t = l_2 alpha_st`
`rArr l_2/l_1 = alpha_a/alpha_s rArr (l_2)/(l_1) + 1 = (alpha_a)/(alpha_s) + 1`
`therefore (l_2 + l_1)/(l_1) = (alpha_a + alpha_s)/(alpha_s) rArr (l_1)/(l_1 + l_2) = (alpha_s)/(alpha_a + alpha_s)`