Two rods, each of coefficient of linear expansion alpha_(2) and length l_(2), form the two sides of an isosceles triangle and the base is formed by another rod of length l_(1) and coefficient of linear expansion alpha_(1). The base is fixed horizontally at its mid-point. What should be the relationship between l_(1) " and " l_(2) so that the distance of the vertex from the mid-point of the base, does not change for any increase in temperature?

Two rods, each of coefficient of linear expansion alpha_(2) and length

1.	Two rods, each of coefficient of linear expansion alpha_(2) and length l_(2), form the two sides of an isosceles triangle and the base is formed by another rod of length l_(1) and coefficient of linear expansion alpha_(1). The base is fixed horizontally at its mid-point. What should be the relationship between l_(1) " and " l_(2) so that the distance of the vertex from the mid-point of the base, does not change for any increase in temperature?
Answer» Solution :ABC is the isosceles TRIANGLE and D is the fixed POINT on the base [Fig. 5.8]. From the figure, `""AD=sqrt(l_(2)^(2)-(l_(1)/2)^(2))=x` (say) `therefore "" x^(2)=l_(2)^(2)-(l_(1)/2)^(2)` Suppose with the RISE in temperature by t, the length of AB and AC change to `l^(.)""_(2),BC " to " l^(.)""_(I)` and AD to y. `therefore "" l^(.)""_(2)=l_(2)(1+alpha_(2)t),l^(.)""_(1)=l_(1)(1+alpha_(1)t)` `therefore "" y^(2)=l^(.)""_(2)""^(2)-((l^(.)""_(1))/2)^(2)={l_(2)(l+a_(2)t)}^(2)-(l_(1)^(2)(1+alpha_(1)t)^(2))/4` `""~=l_(2)^(2)(1+2alpha_(2)t)-l_(1)^(2)((1+2alpha_(1)t)/4)` `""`[neglecting higher powers of `alpha_(1) " and "alpha_(2)`] By condition, length AD should remain the same, i.e. `"" x^(2)=y^(2)` or, `""l_(2)^(2)-l_(1)^(2)/4=l_(2)^(2)(1+2alpha_(2)t)-1/4l_(1)^(2)(1+2alpha_(1)t)` or, `"" 2l_(2)^(2)alpha_(2)t=1/2l_(1)^(2)alpha_(1)t` or, `"" l_(2)^(2)/l_(1)^(2)=1/4*alpha_(1)/alpha_(2) " or, " l_(2)/l_(1)=1/2sqrt(alpha_(1)/alpha_(2)).`