Two metal rods having the length l and 2l having the coefficients of linear expansion a and 2a. These 2 rods are expanded. What is the coefficient of linear expansion of a composite rod

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Solution

$L e t t h e i n i t i a l l e n g t h s o f t h e r o d s b e l_{0} a n d 2 l_{0} . T h e r e f o r e l = l_{0} (1 + a t) a n d 2 l = 2 l_{0} (1 + 2 a t) L e t t h e c o m p o s i t e l e n g t h o f t h e r o d s b e L, t h e n L = l + 2 l = l_{0} (1 + a t) + 2 l_{0} (1 + 2 a t) = l_{0} (3 + 5 a t) o r L = 3 l_{0} (1 + \frac{5}{3} a t) o r L = L_{0} (1 + \frac{5}{3} a t) W h e r e L_{0} = 3 l_{0} a n d n e w c o e f f i c i e n t o f l i n e a r e x p a n s i o n i s \frac{5}{3} a .$

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