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))/(2)`
C. `sqrt(alpha_(1)alpha_(2))`
D. `alpha_(1)-alpha_(2)`

Answer

B

Explanation

`dy=(dx)alphaDeltaT`
`alpha=alpha_(1)+((alpha_(2)-alpha_(1))/(l))x`
`dalpha=dx((alpha_(2)-alpha_(1))/(l))," " dy=(alphaDeltaTldalpha)/(alpha_(2)-alpha_(1))`
`y=(DeltaTl)((alpha_(2)^(2)-alpha_(1)^(2)))/(2(alpha_(2)-alpha_(1)))=DeltaTl((alpha_(2)+alpha_(1)))/(2)` , Now y is total elongation in rod.
`So,DeltaTlalpha_(eq)=DeltaTl((alpha_(2)+alpha_(1))/(2))`
`alpha_(eq)=(alpha_(1)+alpha_(2))/(2)`