question_answer

A bimetallic strip is formed out of two identical strips, one of copper and other of brass. The coefficients of linear expansion of the two metals are \[{{\alpha }_{C}}\] and \[{{\alpha }_{B}}\]. On heating, the temperature of the strip goes up by DT and the strip bends to form an arc of radius of curvature R. Then R is [IIT-JEE (Screening) 1999]

A) Proportional to DT             

B) Inversely proportional to DT

C) Proportional to \[|{{\alpha }_{B}}-{{\alpha }_{C}}|\]

D) Inversely proportional to \[|{{\alpha }_{B}}-{{\alpha }_{C}}|\]

Correct Answer: D

Solution :

 Let L0 be the initial length of each strip before heating. Length after heating will be \[{{L}_{B}}={{L}_{0}}(1+{{\alpha }_{B}}\Delta T)=(R+d)\theta \] \[{{L}_{C}}={{L}_{0}}(1+{{\alpha }_{C}}\Delta T)=R\theta \] Þ \[\frac{R+d}{R}=\frac{1+{{\alpha }_{B}}\Delta T}{1+{{\alpha }_{C}}\Delta T}\] Þ \[1+\frac{d}{R}=1+({{\alpha }_{B}}-{{\alpha }_{C}})\Delta T\] Þ \[R=\frac{d}{({{\alpha }_{B}}-{{\alpha }_{C}})\Delta T}\] Þ \[R\propto \frac{1}{\Delta T}\] and \[R\propto \frac{1}{({{\alpha }_{B}}-{{\alpha }_{C}})}\]