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Two metal plates of length l_(0) and width x are joined together at temperature t by rivetting them in such a way that the edges of the plates coincide. The coefficients of linear expansion of the materials of the plates are alpha_(1) " and " alpha_(2)(alpha_(1) gt alpha_(2)). When the bimetallic strip is heated to (t+deltat) it bends and forms an arc of a circle. Find the radius of curvature of the strip. |
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Answer» Solution :A BIMETALLIC strip bends on heating due to the DIFFERENCE in values of `alpha` for the two metal plates forming the strip. With the increases in temperature, SUPPOSE the lengths of the metal plates AB and CD change to `l_(1) " and " l_(2)`, and the radii of curvature are `r_(1) " and " r_(2)` respectively [Fig. 5.9]. Let the angle subtended at the centre by the arc be `phi`.  `therefore "" l_(1)=l_(0)(1+a_(1)deltat)=r_(1)phi "...(1)"` and `"" l_(2)=l_(0)(1+a_(2)deltat)=r_(2)phi "...(2)"` `therefore "" phi(r_(1)-r_(2))=l_(0)(alpha_(1)-alpha_(2))deltat` `therefore "" phi=(l_(0)(alpha_(1)-alpha_(2))deltat)/(r_(1)-r_(2))=(l_(0)(alpha_(1)-alpha_(2))deltat)/(2x) "...(3)"` `""[therefore r_(1)-r_(2)=2x]` Adding (1) and (2), `(r_(1)+r_(2))phi=2l_(0)+l_(0)(alpha_(1)+alpha_(2))deltat` Let the average RADIUS of curvature be r. `therefore"" r=(r_(1)+r_(2))/2=(2l_(0)+l_(0)(alpha_(1)+alpha_(2))deltat)/(2phi)` `""=(2l_(0)+l_(0)(alpha_(1)+alpha_(2))deltat)/(2l_(0)(alpha_(1)-alpha_(2))deltat) times 2x=({2+(alpha_(1)+alpha_(2))deltat}x)/((alpha_(1)-alpha_(2))deltat).` |
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