The coefficient of linear expansion of an in homogeneous rod change linearly from `alpha_(1)` to `alpha_(2)` from one end to the other end of the rod. The effective coefficient of linear expansion of rod is

Options

A. `alpha_1 - alpha_2`
B. `alpha_1 +alpha_2`
C. `((alpha_1 + alpha_2))/(2)`
D. `sqrt(alpha_1 alpha_2)`

Answer

C

Explanation

Let us consider an element dx at a distance x from end A of rod of length L. Increase in coefficient of linear expansion per unit
Length ` = (alpha_2 -alpha_1)/(L)`
Coefficient of linear expansion at a distance x is
`alpha_x = alpha_1 = ((alpha_2 - alpha_1) )/(L) x `
When the temperature is increased by `Delta T` , increase in length of the element dx is
`dL = alpha_x dx Delta T`
` therefore ` Increasing in length of the rod is
` Delta L = int_(0)^(L) alpha_x dx Delta T`
` = Delta T int_(0)^(L) { alpha_1 ( (alpha_2 - alpha_1)/(L) ) x " dx`
` = Delta T [ alpha_1 x + ( (alpha_2 - alpha_1)/(L) ) (x^2)/(2) ]_(0)^(L)`
` = Delta T [ alpha_1 L + ( (alpha_2 - alpha_1)/(2) )L ]`
` = ( (alpha_2 - alpha_1 )/(2) ) L Delta T = alpha_(eff) L Delta T`
So effective coefficient of linear expansion of the rod is ` alpha_(eff) = ( (alpha_1 + alpha_2)/(2) )`