Question

Two rods one of aluminium and other of steel, having initial

length l

1 and l2 are connected together to form a single rod of

length (l1 + l2

). The coefficients of linear expansion for

aluminium and steel are αAl and αS

respectively. If the length of

each rod increases by the same amount when their

temperature are raised by t

oC then the ratio l

1

/(l1 + l2

) is:​

Answer 1

Question:-

Two rods, one of aluminium and other of steel having initial length $\sf{l_1}$ and $\sf{l_2}$ respectively, are connected together to form a single rod of length $\sf{l_1+l_2.}$ The coefficients of linear expansion of aluminium and steel are $\sf{\alpha_a}$ and $\sf{\alpha_s}$ respectively. If the length of each rod increases by the same amount when their temperature are raised by $\sf{t^oC}$ each, then find the ratio $\sf{\dfrac{l_1}{l_1+l_2}}$ in terms of $\sf{\alpha_a}$ and $\sf{\alpha_s.}$

Answer:-

$\Large\boxed{\sf{\dfrac{l_1}{l_1+l_2}=\dfrac{\alpha_s}{\alpha_s+\alpha_a}}}$

Solution:-

Let extensions in aluminium and steel be $\sf{\Delta l_1}$ and $\sf{\Delta l_2}$ respectively. Given that,

$\longrightarrow\sf{\Delta l_1=\Delta l_2}$

$\longrightarrow\sf{l_1\alpha_at=l_2\alpha_st}$

$\longrightarrow\sf{l_1\alpha_a=l_2\alpha_s}$

$\longrightarrow\sf{\dfrac{l_1}{l_2}=\dfrac{\alpha_s}{\alpha_a}}$

By rule of componendo we get,

$\longrightarrow\sf{\underline{\underline{\dfrac{l_1}{l_1+l_2}=\dfrac{\alpha_s}{\alpha_s+\alpha_a}}}}$