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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» <html><body><p></p>Solution :Suppose at `0^(@)C` the lengths of the <a href="https://interviewquestions.tuteehub.com/tag/rods-1190662" style="font-weight:bold;" target="_blank" title="Click to know more about RODS">RODS</a> are l and the coefficient of linear expansion of the metal of AD and BD is `alpha.` The inner rod <a href="https://interviewquestions.tuteehub.com/tag/ab-360636" style="font-weight:bold;" target="_blank" title="Click to know more about AB">AB</a> has no expansion. <br/> If `l_(1)` is the length of each of the metal rods AD and BD at `100^(@)C, l_(1)=l(1+100alpha).` <br/> A perpendicular DO is <a href="https://interviewquestions.tuteehub.com/tag/drawn-2062388" style="font-weight:bold;" target="_blank" title="Click to know more about DRAWN">DRAWN</a> from the vertex D on AB [Fig. 5.7]. <br/> From the trianlge ADO, we <a href="https://interviewquestions.tuteehub.com/tag/get-11812" style="font-weight:bold;" target="_blank" title="Click to know more about GET">GET</a>, <br/> `"" (l/2)/sin(pi/6-theta/2)=l_(1)/(sin90^(@))` <br/> or, `"" l/(2sin(pi/6-theta/2))=(l(1+100alpha))/1` <br/> or, `"" sin(pi/6-theta/2)=1/(2(1+100alpha))` <br/> or, `"" sin""pi/6cos""theta/2-cos""pi/6sin""theta/2=1/(2(1+100alpha))` <br/> or, `"" 1/2-sqrt(<a href="https://interviewquestions.tuteehub.com/tag/3-301577" style="font-weight:bold;" target="_blank" title="Click to know more about 3">3</a>)/2*theta/2=1/(2(1+100alpha))` <br/> `""` [Since `theta` is very small, `sin""theta/2 rarr theta/2 " and " cos""theta/2 rarr 1`] <br/> or, `"" sqrt(3)/2*theta/2=1/2-1/(2(1+100alpha))=(100alpha)/(2(1+100alpha))` <br/> or, `"" (sqrt(3)theta)/2=(100alpha)/(1+100alpha)=100alpha` <br/> `""`[As `100alpha` is very small compared to 1, it is negligible] <br/> or, `"" alpha=(sqrt(3)theta)/200""^(@)C^(-1).` <br/> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/CHY_DMB_PHY_XI_P2_U07_C05_SLV_021_S01.png" width="80%"/></body></html> | |