Structure of an equilateral triangle ABC is made using three thin rods. Another rod AD connects the vertex A with D, the mid-point of BC. Coefficient of linear expansion for AB and AC is alpha and that for the base BC is beta. Show that, if the coefficient of linear expansion for the rod AD is 1/3(4alpha-beta), the arms of the system will not show any tendency to bend, for a small rise in temperature.

Structure of an equilateral triangle ABC is made using three thin rods

1.	Structure of an equilateral triangle ABC is made using three thin rods. Another rod AD connects the vertex A with D, the mid-point of BC. Coefficient of linear expansion for AB and AC is alpha and that for the base BC is beta. Show that, if the coefficient of linear expansion for the rod AD is 1/3(4alpha-beta), the arms of the system will not show any tendency to bend, for a small rise in temperature.
Answer» Solution :LET the initial length of each side of the triangle ABC be 2l [Fig. 5.11]. Hence, BD = CD = L `therefore "" AD=sqrt(AB^(2)-BD^(2))` `""=sqrt(4l^(2)-l^(2))=sqrt(3)l` Let `gamma` be the coefficient of linear expansion for rod AD. For a rise in temperature t, `""` changed length of `AB=2l(1+alphat)`, `""` changed length of `BD=l(1+betat)` and changed length of`AD=sqrt(3)l(1+gammat)` If the sides do not bend, the triangle ABD continues to be a right angled triangle, i.e., `""[2l(1+alphat)]^(2)=[l(1+betat)]^(2)+[sqrt(3)l(1+gammat)]^(2)` or, `"" 4l^(2)(1+2alphat+alpha^(2)t^(2))=l^(2)(1+2betat+beta^(2)t^(2))+3l^(2)(1+2gammat+gamma^(2)t^(2))` or, `" " 4(l+2alphat)=1+2betat+3(1+2gammat)` (Neglecting the terms CONTAINING `alpha^(2), beta^(2) " and " gamma^(2)` for their very small values) or, `""4+8alphat=1+2betat+3+6gammat " or, " 8alphat=2betat+6gammat` or, `" " 4alpha=beta+3gamma " or, " gamma=1/3(4alpha-beta).