At 0^(@)C, three rods of equal length form an equilateral triangle. Among the three rods, one is made of invar (with neligible expansion) and the other two rods are made of some other metal. When the triangle is heated up to 100^(@)C, the angle between the two rods of the same metal changes to (pi/3-theta). Show that the coefficient of linear expansion of the metal is (sqrt(3)theta)/200""^(@)C^(-1).

At 0^(@)C, three rods of equal length form an equilateral triangle. Am

1.	At 0^(@)C, three rods of equal length form an equilateral triangle. Among the three rods, one is made of invar (with neligible expansion) and the other two rods are made of some other metal. When the triangle is heated up to 100^(@)C, the angle between the two rods of the same metal changes to (pi/3-theta). Show that the coefficient of linear expansion of the metal is (sqrt(3)theta)/200""^(@)C^(-1).
Answer» Solution :Suppose at `0^(@)C` the lengths of the RODS are l and the coefficient of linear expansion of the metal of AD and BD is `alpha.` The inner rod AB has no expansion. If `l_(1)` is the length of each of the metal rods AD and BD at `100^(@)C, l_(1)=l(1+100alpha).` A perpendicular DO is DRAWN from the vertex D on AB [Fig. 5.7]. From the trianlge ADO, we GET, `"" (l/2)/sin(pi/6-theta/2)=l_(1)/(sin90^(@))` or, `"" l/(2sin(pi/6-theta/2))=(l(1+100alpha))/1` or, `"" sin(pi/6-theta/2)=1/(2(1+100alpha))` or, `"" sin""pi/6cos""theta/2-cos""pi/6sin""theta/2=1/(2(1+100alpha))` or, `"" 1/2-sqrt(3)/2theta/2=1/(2(1+100alpha))` `""` [Since `theta` is very small, `sin""theta/2 rarr theta/2 " and " cos""theta/2 rarr 1`] or, `"" sqrt(3)/2theta/2=1/2-1/(2(1+100alpha))=(100alpha)/(2(1+100alpha))` or, `"" (sqrt(3)theta)/2=(100alpha)/(1+100alpha)=100alpha` `""`[As `100alpha` is very small compared to 1, it is negligible] or, `"" alpha=(sqrt(3)theta)/200""^(@)C^(-1).`