Three rods of equal length are joined to from an equilateral triangle PQR.O is the mid-point of PQ. Distance OR remains same for small change in temperature. Coefficient of linear expansion for PR and RQ is same i.e., `alpha_(2)` but for PQ is `alpha_(1^.)` Then

Options

A. `alpha_(2) = 3 alpha_(1)`
B. `alpha_(2) = 4 alpha_(1)`
C. `alpha_(1) = 3 alpha_(2)`
D. `alpha_(1) = 4 alpha_(2)`

Answer

D

Explanation

`(OR)^(2)=(PR)^(2)-(PO)^(2)=l^(2)-(l/2)^(2)`
`" "=[l(1+alpha_(2)t)]^(2)-[1/2(1+alpha_(l)t)]^(2)`
`l-(l^(2))/4=l^(2)(1+alpha_(2)^(2)t^(2)+2alpha_(2)t)-(l^(2))/4(1+alpha_(1)^(2)t^(2)+2alpha_(l)t)`
Neglecting `alpha_(2)^(2)t^(2)and alpha_(1)^(2)t^(2)`
`" "0=l^(2)(2alpha_(2)t^(2))-(l^(2))/4(2alpha_(l)t)`
`rArr" "2alpha_(2)=(2alpha_(1))/4`
`rArr" "alpha_(1)=4alpha_(2)`